Is Cantor wrong about more than one infinity
I'm not sure if the following is a proof that cantor is wrong about there being more than one type of infinity. This is a mostly geometric argument and it goes like this.
1)First convert all numbers into binary strings.
2)Draw a square and a line down the middle
3) Starting at the middle line do the following .If the digit in your string is a 1 move half the distance to the next line to the right. If the digit is a 0 move half the distance to the next line to the left.
4) Repeat this for the rest of the digits in the string until all digits are gone.
If you do this then your terminating line will always be some distance 1/2 x some a/b because we are simple halving distance all the time. Also each number will have a unique coordinate, i.e some unique a/b. Now even if the string is infinite in length it will still terminate on a multiple of a 1/2. We can now order our numbers and put all these numbers in a list because each has a unique a/b coordinate. This would order all the reals and hence prove cantor wrong.
1)First convert all numbers into binary strings.
2)Draw a square and a line down the middle
3) Starting at the middle line do the following .If the digit in your string is a 1 move half the distance to the next line to the right. If the digit is a 0 move half the distance to the next line to the left.
4) Repeat this for the rest of the digits in the string until all digits are gone.
If you do this then your terminating line will always be some distance 1/2 x some a/b because we are simple halving distance all the time. Also each number will have a unique coordinate, i.e some unique a/b. Now even if the string is infinite in length it will still terminate on a multiple of a 1/2. We can now order our numbers and put all these numbers in a list because each has a unique a/b coordinate. This would order all the reals and hence prove cantor wrong.
Comments (203)
You can reject the axiom of infinity and then proceed to say that an infinite set does not exit but if you accept this axiom, cantor's proof of the fact that the cardinality of the uncountable set is greater than the cardinality of the countable set remains true.
We will map the number 5.7
First we turn the 5 into binary i.e 101. Then we map the .7 which equals 111
this creates the number 101.111
Now we draw a line of length 1 with the halfway point marked. We can represent this by three points
A B C
B is the halfway point.
Now we map the number. If the digit is a 1 then we left shift from our current point half way to the next line. If it is a 0 we right shift
101.111
We start at B
First digit is a 1 so we left shift halfway to the next point and create point D as follows
A D B C
D is halfway between A and B
Next digit is a 0 so we right shift halfway to the next point which is B giving
A D E B C
Next digit is a 1 so we left shift halfway to the next point which is D giving
A D F E B C
For arguments sake we will say the line terminates here at point E
Now I measure the distance from A to E. This distance will be some multiple of a 1/2 x some a/b
We know this has to be the case because we are always dividing our distances by 1/2 so the final distance will be some multiple of a 1/2 x a/b. This will be true even if the number has an infinite number of digits
Now each number will produce a unique a/b coordinate.
This produces a set of fractions that I can then list in a 1-1 correspondence with the natural numbers
And then Cantor can read down your list and diagonally generate a real number that isn't already on it.
Cantor's proof begins with a supposition that you've somehow produced a supposedly complete list of all the reals, and then shows how from such a list you can always generate yet another one that wasn't already on it.
Second, Cantor's proof is a theorem. The only way that you can invalidate it is if you find an error in his proof (good luck with that). If you cannot do that, then the right question to ask is not "Is Cantor wrong?" but "How is my proof wrong?" Is this what you wanted to ask?
No, I haven't read your proof. I don't need to, because I have read and understood Cantor's diagonal proof. That's all I need to know that Cantor is right. Unless you can show how the diagonal proof is wrong, Cantor's result stands.
Just so you know, there's a bazillion cranks out there doing just what you are trying to do: attempting to prove Cantor wrong by proving something contrary to his result. They've been at it for decades: even before the Internet they've been inundating mathematicians and mathematical journals with their proofs. It is something like a perpetuum mobile for mathematical cranks. But none of them have managed to invalidate Cantor's proof yet.
Also a tip, since you are new on the forum: if you reply by clicking what looks like a crooked arrow underneath a post, or select some section of text and click on the "quote" prompt, then the person to whom you reply will get a notification about a reply in this thread.
Quoting SophistiCat
How odd, you dismiss an argument you don't understand and don't even try to. That sounds like some sort of dogma to me. The argument once you understand it is very straight forward and you can put the reals into a 1-1 correspondance with the natural numbers using it
It's not odd and it's not dogma. It's just straightforward logic: If Cantor's proof is correct, then his result is a theorem and therefore it is right. Cantor's proof is demonstrably correct, therefore his result is a theorem. You do understand that your result cannot be right if a theorem exists, according to which it is wrong, do you? If you are so confident about your result, then show us how Cantor's proof is wrong. It's as simple as that.
Quoting SophistiCat
Actually, if two proofs prove contradictory things, then there is a problem with one or both of the proofs. To say I understand one, but not the other, and I accept the one that I understand, therefore the other is wrong, as SophistiCat did, is illogical because the acceptance of the one may be based in a failure to see that its unsound, a mistaken understanding. Until you can exclude the possibility of mistake from your understanding, it is illogical to reject demonstrations which would show that your understanding is mistaken. .
I think your procedure does produce an injection between the sets, but the initial set you're feeding into the injection is actually uncountable. You're mapping the real numbers to the real numbers rather than the real numbers to the rationals.
If you wanna see this, there's an uncountable infinity of real numbers whose first digit right of the decimal point is 1 in the binary expansion. The same goes for any binary digit. I think you're not registering the distinction between "the set of sets of real numbers with x in a given digit in their decimal expansion" and "this real number has x with a given digit in their decimal expansion".
Sophisticat is right though. The diagonal argument does establish that no injection from the reals to the rationals exists. If your claim is correct, set theory is inconsistent (as it proves a contradiction), but it is provably consistent (within larger theories).
Quoting Umonsarmon
Even if you grant your whole argument, it doesn't stop Cantor's theorem about powersets and cardinality from going through. And the broader claim about consistency rears its head again.
That's what this should be about. Either all the mathematicians since Cantor have been idiots and retarded to not notice the fatal flaw in the proof or OP is wrong. What's the probability for each case.
50/50.
" Err no. I only need one infinite quantity to map anything. If you can't map something when you have an infinite quantity then you are doing something wrong. Now you can have an infinity of infinities but I can map that to a single infinity "
Well it seems theoretical physics seems to disagree with you there. The infinitely iterated infinite universe is exactly what they are proposing.
Sure, if you take Cantor and OP. If you take the countless mathematicians on one hand and OP on other hand, the outcome isn't that favourable. :wink:
Actually, there is a third possibility, as I said, and that is that both Cantor and the op are wrong. This raises the probability that the mathematicians are idiots to about 67 percent. The number of mathematicians involved is irrelevant, as herd mentality demonstrates. The number of people carrying out an action has no bearing on whether the action is correct.
By the same token, my hypothetical acceptance of the contrary proof could be "based in a failure to see that its unsound, a mistaken understanding." If that's the standard by which you propose to decide between the two proofs, then it cannot resolve anything. Examining the other proof wouldn't tell me anything that I didn't already know: that whatever opinion I form about the soundness of each proof, it might be mistaken.
Of course the possibility of mistake still exists, and if "resolution" requires removing that possibility, it would remain unresolved. But that's why I retained the option that both are wrong. However, broadening the mind to numerous options, presented from numerous different perspectives increases one's understanding of the subject, thereby lowering the probability of misunderstanding, even when there is no "resolution".. Therefore you are wrong to assert that doing this wouldn't tell you anything you didn't already know. The mistake is to assume that there is a "resolution" when there is not.
Is Cantor wrong about more than one infinity
Yes. Theoretical physics seems to demonstrate the infinitely iterated infinite universe.
If someone came up to me and presented a proof that 1=2, l would immediately discard the proof. The OP obviously didn't present something that ridiculous but it does amount to saying that the prove Cantor gave was wrong as it proves the opposite.There isn't a third possibility here. It isn't about herd mentality here since it is mathematics.In mathematics, we stand on the shoulders of giants and it does not tolerate any weakness that we find in philosophy, religion or social sciences. I understand where you are coming from but you have to see for yourself that in this sub section, we need to be more objective and avoid beating around the bush as we normally do.
Well as far as I can tell any number fed into this procedure should result in a terminating rational length which will produce a set of rationals which map to the natural numbers regardless of whether it is an irrational number or not. Now I understand the point that you could argue that the set your feeding in is uncountable but this leaves us in a strange position because the two proofs directly contradict. I think its a problem with infinity personally. Now a better way to do this would be to feed in just the values between 0 and 1. You then have terminating values a/b for +1/x -a/b for -1/x, ai/b for x and -ai/b for -x which can all be grouped and listed hence covering the reals. I must admit to being a little hasty in thinking it could map the complex numbers. I can interleave the two parts of a complex to create a single unique value which covers a +bi and -a-bi but as of yet I cannot map in -a+bi and a-bi cannot be covered.
If you think about it though Cantors proof is really paradoxical because if you have an infinite quantity then the only thing that determines your ability to list those values is the algorithm you use to sort the numbers. .
eply="ovdtogt;360080"]
Sorry but mapping an infinite number of infinities to 1 infinity is easy.
Here is how you do it.
Create from the natural numbers lists composed only of a prime number and its powers. for each prime number. This creates an infinite number of lists each with an infinite number of numbers with no number in any list overlapping. I can repeat this process again with each of those lists by mapping them back to the total number of natural numbers and creating a 2nd tier of infinite infinities. I can do that infinetly All of this though is contained in the infinity of the natural numbers.
If that were the case you would only need 1 infinite Universe to map our Universe.
Well it seems theoretical physics seems to disagree with you there. Apparently you need infinitely iterated universes to map our Universe.
I would be a bit cautious about statements about dismissing proofs like that. I can easily produce an argument using triangles and how they combine that proves 1+1 = 3. I would even go as far to say that using a very simple technique that involves very simple geometry and counting that 4=5 in some circumstances.. People get fixated with what they treat as absolutes and they will come a cropper later.
1 dollar is worth 2 50 cent pieces.
We can imagine the universe in two dimensions as a plane, which is flat and infinite (like an infinite piece of paper). ... However, with expansion, it is possible that even if the universe just has a very large volume now, it will reach infinite volume in the infinite future
Well rather than trying to deal with phantom probabilities why don't you just read the proof. The point you are forgetting is that how many people have tried to debunk Cantors proof after it was published, most people just accept it as fact. The logic is very water tight but set theory has produced paradoxes in the past and this just might be one of those cases.I only stumbled on it when i was working on a different problem.I am not closed to it being wrong it just I don't see many people trying to understand the proof. I can explain it in an even simpler way if need be.
