When/How does Infinity Become Infinite?
As people have pointed out - and I acknowledge - maths is not my strong point. So I would be grateful to get input from those will more skill in the discipline on the following arguments:
1. [math]\infty=\sum_{n \to \infty} 1[/math]
2. [math]= 1 + 1 + 1 + 1 + ... [/math]
3. The first term in the series [2] is finite
4. If n is finite, then n+1 is finite
5. So by induction, the sequence [2] is finite for all natural numbers n
6. What about for n=?? Well for no individual (n+1) step on the way to ?, does a finite number ever become an infinite number. And infinite steps are just the sum individual steps, so it seems ? retains the property ‘is finite’ even at 'infinity'?
I think step [6] above is no doubt questionable, but it brings out the point: how exactly does a finite number ever become infinite? - We have no basic arithmetical operators to convert finite numbers into infinite numbers. To focus on this aspect, here is a similar argument that more graphically brings out the discontinuity between natural numbers and infinity:
A. [math]\infty=\sum_{n \to \infty} 1[/math]
B. [math]= 1 + 1 + 1 + 1 + ... [/math]
C. 1 is 0% of the way to counting all numbers / reaching infinity (because 1/?=0%)
D. If n is 0% of the way to counting all numbers / reaching infinity, then so is n+1 (n+1/?=0%)
E. (eg 100^100/?=0% of the way to counting all numbers / reaching infinity)
F. So by induction on [C] and [D], for no natural n are we greater than 0% of the way to counting all numbers / reaching infinity
G. ?/?=UNDEFINED, so infinity is never in this sense 'completed'. In fact there is no number n such that n/?=1%, n/?=10%, n/?=50%, so the sum of the series never gets anywhere near infinity / counting all numbers - there seems to be a massive discontinuous 'train wreck' at the end of the series associated with 'infinity' - it is unattainable?
1. [math]\infty=\sum_{n \to \infty} 1[/math]
2. [math]= 1 + 1 + 1 + 1 + ... [/math]
3. The first term in the series [2] is finite
4. If n is finite, then n+1 is finite
5. So by induction, the sequence [2] is finite for all natural numbers n
6. What about for n=?? Well for no individual (n+1) step on the way to ?, does a finite number ever become an infinite number. And infinite steps are just the sum individual steps, so it seems ? retains the property ‘is finite’ even at 'infinity'?
I think step [6] above is no doubt questionable, but it brings out the point: how exactly does a finite number ever become infinite? - We have no basic arithmetical operators to convert finite numbers into infinite numbers. To focus on this aspect, here is a similar argument that more graphically brings out the discontinuity between natural numbers and infinity:
A. [math]\infty=\sum_{n \to \infty} 1[/math]