Well the number 51.32 would be broken up into the binary for 51 and the binary for 32 and then combined into a single string.
Pi would just have a never ending trail of digits but the procedure is exactly the same. That's just an infinite binary string.Cantors argument relies on numbers with infinite numbers of decimals and thats what he uses in his argument as well so if you want to cobble me for that then you have to cobble Cantor as well.Cantors argument specifically relies on having infinite strings with his slash argument. It wont work with finite strings because these can all be converted into rational fractions which we can list
Just to explain myself more clearly with pi you can simply covert each digit of pi into binary and do this with all the other numbers as well to generate your string.
i.e 3.141 etc = 3 in binary+1 in binary +4 in binary + 1 in binary etc etc
Yes! You can quite happily map a countable set of irrationals to any other countable set.
Quoting Umonsarmon
Quoting Umonsarmon
Convert all real numbers to binary strings!
Quoting Umonsarmon
The line is an uncountable set.
Quoting Umonsarmon
You have to apply this to every real number. You're mapping an interval of reals to an interval of reals with (what looks to me like) an injection.
I will explain the proof again. I am aware that my presentation of it is not brilliant so here goes.
Firstly Cantor himself uses infinite decimal numbers in his diagonal argument so I don't understand why you cut him some slack but not me :)
Now first convert the real number into a binary number. We can convert all numbers to base 2 people if we need to.or we could just turn each digit into a base 2 representation
We draw a line of length 1. Point A = 0 Point B = 1/2 and point C = 1
To keep this simple I will use the binary string 101.
A B C
0 1/2 1
Now if the digit in your number is a 1 we left shift 1/2 the distance to the next point.
If the digit is 0 right shift 1/2 the distance to the next right point.
We start at B
So the first digit is 1. We left shift to the new point D
A D B C
0 1/4 1/2 1
Next digit is 0 so we right shift from D halfway to the next point which in this case is B
A D E B C
0 1/4 3/8 1/2 1
Next digit is 1 so we left shift from E halfway to the next point which in this case is D
A D F E B C
0 1/4 5/16 3/8 1/2 1
Number terminates here at 5/16 which is a rational number a/b.
Each binary number will terminate on its own unique rational a/b as will each irrational number. This can then be mapped into the natural numbers, in other words each number fed into the system comes out with a rational key regardless of how many numbers you feed into it.
The sequences don't terminate for any real number which has an infinite (non-repeating) binary expansion. EG pi/10 would never terminate. The consequence of allowing infinite sequences there means the function is simply from binary expansions to real numbers - essentially a way of encoding binary expansions. The representation of a set doesn't change any of its cardinality properties though.
I don't think this is right. Infinite convergent sequences of rationals typically converge on non-rational reals, even though all the finite sums and elements are rational. The incompleteness of the rationals demonstrates this. (Well, more precisely, convergence doesn't make much sense for infinite sequences of rationals...)
I bring this up because the incompleteness property of the rationals shows that infinite convergent sequences of rationals do not have to converge to rationals. (or more precisely that convergence breaks when you don't have ensured existence of suprema and infima)
Try and prove it! This discussion might be helpful.
I would cut you more slack if you demonstrated more understanding. You've been thoughtful, but the rigour's lacking. This shows whenever someone spells out a mathematical idea with more precision and it turns out that the concepts (as they use them) contradict established theorems or the intuitions (conceptual understanding/imaginative background) that support them.
Specifically, your understanding of convergence needs refining. I also suspect you would discover more difficulties with your argument if you tried to define your function formally (what's its domain, what's its image, is it injective, surjective...) - one presentation of it is a way of associating reals with infinite sequences of rationals (if they do converge in the usual sense, you will successfully associate with only rationals, making the image countable, if they don't converge in the usual sense your argument doesn't work).
Finite sequences don't automatically have the same properties as infinite sequences. It doesn't boil down to that at all.
Quoting Umonsarmon
[math]\sum_{i=0}^{N}(-1)^{i}\frac{(2i-3)!!}{2i!!}[/math]
where [math]x!![/math] is all the even numbers up to [math]x[/math] multiplied together. Is always rational for any [math]N[/math].
[math]\sum_{i=0}^{\infty}(-1)^{i}\frac{(2i-3)!!}{2k!!}[/math]
Is [math]\sqrt{2}[/math]
But then zero would be mapped to -(1/2) and -1 too would be mapped to -(1/2) which would result in a failure of the necessary bijection. There would be two real numbers (0 and -1) mapped to only one point -(1/2) in your scheme.
I don't know the derivation for that one, just looked on the wiki page for series expansion of root 2. How about:
[math]e^x = \sum_{i=0}^{\infty}\frac{x^i}{i!}[/math]
sub in [math]x=1[/math].
You can do that one yourself. Derive the Taylor series of [math]e^x[/math], show the radius of convergence includes [math]x=1[/math] (it's actually infinite, the function is analytic). Sub in [math]x=1[/math]. All the finite sums are rational. The infinite sum is the irrational [math]e[/math].
Maybe I've got it wrong but here's what my understanding is
1. You convert all reals into binary so that we have only 1's and 0's
2. You need direction to distinguish between 1's and 0's. Left or right depending on what the digit is, 0 or 1.
Now I want you to follow your scheme and map the following numbers
a) 1
b) -1
c) 0
These numbers are simple because they only require 1 step i. e will map to 1/2. There are two types of 1/2. One is +1/2 and the other is -1/2 and we have 3 numbers to consider +1, -1 and 0. It's impossible.
With this procedure you can produce rationals arbitrarily close to irrationals but not irrationals. If you want the procedure to be infinite, you're no longer dealing with rationals (since the series doesn't converge in the rationals, it only converges in the reals).
1 goes to 1/4 (i.e halfway between 0 and 1/2
-1 just gets represented as the key -1/4
0 right shifts from 1/2 to 3/4
It would've been easier to say 0 maps to 0 but that would break your rule or demands tweaking it a little bit.
If you graph your rule the output of the function (1/2)^x approaches zero which raises the problem that two real numbers 0 and infinity map to zero. Do you find that problematic?
If we have the sequence 1/4 + 1/8 + 1/16 etc etc no matter how many numbers I remove from that list the product will be rational. If we sum them all up then the product is 1/2. If the sum of the total is rational than any total in the sequence at any point also has to be rational. If I knock out numbers from that sequence than that still leaves me with a rational total otherwise I could add up those numbers in a different order and somehow produce an irrational number. This is impossible from that perspective. If the sum of the total is rational than how can we subtract an amount from this to create an irrational number if the only numbers we are subtracting are rational numbers. So no I stand by my proof ;)
[math]e^x -1 = \sum_{i=1}^{\infty}\frac{x^i}{i!}[/math]
No.
The sum is 1/2 + sum of positive fractions - negative fractions
The maximum sum of the positive fractions is a 1/2 from the sum 1/4 + 1/8 +1/16 etc
Now if the sum is a rational number then at no point in the sequence 1/4+1/8 etc can the sum be an irrational number otherwise the sum of the whole sequence will be irrational.
Now I can rearrange this sequence in all possible combinations and at all points in all these rearrangements the sum is rational. If it were not then the product would be irrational ok.
Now I can arrange the sequence 1/4+1/8+1/16 in such a way that the positive fractions in our sum are in a sequence. But this sequence has to sum to to a rational number from what was argued earlier. This means that the sum of the positive fractions is always rational...Hence the negative fractions are also rational because the produce is a 1/2.
No.
1 + root(2) + 1/2 - root(2) + 1/4 + 1/8 ...
Also, if your sequence must output a rational, it can't output an irrational.
But what is uncountable is the reals, which include irrational numbers like Sq root 2, which was discovered to be irrational by the Ancient Greeks, much to their consternation. So where are root 2 and pi in your list?
(1) The sequence contains irrationals. The infinite sum remains rational.
(2) The sequence can consist only of rationals. The infinite sum can be irrational.
(3) The sequence can consist only of rationals, it can be strictly increasing or decreasing, but not converge in the rationals. (see 2)
(4) The sequence can consist only of irrationals, it can be strictly increasing or decreasing, but converge to a rational.
You don't have the sequence/series facts down.
0.1 would be 1/2 to the right and 1/2 to the left to give us 1/4
1.1 would be 1/2 to the right and again 1/2 to the right to give us, again, 1/4
Both 0.1 and 1.1 are mapping on to the same fraction 1/4.
There is no bijection and the proof is faulty.
1.1 would give the number 11 in binary. This means we either shift twice to the left or twice to the right depending on what direction you give the 1,s and 0,s . Lets suppose its left. We start at 1/2 and left shift to 1/4 then we left shift again to 1/8
-1.1 would just be represented by the key -1/8
0.1 would shift right for zero going to 3/4 and then shift left to halfway between 3/4 and a 1/2
i.e A=0 B=1/2 C=1
This goes to point D halfway between B and C
A=0 B=1/2 D=3/4 C=1
The next shift is left so we have
A=0 B=1/2 E=5/8 D=3/4 C=1
So it terminates on 5/8
1.1 would be 1/2 to the right and again 1/2 to the right to give us, again, 1/4
Both 0.1 and 1.1 are mapping on to the same fraction 1/4.
Bijection fails.
Lets go through this proof again.
We have the sum 1=1/2 + sum of positive fractions - sum of negative fractions
So where does the 1 come from you might ask. Well if you take all shifts as positive regardless of direction then the distance the line travels = 1/2 +1/4 +1/8 etc which sums to 1.
Now this is equal to 1/2 + sum of the positive fractions - sum of negative fractions ok.
This gives 1/2 = sum of positive fractions - sum of negative fractions.
Now the sum 1/4 + 1/8 + 1/16 etc sums to a 1/2
I hope you would agree that no matter how I reorder this sum then it always = 1/2 ok
Also no fraction of the value 1/2^y will be irrational on its own
All numbers in the sum are rational
Now at all points in that sum the sum is always rational at any point we sum up to regardless of the order of the numbers. If this were not the case then the sum would not be rational.
If at all points in all sequences of this sum the product is rational then we can find a particular ordering with only the positive fractions being summed which will be rational from what has just been argued and with the remainder of the sequence being the negative fractions which must also sum to a rational value in order for the product (1/2) to remain the product.
Hence all irrational numbers are mapped to rational keys and my proof stands.
01 goes first right which means it goes 1/2 the distance between 1/2 and 1 which are points on a line to first give 3/4. Then it left shifts halfway to the next point which is 1/2. This means it sits halfway between
1/2 and 3/4 which is 5/8. This value is halfway between 1/2 and 3/4. This is why I would normally use a square with a line going halfway down the middle. Each shift creates a new line and you only move halfway to your next line ok
Yes there clearly is a third possibility, it's not a case of either Cantor is right, or the op is right, because both Cantor and the op might be wrong.
Quoting unenlightened
Actually I think it was Pythagoras who first proved that the square root of two is irrational. And I think it really frustrated him because it demonstrates that the geometrical figure, the square, cannot be a real figure, The two perpendicular sides of a square are incommensurable. This is similar to the problem with pi. It appears like there is an incommensurability between one dimension and another which makes two dimensional figures inherently irrational. It's very odd when you think about it because it casts doubt onto our understanding of spatial existence in terms of dimensions. It may well be that spatial existence would be better represented in another way.
Just looking at this it appears you are rearranging a conditionally convergent series and expecting the same sum. If this is the case you can draw no logical conclusion.
https://en.wikipedia.org/wiki/Riemann_series_theorem
But I haven't followed the discussion so I may be misinterpreting your argument. If so, I apologize.
Please tell me how you post mathematical expressions on this forum. Thanks.
Here..!
[math]\int\limits_{0}^{1}{\phi (z,t)dt={{\lambda }^{2}}}[/math]
Aha! The Wikipedia cut/paste on Mathtype with square vs angle "math"
Bear with me but I still think there's a problem.
1. For reals less than base-ten 1 you'll get something like 0.xyz...
2. Suppose you take base-ten 0.5 whose binary representation is 0.1
3. You will have to take the "0" in the 0.1 into account or else there's no way you can get a unique fractional counterpart for 1 and 0.1 because if you ignore the "0" you'll be left with 1 for both numbers 1 and 0.1 and that maps to 1/2 to the right in your system.
4. If you take the "0" into account for binary numbers like 0.xyz... then how would you map the numbers 10base2 and -10base2?
5. So 0.1base2 would be 1/2 to the left and then 1/2 of 1/2 to the right giving us 1/2 - 1/4 = 1/4
5. Where would base-ten +2 i.e. 10base2 and base-ten -2 i.e. -10base 2 maps to?
10base2 would be 1/2 to the right and 1/2 of 1/2 to the left i.e. 1/2 - 1/4 = 1/4
-10base2 would be just as above but with a negative sign i.e. -1/4
As you can see three numbers (+2, -2, and 0.5 all base 10) are mapped onto only 2 points. There is no bijection.
[math]1/2+\sum_{i=2}^{\infty}(-1)^{k_i}(\frac{1}{2})^i[/math], where [math]k_i \in \{0, 1\}[/math]
ETA: Corrected the formula. Thanks
It can be easily shown that the series is convergent by Cauchy's criterion (yes, I just looked up the name - hey, I am three decades out of practice, you guys should be doing this :)). I suspect that it is also order-invariant (if that's the right term), but I won't attempt a proof.
The series can converge to any real number in the interval [0, 1]. There is a simple root-finding numerical method - interval halving, or bisection method - that can be used to demonstrate this. Take the function f(x) = x - r, where r is any real number between 0 and 1. Finding its root in the interval [0, 1] using the interval halving method will produce a series of the above form.
Hmmm. The series goes complex for some [math]k_i[/math], if you meant the interval of rationals [math](0,1)[/math], I guess you mean [math]k_i[/math] is either [math]0[/math] or [math]1[/math] for all [math]k_i[/math]. But that makes sense!
Quoting SophistiCat
So the procedure says:
"For all input sequence k, the series will converge to a real number in [0,1]"
Now the questions are:
"For all r in the reals, there exists an infinite sequence of inputs such that the series will converge to r?" (this would show the procedure is surjective)
Let's assume that's true. Then the surjection is from infinite binary sequences to infinite binary sequences... Which is unsurprising. It's a surjection from the unit interval of reals to the unit interval of reals, rather than from the unit interval of rationals to the unit interval of reals.
[math]\frac{1}{2}+\sum_{i=2}^{\infty}(-1)^{k_i}(\frac{1}{2})^i[/math], where [math]k_i[/math] is 0 or 1
We should be able to prove a stronger claim that a series composed of only positive fractions can converge to any real number between 0 and 1:
[math]\sum_{i=1}^{\infty}g_i(\frac{1}{2})^i[/math], where [math]g_i[/math] is 0 or 1
We can demonstrate this by a variant of the interval halving method. Let r be a real number between 0 and 1. If r < 1/2, then g[sub]1[/sub] = 0, otherwise g[sub]1[/sub] = 1. Take the next bracketing interval - [0, 1/2) or [1/2, 1] - and repeat the procedure to find the next g[sub]i[/sub]. Since each consecutive bracketing interval is half as wide as the previous one, then for any ? we can find n such that
[math]\sum_{i=1}^{n}g_i(\frac{1}{2})^i \in (r-\epsilon, r+\epsilon)[/math]
(I beg your indulgence. It pleases me that I can still solve an elementary calculus problem decades after I took the class :) This thread should probably be moved, since it doesn't really contain any philosophy.)
:up: Looks good to me. I've never even thought of using numerical methods in proofs! That's very cool.
Think it's curly braces. [math]\{\}[/math] They need \ in front of them in math mode.
Note that this is rational number; more specifically, a rational number whose proper-form denominator is equal to (2^n)*(5*m) for integers n and m. The important characteristic is that there is a finite number of digits on each side of the decimal point.
But what if the process does not terminate? Then this won't always be true.
If I understand your poorly-described algorithm, what you really get is a terminating sum that looks like this, for a set of increasing integers {n1,n2,n3,...,nk}:
If that isn't exactly right, it's something similar. Yes, this is a rational number.
But if the set of integers does not end, it does not have to be rational. And if the number you trace is irrational, your algorithm will not terminate. So no, you did not disprove Cantor.
+++++
A lot of people feel there is something wrong with Cantor's Diagonal Argument. That's because what they were taught was an invalid version of CDA. Some of the issues are just semantics (ex: it didn't use real numbers), and some are correctable (if you use numbers, you need to account for real numbers that have two decimal representations).
But one invalidates what they were taught as a proof. The correct proof is right, though. Here's a rough outline:
This proves that every listable portion of T is not all of T.
Most of the times CDA is taught, it is assumed that all of T is put into a list. Then, by proving that there is an element that is not in the list, it contradicts that you have a complete list. Thus proving by contradiction that all of T cannot be listed.
This is an invalid proof by contradiction. You have to use all aspects of the assumption to derive the contradiction. The derivation does not use the assumption of a complete list, so it cannot disprove completeness by contradiction. But it did disprove it directly.
What seems to have started the confusion was the second part of Cantor's version. If S=T, then s0 both is, and is not, an element of T. This is a proper proof by contradiction.
I think most people understand the reductio ad absurdum proof. What the big problem is what then?
You see there is an inherent structural difference between giving a direct proof and having only the possibility of giving an indirect proof. What can you say about mathematical objects, that can only be shown to exist, to be true, by an indirect proof? This is where critique about Cantor really lies. Can you just assume "larger" infinities, if you indirectly prove N < R? Everything fine and dandy after that and you can assume aleph-2, aleph3, aleph4...?
The problem is that CDA isn't a reductio ad absurdum proof; at least not as people think. The common presentation of it as reductio fails logically. As I described.
It also wasn't intended to be the focus of Cantor;s effort. It was a demonstration of the principle with an explicit set. What is known as Cantor's Theorem uses an abstract set, and proves that its power set must have a greater cardinality. (And, btw, it is a correct use of reductio.)
And we don't prove existence. Axioms do. The Axiom of Infinity establishes that the set of all natural numbers exists. The Axiom of Power Set says the power set of any existing set also exists. With those axioms, Cantor's Theorem proves that an infinite number of Alephs exist.
The problem with your algorithm is that the binary representation of a natural number is finite. Therefore, your algorithm picks out only those numbers that can be expressed through finite bisections of an interval. It turns out that the unit interval is isomorphic to the set of infinite binary sequences. The diagonalization argument establishes that the set of finite binary sequences does not surject the set of infinite binary sequences. It therefore deserves the name "uncountable." However, because there is an injection from the set of finite binary sequences to the set of infinite binary sequences that does not surject the set of infinite binary sequences, this "uncountable" set is strictly larger than the "countable set," despite the obvious fact that both sets are infinite.
I wouldn't agree on this. Axioms don't give proofs. Perhaps we are just thinking of this a bit differently.
A proof shows that something is true. If a mathematical object, a hypothesis, theorem, lemma is shown to be true, hence it is said to exist in the mathematical realm. It's hard to argue that something is illogical or false in math, if someone has given a logical proof for it. Axioms on the other hand are just given: "a statement or proposition which is regarded as being established, accepted, or self-evidently true." Or defined in a more mathematical way:
Mathworld Wolfram
I'm sorry, I worded that poorly.
We don't establish the existence of these sets by proof. We do it by the axioms we choose to accept. And since all proofs in mathematics are based on such axioms, there is no difference in the validity of either method.
Just look at how much debate here in this forum there is about infinity. Axiom of Infinity is anything but established and self-evidently true. The discussion here ought to show it. We just don't know yet! Bizarre to think that there are these gaping holes in our understanding of math, but they are there.
And yet it sounds so simple. A set is a collection of "elements" or objects and the axiom says we can consider the collection itself as an object, but perhaps not the same kind of object - but wait, can a set be considered an element of itself??? :roll:
If I remember correctly, some use that as a definition of infinity:
A set is infinite if and only if it is equivalent to one of its proper subsets.
With all due respect, if you want "actual truth", then you do not understand the purpose of an axiom in mathematics. The point is that mathematics contains no concept of actual truth. We define different sets of "truths" that we accept without justification. A set is invalid only if it leads to internal inconsistency, not if you think it violates an "actual truth" that is not in that set.
If we accept as true that there exists a set that contains the number 1 and, if it contains the number n then it also contains the number n+1? Then we can prove the existence of the cardinalities we name aleph0, alpeh1, aleph2, etc. Any inconsistencies you may think you have found are due to things you assume are actual truth but are not included in this mathematics.
What is bizarre is that some don't understand that everything proven in mathematics is based on unproven, and unproveable, axioms.
See the argument given in this OP:
https://thephilosophyforum.com/discussion/7309/whenhow-does-infinity-become-infinite/p1
IE induction leads to the conclusion that aleph0 must be finite.
IMO, the axiom of infinity is nonsensical and leads to absurdities like Galileo's paradox and Hilbert's hotel.
My point was that axioms can be possibly false. Our understanding can change. Best example of this was that until some Greeks found it not to be true, people earlier thought that all numbers are rational. Yet once when you prove there are irrational numbers, then the 'axiom' of all numbers being rational is shown not to be true.
And not all axioms are self evident. Just look at what Devans99 wrote about the axiom of infinity above
My point is that they can't. That's why they are axioms.
There are no absolute truths in Mathematics, only the concepts we choose to accept as true. While we can find that a set of axioms is not self-consistent, that does not make any of them untrue.
Best example is from Euclid, who first proposed an axiomatic mathematics. He thought that he should be able to prove a self-evident fact from his axioms in plane geometry. That given a line and a point not on that line, then there must exist exactly one line in the plane they define that passes through the point and is parallel to the line. It turns out that this needs to be another axiom. And that you can define consistent geometries if that axiom says there is one, many, or none.
Axioms don't need to be self-evident; in fact, that's what requires them to be axioms.
The normal definition of axiom:
'a statement or proposition which is regarded as being established, accepted, or self-evidently true.'
The mathematical definition of an axiom:
'a statement or proposition on which an abstractly defined structure is based'
The definition of a mathematical statement:
'A meaningful composition of words which can be considered either true or false'
So an axiom can be false; invaliding any results derived from that axiom.
?
I already gave an example of what was thought to be an axiom that wasn't. Greeks thought that all numbers were commensurable. The thought was for them self evident: math was so beautiful and harmonious. Yet all numbers weren't commensurable.There exists irrational numbers (and even transcendental numbers). What we had accepted to be true wasn't the case.
As Reuben Hersh says: "Mathematical knowledge isn’t infallible. Like science, mathematics can advance by making mistakes, correcting and recorrecting them."
Also a good point!
Mathematical set theory and foundations provide a playground for combative metaphysicians. And if you contemplate my question you might see why.
You put 'axiom' in inverted commas for good reason, even if you didn't understand it. That all numbers are rational wasn't an axiom - it was a definition, an informal intuition, or a conjecture, depending on how they approached numbers in their thinking. And accepting or rejecting an axiom does not amount to judging it to be true or false; as has been repeatedly explained to you, that doesn't even make any sense.
You gave an example of a near-religious belief. It was never an axiom in a consistent mathematics.This is similar to the belief that we can't treat aleph0 as a valid mathematical concept.
And I gave you an example where three different, consistent mathematics have been created based on three different axioms that contradict each other. The point is that neither mathematics, nor the axioms any specific field of mathematics is based on, are intended to represent absolute truth. You cannot win this debate with a nonsense claim like "the axiom is wrong."
Your reasoning of it being an conjecture or an informal intuition can be done only in hindsight. The definition of an axiom is "A self evident proposition requiring no formal demonstration to prove its truth, but received and assented
to as soon as mentioned". People were thinking about numbers and their commensurability exactly like as an axiom: a self evident proposition requiring no demonstration, something that was evident. Before it wasn't. And now it's an 'informal intuition'. Math simply is similar to science: we can make mistakes.
Quoting JeffJo
Yet what you are stating is a philosophical view of mathematics. What you are basically saying is that: "You cannot win this debate because you don't accept formalism!"
Formalism goes exactly on these lines you said above: Math isn't a body of propositions representing an abstract sector of reality, but more akin to a game. Nothing to do with ontology of objects or properties and something more like chess. The truths expressed in logic and mathematics tell us hardly anything about numbers or sets themselves. It's just basically a game where axioms are just premises that define the system used and how useful the system is or isn't doesn't matter. Hence if you have a math system where 0=1, then it means just that anything goes, right? That's formalism.
And lastly, I'm not here to win anything, but to have a conversation from which I hopefully can learn something.
An axiom is a statement - statements are true or false. End of story.
Treating aleph0 as a mathematical concept leads to paradoxes, eg:
https://en.wikipedia.org/wiki/List_of_paradoxes#Infinity_and_infinitesimals
What is a paradox? It's a contradiction. So a paradox indicates the presence of an reductio ad absurdum argument - IE we have made a false assumption (one of our axioms is false) and it has lead to an absurd conclusion.
In the examples linked above, that false assumption is the axiom of infinity.
The actually infinite is just a mental convenience - it is merely economical mentally to talk about the number of reals in [0,1] as being actually infinite. Of course in reality, it is no such thing - it is an example of potential infinity - those numbers do not take on real existence until we list them (actualise them).
So the axiom of infinity works in our minds, but does not work in any sort of reality - hence all the paradoxes.
Many things work in our minds (square circles, talking trees, levitation) but have no basis in reality. Actual infinity is one such thing.
That's not the definition of an axiom, as you ought to have learned by now if you were paying attention.
Seems like then you have your your own definition...
Definitons of axiom:
"An axiom is a proposition regarded as self-evidently true without proof. The word "axiom" is a slightly archaic synonym for postulate. Compare conjecture or hypothesis, both of which connote apparently true but not self-evident statements." Axiom - Wolfram Mathworld
"An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. " Axiom - Wikipedia
"1) a statement accepted as true as the basis for argument or inference. 2) an established rule or principle or a self-evident truth" axiom - Merriam Webster dictionary
No, seems like you are only interested in playing dictionary games. You can join the other idiot then, I am not interested.
Not the end of the story. Definitions are not commutative. An axiom is indeed a "proposition regarded as self-evidently true without proof." That does not mean that every "proposition regarded as self-evidently true without proof" is an axiom.
Looking further at Wolfram, an[ i]Axiomatic System[/i] is a "logical system which possesses an explicitly stated set of axioms from which theorems can be derived." So once again, the statement you claimed was an axiom was never stated as part of such a set, from which theorems could be derived. It was not an axiom, it was a near-religious belief.
No. What I am saying is that theorems in a field of mathematics need to be based on some set of accepted truths that are called the axioms of that field. Such a set can be demonstrated to be invalid as a set by deriving a contradiction from then, but not by comparing them to other so-called "truths" that you choose to call "self-evident."
What I am explicitly saying is that any arguments not based in the axiomatic system of a field of mathematics cannot not say anything about that field.
A couple of months ago the forum was infested with bad theology. Now it's bad maths.
Besides, one shouldn't assume that one school of Mathematical philosophy is correct and another is not. I gather that JeffJo thinks on the line of mathematical formalism when it comes to axioms (I assume, of course I could be wrong) while you've just said that I ought to know better...besides being an idiot.
So, we're making progress. :strong:
And you still haven't grasped the very simple fact that no field of mathematics claims to be "correct", or that another is not. Only that no statement is can be shown to be true without first assuming a set of unsupported Axiom, and proving theorems within that framework.
And it is quite clear that you have no interest in any formalism but your own.
Perhaps you can point to an example of a "proposition regarded as self-evidently true without proof' that is not an axiom?
Quoting JeffJo
Yes it is, the axiom of infinity is part of the Zermelo–Fraenkel system of axioms:
"Together with the axiom of choice (see below), these are the de facto standard axioms for contemporary mathematics or set theory."
https://en.wikipedia.org/wiki/List_of_axioms#ZF_(the_Zermelo–Fraenkel_axioms_without_the_axiom_of_choice)
Quoting JeffJo
You are confused. Perhaps another example will help:
Special relativity is derived from two axioms:
1. The laws of physics are the same in all inertial frames of reference
2. The speed of light in a vacuum is the same for all observers, regardless of the motion
If either of these axioms is ever demonstrated (by experiment or a theoretical argument) not to be true, then all the results of special relativity will be invalidated.
Any 'mathematical truth' is only as good as its axioms - if any of the axioms are false, everything derived from that axiom is invalidated.
Quoting ssu
This is an antiquated definition, suitable perhaps as an informal introduction to the topic, but not suitable for today's mathematics. And it's not about formalism vs. intuitionism or whatnot. For one thing, this formulation just isn't accurate and doesn't capture the role of axioms, even in Euclid's original books. For example, the fact that a square with the side equal to 1 cannot be inscribed inside a circle with the radius equal to 1 may be self-evidently true, but Euclid did not make it an axiom. Axioms are those propositions that are specifically chosen as the primitive building blocks of a mathematical theory.
More importantly, if axioms were a matter of self-evident truths, then there would be just the one mathematics, because there is only one truth (at least that's how most people see it). But this hasn't been the case with mathematics since long before people even started contemplating foundational philosophical questions like formalism, logicism, etc. The notion that mathematical axioms are some extra-mathematical truths (truths about what?) has been abandoned.
Perhaps you didn't understand my point.
I'm not looking for some ultimate truth. The question is if a set of axioms, an axiomatic system, is simply consistent. If they aren't consistent, I would in my mind declare then an axiom or axioms to be false, if we can pinpoint the reason for the inconsistency. This kind of "formalism" I do accept. So the question of the possibility of "an axiom being false" would perhaps be better understood from your viewpoint as that "an axiomatic system is inconsistent".
Could that be possible? Let's take the example of the axioms of ZF set theory. Consider the reason just why Zermelo and Fraenkel made those axioms: it was because of Russell's Paradox, which had made Frege's naive set theory, well...."naive". There was a reason why to do it. Yet perhaps the Paradoxes aren't at all an obstacle to be eradicated, but simply part of an answer we haven't fully understood. Because we don't understand infinity clearly, there's still CH you know, our understanding of these issues can change. Then axioms denying Paradoxes (assuming if they would be a part a reductio ad absurdum proof) would be, well, some might say false, others would say that the system wouldn't be consistent.
You might argue that fine, that doesn't matter, lets just form a new Set Theory and leave ZF as it is. But if so, are you OK with an axiomatic system where 0=1? In that system you can prove truly whatever you want! I might argue that the 0=1 is false, but I do understand your formalist point of view. Perhaps you would get angry at me saying that, because for you it's just an axiomatic system as anything else. If you hate the true/false dichotomy and juxtaposition, how about useful / useless then? If we have one axiomatic system, which is very useful to us, we can model extremely many things with it and another that cannot be used in any way, is there something to be said about the axioms.
I think that these questions go to the core of the philosophy of Mathematics.
I think I understand your point. Perhaps my answer to JeffJo above will make my point more clear. It's better to think of axioms as part of axiomatic systems. Yet when it comes to mathematics, is every axiomatic system as useful as the other? We tend to use some systems more than others, at least.
Quoting SophistiCat
What would be so terrible if it would be so? Now it isn't, I agree with that wholeheartedly, but just making a hypothesis here. Could there be an universal foundation for Mathematics?
Unfortunately in Cantors day there wasn't much attention paid to the underlying algorithms used to generate binary sequences, and hence he failed to admit the critical distinctions between
-The Total functions corresponding to the countable set of reals that can actually be constructed by an algorithm that halts to produce a digit for every argument representing a position of the sequence.
-The Provably Total functions corresponding to the subset of constructive reals that are 'a priori' deducible, in the sense that their underlying algorithms can be proved to halt without requiring the algorithms in question to be actually run.
-The Non-Provably Total functions corresponding to the subset of 'a posteriori' deducible constructable reals , whose algorithms in fact halt when run, but whose halting cannot be proved without running the algorithms in question.
-The Partial functions that fail to produce digits for every position of the sequence, and hence fail to represent a legitimate number, a property which is generally unknowable.
Cantor's diagonal argument when applied to the countable set of provably total functions, constructs a 'diagonal' total function, i.e. an additional valid real number, that is no longer provably total, and we also know that the combined set of provably total and non-provably total functions is countable (via enumerating their respective Turing machines).
Yet we cannot apply a diagonal argument to the set of non-provably total functions to produce an additional real number, for their very non-provability forbids us from knowing apriori whether or not our list contains only total functions representing genuine real numbers. Therefore there is no constructively acceptable diagonal method for producing a new real number from an enumeration of all and only the real numbers.
I assume all of you grok the sophisticated presentations on this thread. No need to worry so much about all these technical details! Life will go on.
You guys are so HARD on Cantor! Are you not aware that normal, everyday mathematics produces excellent results? And much of that, done on a computer, treats all numbers as rational. Are you really that concerned with non-computable functions or non-measurable sets? Material like that in math is referred to as "pathological" frequently. :meh:
In mathematics, instrumentalism reigns supreme. Anyone who rejects instrumentalism, opting for truth as a first principle, is accused of "bad maths". However, the terms of "bad" and "good" receive their meaning from morality. So to resolve the issue of whether it is really the instrumentalist, or the truth seeker who has "bad maths", we would need to apply moral principles.
Do you have any moral principles which show that instrumentalism is better than truth seeking?
We ought to treat the existence of non-computability and incommeasurability much more seriously than we do. Yet mathematicians push them aside and think somehow that they are 'negative' or something that ought to be avoided.
I personally think that absolutely everything is mathematical or can be described mathematically. Huge part is just non-computable. When we would understand just what is non-computable, we would avoid banging our heads into the wall with assuming that everything would be computable.
It's clear you don't understand mine. Nor have you tried.
Yes, it is. That is exactly what you have not addressed.
And you would be wrong to do so. All it shows is that the set is inconsistent. Any of the axioms could individually be part of a different, consistent, set. Yet you are calling an axiom, or axioms, "false" in a sense that can only be called "ultimate" or "absolute."
Get this point straight: The Axiom of Infinity cannot be proven to be true, or false, outside of some set of Axioms. I believe your words were that that his discussion should establish whether the AoI is self-evidently true. Nothing is further from the point if this discussion.
Given a consistent set theory that accepts the existence of an infinite set, we require different sizes of such sets. And not believing in infinity cannot change that.
Wiki: incommensurable generally refers to things that are unlike and incompatible, sharing no common ground (as in the "incommensurable theories" of the first example sentence), or to things that are very disproportionate, often to the point of defying comparison ("incommensurable crimes"). Both words entered English in the 1500s and were originally used (as they still can be) for numbers that have or don't have a common divisor.
Not quite the same as mathematical measure theory. But the above may have more relevance to the thread.
Wiki: According to the Church–Turing thesis, computable functions are exactly the functions that can be calculated using a mechanical calculation device given unlimited amounts of time and storage space. Equivalently, this thesis states that a function is computable if and only if it has an algorithm.
Every computable function has a finite procedure giving explicit, unambiguous instructions on how to compute it. Furthermore, this procedure has to be encoded in the finite alphabet used by the computational model, so there are only countably many computable functions. For example, functions may be encoded using a string of bits (the alphabet ? = {0, 1}). The real numbers are uncountable so most real numbers are not computable. See computable number. The set of finitary functions on the natural numbers is uncountable so most are not computable. Concrete examples of such functions are Busy beaver, Kolmogorov complexity, or any function that outputs the digits of a noncomputable number, such as Chaitin's constant.
It's amazing at how well computers have served us, isn't it, given these restrictions? :cool:
Just some folk are. :)
Quoting ssu
There are a bunch of areas in computer science on computability and such, e.g. ...
Computational Complexity Theory (Stanford Encyclopedia of Philosophy)
Computational complexity theory, Computational complexity (Wikipedia)
NP-completeness, NP (complexity), P versus NP problem (Wikipedia)
Within some limits you can write code to handle infinite sets.
Nowhere near what mathematicians routinely do, but some things are possible.
This is a straw-man argument. Just like we cannot escape theories in other fields, we cannot escape axiomatic systems. What my point was that as we have things like CH, we don't understand Infinity yet clearly. Hence there is the possibility that for example the axioms of a axiomatic system that we think is consistent might be proven inconsistent. Just as the fate of naive set theory. I don't understand why you won't believe our understanding of math could continue to change as it has changed in history.
Quoting JeffJo
No, the axioms are inconsistent to each other in the defined axiomatic system.
Quoting JeffJo
Wrong. As I said: "I'm not looking for some ultimate truth. The question is if a set of axioms, an axiomatic system, is simply consistent. I just happen to be such a logicist that I think that something that is inconsistent in math is in other words false.
True jorndoe, in my view it's a field we likely could find something new. The Church-Turing thesis is quite vague in my view. I think the most important issue here in the most simple format is Cantor's diagonalization. It seems with logic has a lot of peculiar things happening.
It is how Mathematics works. Anything that "exists" has to be based on Axioms.
Now that's a strawman argument. You need the AoI before you can even try to understand this thing you want to call "infinity."
And you have proven this? Or are you just supposing it could be so?
Yes, you did say that. You have also said that the AoI could be "wrong" and that we need to discuss whether it is.These statements contradict each other. This makes your axiomatic system inconsistent, and "false" by your definition.
Not ultimately false, or absolutely false, but some other kind of "false"? What kind?
Quite circular reasoning you have there, Jeffjo.
Quoting JeffJo
The axiom of infinity could be wrong in the way that it is inconsistent with the other axioms of ZF, for example. It is you that is making the case of some eternal truth as you don't take into consideration at all that the now used axiomatic systems could be inconsistent. I'm really not making the case for some universal truth here either. My point is that from the historical perspective we have thought about math one way and because of new theorems or observations we have changed our way of thinking about math. Why would you assume that now at this it wouldn't be so as earlier?
Quoting JeffJo
You tell me. All I understand is that if something is inconsistent, we can say it's false.
All you have to do is come up with an example showing this to be the case, rather than argue in an abstract way about it. Maybe you have, as I haven't read all the posts. Good luck.
Computability isn't a mathematical assumption, rather computability refers to the very activity of construction by following rules. Since mathematical logic consists only of the constructive activity of rule-following, the idea that mathematical logic can capture the non-constructive notion of "non-computability" is a contradiction in terms. None of Cantor's conclusions are really captured in his (ironically constructive) syntactical expressions. He is merely projecting his theological intuitions onto logical syntax.
Science on the other hand, attempts to predict the course of nature using a particular set of rules. Yet there is no reason to believe that the course of nature follows any particular set of rules. Hence we could say that nature might be "non-computable", but this "non-computability" cannot become the object of mathematical study on pain of contradiction.
Do think you understand the point of Axioms? Maybe you need to explain what you think it is. Because it is your arguments that are circular.
And Santa Claus could visit my house tomorrow night. But I don't draw conclusions from suppositions like that.
You are the one suggesting that statements could be called true, or false, outside of an axiomatic system. All I'm saying th that the AoI can be part of a consistent system, and you can't conclude anything about "Infinity" outside of one.
Yes, you are.
No, we have not. We may have changed the Axioms.
Define what "something" represents here. Because an Axiom, by itself, cannot be this "something" here yet youy keep treating it as though it can.
A ****SET***** of axioms can be inconsistent, which only means that at least one of them disagrees with one or more of the others. Not that any of them is "false." And claiming otherwise is claiming that a universal truth exists.
Do you understand Turing's answer to the Halting problem? Just as Cantor's diagonal argument shows that not every infinite set of numbers can be put into 1-to-1 correspondence with the Natural numbers, so do the various undecidability results, starting from Church-Turing thesis, show that indeed there are mathematical objects that cannot computed. Not everything can be calculated/computed by a Turing Machine.
Feel free to think that there is nothing that we could understand better in mathematics any time ever. All I said that what one could easily see even from this forum is that we do not understand infinity yet.
So changing the axioms isn't changing the way think about math?
Quoting JeffJo
Right. So are against something the idea that if something is inconsistent (in math/logic), it is false, because that would be a 'universal truth'. I guess you oppose talking about "The Law of excluded middle" because for you it's just one axiomatic system.
It's hard to argue with that. :chin:
And I know that I may not be the sharpest razors here when it comes to math and hence I'm happy if I am shown to be wrong.
Yet I think there still is important things to be discovered in math. Just a hunch...
Maths is constructed. One can do with it as one pleases with the symbols involved. We make the rules up as we go, and we can and do go back and change them as we like. They are not tied to instruments or forms or anything other than themselves.
It's a little more complicated than that. But go ahead and make one up. Should be fun. :smile:
Banno's Game
You said more than that; this is just your go-to defense: to invoke the mysteriousness of infinity, like some invoke the mysteriousness of God. And yet even this seemingly innocuous banality says more than you think it does. It implies that there is some extra-mathematical Infinity that mathematics is trying to grapple with. But mathematics as such doesn't contain anything extra-mathematical. Everything in mathematics exists only to the extent to which it is defined. The Axiom of Infinity is just a name for an axiom (a family of axioms in various systems); it plays by the same rules as every other axiom and doesn't purport to refer to something extra-mathematical - unless you want it to; but that would be an extra-mathematical choice on your part, as would be any use of mathematics to model something extra-mathematical. "Infinity" in mathematics is just a name, a symbol that could be replaced with any other symbol salva veritate. There is nothing inherently pathological about it.
We make rules for a purpose. The rules are instruments, used for obtaining our goals. Rules do not exist independently from their use, as if they are things in "themselves". And "use" implies purpose.
More of a concept, actually. An expression "becomes infinite" if it grows without bound. No need for a symbol. But it's there to be used if one wishes.
Recall that the proofs of Godel and Cantor correspond to Turing-computable algorithms. Are you meaning to suggest that a deterministic machine can follow a set of rules to prove that there exists a theorem that cannot be deterministically derived by those rules? Or that proves in a finite number of steps the existence of a literally uncountable universe? That's what the common (mis)interpretation of Godel's and Cantor's results amounts to.
It is certainly possible that nature is really random in the sense of falsifying any proposed theory-of-everything. But this is to speculate about nature, rather than to deduce a logical conclusion.
That's the first thing you've gotten right. And the fact that you will disagree is why you won't ever understand what I am saying.
Axioms don't define "the way we think about math." They define areas ("fields") within the framework of "how we think about math." There are three different fields that use contradictory Parallel Axioms (hyperbolic, elliptic, and Euclidean geometry), yet the way we think about them in math is the same. Each field is valid, and each Parallel Axiom is true within its field, and false in the others.. Each becomes a false statement (not an Axiom) in the others.
I can repeat this as often as you ignore it, but I'm running out of ways to make it sound different from what you've ignored before. You keep using the indexical word "something" without indicating what it refers to, the statement or the set. Some part of what you say is clearly wrong each time you use the word, but how it is wrong depends on what you mean. And if you understand the difference.
Even if you prove that a ****SET**** of Axioms is inconsistent, that does not mean that any one Axiom in it is "false." And a ****SET**** can't be false, just inconsistent.
Qualities a *****SET***** of Axioms can have include "consistent" and "inconsistent," but not "true" or "false."
Qualities a ****STATEMENT**** can have include "true in the context of a ****SET**** of Axioms," and "false in the context of a ****SET**** of Axioms," but not "true outside the context of a ****SET**** of Axioms," of "false outside the context of a ****SET**** of Axioms."
An Axiom is a special kind of statement that assumed to be true to define the ****SET***. The only quality an Axiom can have is ""true because we assume it in this ****SET**** of Axioms." Outside the context of any ****SET**** - that is, in the absolute or universal sense you deny you are using - it is neither true nor false. It is merely a set of words.
At least I'm trying to understand your point. (Which you think is impossible, I guess)
Quoting JeffJo
That geometry is different in two dimensions and more dimensions is evident yes. Yet we do speak of Geometry, even when there is Euclidean and non-Euclidean geometry.
Quoting JeffJo
Fine. So in this case we will you just the definitions of "consistent" and "inconsistent".
So how much do you do with "inconsistent" axiomatic systems, or as you wrote, "a *****SET***** of Axioms" that is inconsistent?
There is no evidence of it.
What I think, is that you only try to see what contradicts your predetermined idea of universal truth. And then you say the first thing that comes to your mind, that seems to imply what I said as wrong. Example:
This is a classic example of a strawman argument.
I described three different examples of two-dimensional geometry. They can be expanded into more dimensions. Yet I mentioned no such higher dimensions, and their existence is completely irrelevant. There are ***THREE*** ***DIFFERENT***, mutually contradictory versions of ***PLANE*** geometry. Each is a consistent field of mathematics. And the point is that there is no such thing as this universal, capital-G Geometry as you imply. Just consistent fields of geometry based on different sets of axioms. One axiom in particular is different in each field, making the other two false in that field. Yet none of them is true, or false, outside of a set of axioms that describes it.
If you truly are trying to understand my point, understand that one. And then try to see that your claim is preposterous, because it says the opposite. The claim that the Axiom of Infinity can be considered to be true, or false, outside of a field of mathematics that has either accepts it as true, or as false, without justification or proof.
?????
I don't "do" any quantity of whatever it is you are implying with them. I also don't suppose that they could be inconsistent because they contradict a "universal truth" that I want to others to accept as blindly as you do, and dismiss an individual Axiom in the set solely on the basis of that unsupported supposition. Which is exactly what you are doing.
Right. So you don't do anything with them. Well, neither do I.
And that was my point. But from the following it's obvious you don't get it.
Quoting JeffJo
No, the set of axioms are inconsistent when they aren't consistent with each other. You don't compare two different axiomatic systems to each other.
But perhaps for you even to mention that there is the law of non-contradiction is too much like an "universal truth", which you oppose.
s1 000001...
s2 101010...
s3 001100...
s4 100000...
s5 100001...
s6 100111...
s7 101100...
s8 000110...
s9 101010...
Diagonal sequence formed from list starting with s1, then s2, then s3, etc.
d1 110110...
d2 011101...
d3 111011...
...
If d1 is not in the list, neither is d2, and all that follow.
For every s there is a corresponding d missing.
Therefore half of the list is missing!
Which is what I have been saying. When the set of axioms lead to an inconsistency, it is the set is that is inconsistent. No one axiom is inconsistent, or false. Nor is any one axiom inconsistent with another. The set itself is inconsistent.
And I never compared two systems to each other. I listed three different, consistent sets that include three contradictory versions of the parallel postulate. The point wasn't to compare the sets, it was to show that none of these parallel postulates could be called "true" or "false." But I'm beginning to suspect you know this, and are deliberately arguing around the point.
And you still have not demonstrated an inconsistency with Zermelo–Fraenkel set theory, You have supposed it could be inconsistent, and blamed it on the Axiom of Infinity possibly being false. Which is preposterous.
Great, we both agree on something.
Quoting JeffJo
And notice the word "could". Could doesn't have the same meaning as is. I've only said it could be a possibility that in the future it is shown to be inconsistent. You see, there was a purpose for ZF - set theory to be made: It was to avoid the Paradoxes. It was made to avoid the pitfall that Frege's naive set theory had fallen to. I don't blame the axiom, in my view Infinity (and hence an axiom for it) is an integral part of mathematics. All I've said that we haven't understood infinity well. Even if ZF doesn't directly answer Cantor's hierarchial system of ever larger infinities, it's still there. Yet how much has there been use for Aleph-2, for Aleph-3, or Aleph-4? Cantor, a very religious man, thought that there could be an Absolute Infinity, but that was only for God to know. All I'm saying is that there could be surprises and new insights in this issue.
So please understand my point of view: we have gotten new insights on mathematics in history and our understanding of math has greatly changed from what it was during Ancient times and what it is now. Hence what is preposterous is then to think that a) no new insights will be made in mathematics in the future and b) these new insights won't change our understanding from the one we currently have. In science we admit this and talk just about theorems.
Yes, you might argue in your formalism that then these new insights would be just are new axiomatic systems separate of others. But if something is shown to be inconsistent, some people would dare to say there's something wrong then, it's false. Some would even dare to say that it would change our understanding of math as ZF is commonly seen (at least by some) as part of the foundations of mathematics. I do understand your point if you disagree with this, but still argue that this is a philosophical disagreement we have here.
So the basic argument we have had has been about inconsistency and falsehood.
Not only do the foundations shift, but mathematics rolls along like a giant intellectual snowball, gathering layer after layer of new concepts and theory, a plethora of results that can be bewildering even to an expert in a specific area. I was in a classical area, complex analysis, for years, and still dabble in elementary research, but these days I can hardly understand the titles of papers in that subject.
Did you notice the word "could" also? Anything "could" happen.The sun could explode tomorrow, ending life as we know it. Do you discuss that possibility? I personally am not even considering emptying my bank account, to use it before I lose it. mathematics is not about what "could" be true, it is about what is shown to be true within a given framework of axioms.
And did you understand that finding that a ****SET**** of axioms is inconsistent does not mean that any one axiom in the set is "false"? Or that claiming that an axiom could be false is an oxymoron?
And all I've said is that this is a nonsensical statement. You, on the other hand, said:
Quoting ssu
That would be a pointless discussion. An axiom is not, and cannot be, inherently "self-evidently true." We cannot "prove" it, and no amount of discussion will shed any light on it. It is because it cannot be shown to be self-evidently true, or false, that we assume it is self-evidently true. So we can lay the groundwork for a specific field of mathematics.
The insights you refer to all apply within a particular set of axiom, The only insights on mathematics in general that we have gained, are how to demonstrate that a field is internally consistent, and that there is no such thing as truth *in* a field from supposed truths *outside of* that field.
I agree.
Even if math follows it's own logic (no pun intended), it's still something that people do and it does evolve. Early 19th Century Mathematics and present day mathematics are taught differently and are different, even if a large part is totally same. It would be naive to think that the intellectual snowball, as you put it, would now stop. Obscure fields of mathematics can become important once people can use the field to build models and compute things. As one physicist once remarked, he just hopes math will give us new tools to use. I'm optimistic that those new tools will be invented/found.
Nobody has said otherwise. (Well, other than "what people do" is completely ambiguous.)
What was said, is that Math accepts no absolute truths.The entire point is that there should be no need to discuss what is, or is not, self-evidently true.
Math says "If the statements in the set of axioms A are accepted as[i] true, then the statements in the set of theorems T follow logically from them. The point of the evolution you misinterpret is to determine if the set T is internally consistent. If it is not, then the set A is invalid as a set of Axioms. But [i]no one axiom in A is invalid, and any one of them can be in another set A' that is consistent.
Quoting JeffJo
No. It's the inconsistency between two or more axioms in the axiomatic system, which make the system inconsistent. Your assumptions what others think are incorrect here.
Quoting JeffJo
Hmm. And in just what category would you put your idea presented here btw? :wink:
I think you fail to get the point so this discussion isn't productive. There is a thing called the philosophy of mathematics and there are various schools of thought in philosophy of math, you know.
No, I really don't think you do. Or at least, you have shown no evidence of it.
The point of the evolution you misinterpret is to determine if the set T is internally consistent. — JeffJo
And how is that not what I said?
But my point, that you have not shown you understand, is that finding such an inconsistency does not mean any of these two-or-more axioms is false.
And it is that there are no pre-determined truths, only truths that follow from one's axioms which are assumed to be true without proof.
It is a statement about philosophy, not a statement in math. "True" statments in math are either axioms, or theorems that follow from axioms. Unlike what you want here:
Quoting ssu
Yeah, Jeffjo, how isn't it what you said? (Hint: see first line in the quote above)
Quoting JeffJo
Great! So you admit that what you said was a philosophical statement.
FYI, it's the Logic & Philosophy of Mathematics part of the Philosophy Forum, so you shouldn't be surprised that we debate here questions of philosophy.
Quoting Umonsarmon
Here is where your mistake lies. The numbers a and b can become ever bigger as the string gets longer. When the string gets endlessly long, so can the numbers a and b. But for a/b to lie in the countable set of rationals, a and b must both be finite. Therefore, what you say is untrue for some strings of endless length. That’s the case for irrational numbers, such as the square root of 2. The latter’s irrationalness can be very easily proven, which was done more than two thousand years ago. In particular, the distance corresponding to sqrt(2) is not a rational multiple of ½.
Cantor’s diagonal proof, on the other hand, is perfectly sound. So, Cantor most certainly is right after all.
I am not so sure Tristan. You are assuming a priori that countable infinities are not equal to uncountable. That's one of your premises, but it's left unproven. IF it's true, the diagonal proof works, but not otherwise. Aristotle said all infinities are equal, so i've read
There can potentially be more or less. I would say four is a powerful number.
Infinity is also a wider concept of a beginning and middle and end.
It's end being that which is progression from a point that can continue infinitely, or it's a illogical theory of infinity. Such as a breathing, it reaches a point and that's it's end, but breathing continues because the shape is infinite.
A Klien bottle is a type of abstract infinity of air.
The proof that sqrt(2) is irrational and thus not a rational multiple of 1/2 doesn't depend on the existence of uncountable infinities or their difference from countable ones. Therefore, the OP Umonsarmon’s argument is invalid and flawed.
Regarding uncountable infinities:
First of all, “uncountable” means “not countable” by definition, so what you want to say likely is that I am assuming the existence of uncountable infinities. That’s not the case at all. Cantor proved that the set {0, 1}^IN of all functions from the set IN of natural numbers to the finite set {0, 1} is uncountable in the sense that there cannot be a bijection between {0, 1}^IN and IN. So, the only thing that I assume is that IN and {0, 1}^IN exist, and with this assumption, the diagonal proof works. More generally, the diagonal proof shows that if every set has a power set (which is a very reasonable assumption that does not postulate the existence of infinities, let alone differences between them), then there is no bijection between that power set and the original set. Thus, since there is obviously an injection from the original set to its power set, the power set is bigger than the original one. That’s how size for sets is defined. Therefore, if there is any infinite set at all (and we can be pretty sure that IN exists and is infinite), then the power set of that infinite set must be infinite and bigger than the original set.
I trust this hard mathematical, logical and sound reasoning more than what some philosopher who advocated severe ethnocentrism and a geocentric world view and who rejected atomism said more than two thousand years ago (that’s an attack aimed not at you, but at Aristotle. I find your search for unproven premises very useful and important, and I would like you to correct me if my reasoning regarding different infinities contains other premises which I’ve not explicitly stated or if my reasoning went wrong somewhere). In fact, Aristotle even ruled out the existence of an actual infinite on the grounds that such a thing would be “bigger than the heavens” – a very logical argument (scoffing), which also shows that Aristotle was thinking in much too concrete a way (mathematical objects are abstract and cannot be compared in size to the heavens any more than oddness can be compared to an orange). However, as long as every natural number is real and has a successor bigger than itself and all smaller natural numbers, the set of all natural numbers is also real and actually infinite. In fact, being abstract, it is at least as real as every concrete object.
Summing up, the diagonal proof that there are different infinities only rests on the premise that there is an infinite set which has a power set. Do you agree?
No, by definition, x is uncountable if and only if x is not countable. That's simply a definition of 'uncountable'.
Then we prove that there do indeed exist sets that are uncountable. And we prove that, in particular, the set of real numbers is uncountable.
The diagonal proof does not assume what it proves. Rather, it assumes some quite basic axioms of set theory, then it proves from those axioms that the set of real numbers is not countable (i.e. that it is uncountable). One may wish not to accept the axioms used, but that is a different matter.
No, I don't agree with your argument. The odds numbers don't line up with the whole numbers (you say), but you say they are equal infinities. You can prove "uncountable" infinities don't line up with the whole numbers either, but maybe they are equal as well. Until you prove that "uncountable" cannot be lined up with the wholes you haven't proven Cantor right. The diagonal shows that there are numbers not in the wholes, but there are evens not in the odds. I don't see the argument for why you can't just start at zero and line any infinity up with any other
We DEFINE them to have equal cardinality because there is at least one way of matching them up bijectively.
Quoting Gregory
Again, this follows by definition. If a countable set is defined as one that can be bijected to the natural numbers, then BY DEFINITION an uncountable set is one that can't. So if a set is uncountable, by definition it can not be bijected with the naturals.
That doesn't in and of itself prove that there ARE any uncountable sets; only that if there were one, it could not be bijected to the naturals. And of course Cantor proved that the reals are one such uncountable set.
Virtually none. There are proofs all over the Internet.
The diagonal argument is the one people usually see. But I think the proof of Cantor's theorem is simpler and more beautiful. It shows by way of a short and simple argument that there can never be a bijection between a set and its powerset.
So the powerset of the natural numbers must be uncountable. It's not difficult to show that there is a bijection between the powerset of the naturals and the set of real numbers (think binary strings) so this shows that the reals are uncountable without the confusion usually generated by the diagonal argument.
I'm afraid I don't know Hegel. I've heard he's difficult to read. I'm not much for the classical philosophers, my limitation. That said, I think it's better, when studying mathematical infinity, to put aside prior notions of philosophical conceptions of infinity.
Mathematical infinity starts from the counting numbers 0, 1, 2, 3, ... that we all have an intuition of as being unending. Given that, there are interesting things we can say. But none of this is intended to resolve any philosophical issues regarding being or the world or heavy things like that. It's only math.
If you can take the math on its own terms, the study of mathematical infinity is interesting and beautiful. Those are the criteria for what mathematicians care about.
It's true that Cantor himself believed that after Aleph-0, Aleph-1, and so forth, was an "absolute infinity" that he called God. Today, Cantor's religious beliefs are not much remembered except historically.
All in all, when approaching this material it's better to put aside all philosophical preconceptions. Mathematical infinity is not attempting to resolve any philosophical issues about the world or God or metaphysics.
In particular, there's no reason to believe the world is made up of dimensionless mathematical points in the same way the real number line or Euclidean space are. The question of how many points are in your pineal gland is meaningless. The pineal gland is made up of organic molecules, or atoms, or electrons, or quarks, or quantum probability waves; depending on the level of discourse. But nothing in the body, or in the world, is made up of mathematical points. Mathematical points are purely conceptual entities, like justice; or fictional entities like chess pieces.
Is this for me? Can you please Quote a bit of my text by selecting it and hitting the Quote button that appears? That way I get notified rather than having to keep coming back to the thread. Thanks.
I just said that mathematical infinity has nothing to do with heaven or eternity. Or people for that matter. Or religion. So if this post was for me, I surely don't have any idea.
Quoting Gregory
In 1638 Galileo noted that you can put the whole numbers 0, 1, 2, 3, 4, ... into 1-1 correspondence with the perfect squares 0, 1, 3, 9, 16, ... You can see that this is true, right? Without overthinking it. You can line up the whole numbers and line up the square numbers and connect them with lines such that every whole corresponds to a unique square and vice versa. Without trying to figure it out or overthink it, you can see this is true, right?
Quoting Gregory
We're all on that path. I'm not smart enough to understand all the math I wish I knew but I like to read about it anyway. Remember Wiles spent seven years of evenings after his regular day job as a math professor, working on Fermat's last theorem. Seven years of confusion and hard work and struggling to learn all the areas of math he needed in order to figure the thing out. Math is beyond everyone that way. You have to work at it. As Euclid said when the King asked him for an easy way to understand Euclid's great book The Elements: "Sire, there is no royal road to geometry."
Liking it's a good place to be.
Sorry about that.
Anyway, with infinity there are
1) cardinality
2) density
3) measure
It seems to me you have to consider all three of these in comparing uncountable to countable and all the other comparisons. Are we sure there are even only these 3 ways of assessing infinity?
Oh there are lots of ways. There's the subset relation, we can say that the set of odd naturals is smaller than the set of naturals because the odds are a proper subset of the naturals. This way of thinking is incompatible with bijections, but you can use it as a definition if you like. Bijection gives you a more interesting theory so that's why it's so common.
There's natural density, that's the idea that the density of the even numbers in the naturals must be 1/2, because the limit of the number of evens in the first n naturals goes to 1/2 as n goes to infinity.
There are probably other ways.
Quoting Gregory
When did I say that the odd numbers don’t line up with the whole numbers? Never, and that’s because the not-negative odds, the naturals, the odds and the wholes do line up perfectly:
Not-negative odds: 1, 3, 5, 7, 9, 11, 13, 15, 17, ...
Naturals: 0, 1, 2, 3, 4, 5, 6, 7, 8, ...
Odds: 1, -1, 3, -3, 5, -5, 7, -7, 9, ...
Wholes: 0, 1, -1, 2, -2, 3, -3, 4, -4, ...
Therefore,
Quoting Gregory
does not follow from anything that I have said. In fact, “uncountable” means “cannot be aligned with the natural numbers” by definition, and “have the same cardinality” is defined to mean “can be lined up with each other”.
Quoting Gregory
The diagonal proof does not show that there are numbers not it the wholes. That would be trivial. The diagonal proof shows that it is not possible to find a bijection between the naturals or the wholes and the reals, that is, it shows that you cannot line up the naturals or the wholes with the reals. By definition, that means that they have different cardinalities. More generally, it shows that there is no set for which a bijection between that set and its power set (the set of all subsets of the original set) exists.
Here is how it works. Assume that there is a bijection f from the set IN of all natural numbers to the set {0, 1}^IN of all functions from IN into {0, 1}. {0, 1}^IN is the set of all sequences of binary digits indexed by the naturals. Now we define the sequence s as the function from IN into {0, 1} which sends each natural number n to 1 – f(n)(n) (remember that f(n) is a function from IN to {0, 1}). Then for every natural number n, if f(n) = s, then f(n)(n) = s(n), which in turn implies that f(n)(n) = 1 – f(n)(n), giving f(n)(n) = ½. This contradicts the fact that f(n) sends every natural number to either 0 or 1 and never to ½. Therefore, there can be no natural number which is sent to s by f, contradicting the fact that s lies in {0, 1}^IN and the assumption that f is a bijection between IN and {0, 1}^IN. This contradiction shows that the assumption that there is a bijection between IN and {0, 1}^IN is untrue, since for any such bijection f, a contradiction follows.
There you have a beautiful proof that IN and {0, 1}^IN do not have the same cardinality. For the more general proof, you simply have to replace IN by an arbitrary set S and realize that there is a one-to-one-correspondence between the power set P(S) and the set {0, 1}^S; each member s in {0, 1}^S corresponds to the set of all members of S which are sent to 1 by s, and each subset R of S corresponds to the function from S into {0, 1} which sends every member of R to 1 and all other members of S to 0.
Quoting Gregory
And... voila, as we have shown above, Aristotle’s assumption leads to a contradiction. As Fishfry said, you can understand the above proof with almost no prior mathematical knowledge.
Quoting fishfry
Though I agree with you on many other points, I strongly disagree with you on this one. Mathematical points and mathematical objects in general, as well as all other abstract entities, are not conceptual at all. They are neither physical nor mental and exist independently of space and time. In fact, if anything, they are more real than any concrete entity. For example, Justice itself comes before every just individual, act and country, for if there were no Justice, nothing could be just, yet if all just individuals were slain, all just acts stopped and all just countries overthrown, Justice itself would still exist totally unaffected. In fact, without Justice itself, the very fact that all concrete just things have been destroyed couldn’t exist. Also, without Justice itself, it wouldn’t make sence to jointly call certain fellows, acts and countries just.
When frishfry asked me if it Galileo's conclusion sounded rational to me "without overthinking it", I really didn't know what to say. You have proven that the uncountable are not equal to the countable. But that doesn't prove every countable is equal to every other countable. I think we have to base this on geometry. The pineal gland has as many points as the body, but it is a lesser infinity. There is something you guys are missing in this. The bijection argument that the natural and the wholes and the odd numbers are all equal infinities forgets about density first of all, and probably many other things. I think we have to posit a "basic uncountable infinity" such that this infinity plus one is suddenly larger (by one). Lining a couple of units of infinity up proves nothing. Yes they then go to infinity, but which infinity? Again, when you compare the odd to the natural numbers you have to move the odd numbers back, disturbing the infinity in the direction to the right. It's an illegal move imo
There are many senses in which mathematicians use the term 'infinity'. But when we get to formal definitions, we distinguish among those senses.
These are basic definitions ['<->' stands for 'if and only if]:
w = the set of natural numbers
x is finite <-> there is a 1-1 correspondence between x and a natural number
x is infinite <-> x is not finite
x is denumerable <-> there is a 1-1 correspondence between x and w
x is countable <-> (x is finite or x is denumerable)
x is uncountable <-> x is not countable
More good understanding about this and related topics is available in any introductory textbook in set theory.
I am saying that is an illegal move
The best explantion (an exellent one) I have found of how mathematical definitions work is:
Introduction To Logic - Patrick Suppes
Can you briefly explain why it's a legal move to move 3's place to line up with 2 when it's realty in-between 2 and 4?
I think that pairing up odd numbers with natural numbers by sliding the former back is an illegal move. Prove otherwise
Define f as the function from the set of all naturals to the set of all odd naturals which sends each natural n to the odd natural 2n+1. Do you agree that this function is well-defined and a bijection, that is, do you agree that f sends each natural to exactly one odd natural, that it sends no two naturals to the same odd natural, and that to each odd natural it sends some natural?
If yes, then by definition the set of all naturals has the same cardinality as the set of all odd naturals. That’s because having the same cardinality is defined as the binary relation on sets which for all sets A, B relates A to B if and only if there is a bijection between A and B. You can easily see that having the same cardinality is an equivalence relation.
By saying that Quoting Gregory, you must mean that f doesn’t exist, which you believe is not the case if you answered “yes” to my question above.
I don't see the relationship between your equation and moving all the odd numbers back without changing its infinity. I mean I get the equation, but it has no operative power to slide back the odds without consequence. The odd numbers are a specific infinity, half in density than the naturals. I am shocked professional mathematician try to compare otherwise by bijection. None of it means anything. You can't put anything you like in place of odd numbers. Maybe the naturals can be defined geometrically, so that the odd of half in density like a banana is bigger than its peel. MAYBE the peel has as many points as the banana, but it's clear which is larger, and that has more truth
We aren’t literally sliding back the odd naturals. After all, they’re abstract entitities and thus can’t be changed. What we can very easily prove is this:
1. The binary relation ~, defined for all sets A, B by
A ~ B if and only if there exists a bijection between A and B,
is an equivalence relation on the class of all sets. Cardinalities are defined to correspond to the equivalence classes generated by the relation ~. So, we can call ~ by the name “having the same cardinality”.
2. The set of all odd naturals is related to the set of all naturals by ~. This I have proved above with my function f.
Therefore, ‘sliding back the odd naturals’ has no consequece with respect to ~, that is, is keeps the set in question in the same equivalence class generated by ~. Of course, that doesn’t mean that is keeps the set in the same equivalence class generated by some other equivalence relation. For example, it obviously doesn’t keep the set in the same equivalence class generated by the equivalence relation of equality, or the equivalence relation of having the same density.
Quoting Gregory
It’s similar to when I say, “The Cologne Cathedral is higher than St. Peter's Basilica”, and then you say, “No, St. Peter's Basilica is longer and wider than the Cologne Cathedral. I’m shocked professional architects compare buildings by their height. Nothing of this means anything.” Your statement about length and width is just as true as mine about height, but as it turns out, height is one of the most useful and important characters of buildings. Same with sets; cardinality is applicable to every set and, as it turns out, gives us very useful information and a rich theory, whereas some others, like measure or density, are only applicable in specific situations. Comparing sets by the subset relation is, of course, universally applicable, too, but it doesn’t give us a totally ordered hierarchy of infinities, unlike cardinality (the latter can be proved, but needs some work).
As a matter of fact, it means very much whether or not two sets have the same cardinality, th.i. (that is) are related by ~. The rich theory of cardinal numbers is proof of that.
Quoting Gregory
Actually, you can. Let me give you the ordered pair (IN, nf), where IN is the set of all natural numbers and nf is the successor function n -> n+1 from IN into IN, and the ordered pair (IN’, nf’), where IN’ is the set of all odd natural numbers and nf’ is the odd successor function n -> n+2 from IN’ into IN’. Then you won’t be able to decide which pair is the ‘true’ structured set of the naturals. That’s because both have exactly the same structure. So, the odd naturals actually can ‘become’ the naturals. The same is true for the other direction.
Quoting Gregory
Again, saying that comparing by equality and the subset relation has more truth than comparing by cardinality is like saying that comparing by length and width has more truth than comparing by height. That is not true. Both are equally valid. That the set of the odd naturals is a proper subset of the set of the naturals is equally true as the proposition that the two sets have the same cardinality.
Quoting Gregory
Actually, Cantor’s proof has nothing to do with that. He proved that every set has a strictly smaller cardinality than its power set.
Quoting Gregory
No, since one set can have the same cardinality as another and still be a proper subset of it and have only half its density.
Quoting Gregory
Firstly, the part cannot be equal to the whole, and Cantor doesn’t imply that. A set is never equal to any of its proper subsets. Rather, some sets (the infinite ones, and no others) have the same cardinality as some of their proper subsets. That does not mean, however, that every two infinite sets can have the same cardinality. We have proven that above. Please don’t just take my word for it, but read Cantor’s proof, which I have given above, and understand every step of it. If you believe that something in there doesn’t seem sound, please tell me. I’d be happy to clarify.
By the way, countable infinities are by definition not equal to uncountable ones. What Cantor proved is that uncountable infinities exist.
If you place a dot on the page and you place a second dot, in relation to each other, one is closer to a different edge of the object. Therefore I'm saying that when placing a dot, you use squaring agility.
Isn't measuring how many infinites there are depend on the potency of infinity? It's agility?
Isn't there also thus a multi-infinity?
Infinity usually takes form of an abyss. Infinite water infinite sound. Infinite human is a proposition at a certain ratio.
I suggest first establishing a firm understanding of the axioms and rules of inference of mathematical proof and definitions - whether in formal first order predicate logic or the informal techniques that are formalized by first order predicate logic. If one understands these conventions for mathematical proof, then one sees that the proofs of, for example, these theorems are incontestably correct:
the set of natural numbers is equinumerous with the set of odd numbers
the set of natural numbers is equinumerous with the set of rational numbers
the set of natural numbers is not equinumerous with the set of denumerable binary sequences
the set of natural numbers is not equinumerous with the set of real numbers
no set is equinumerous with its power set
Of course, one may choose to reject the axioms and rules of inference upon which the proofs depend. But it is incontestable that the theorems are entailed by those axioms and rules.
As I mentioned before, the predicate 'is infinite' is defined as 'is not finite'. You then objected that that is "an illegal move". I explained that definitions are not "illegal moves" and I suggested a book that has an excellent discussion of the methods of mathematical definition. I don't know whether you understand now that definitions are not "illegal moves".
Meanwhile, 'is dense in the ordering' is a separate predicate.
Quoting Gregory
There is no rearrangement in the proof.
As Tristan L has so generously and perspicaciously explained, the proof adduces a one-to-one function from the naturals onto the odds, and that is all that is needed.
Do you know the mathematical definitions of 'function', 'one-to-one', and 'onto'? Understanding those basic concepts, plus some grammar school arithmetic, leads to understanding the proof easily.
This should put an end to the issue. But it won't. :roll:
|>ouglas
That's all I wanted to say. So Douglas can maybe cheer up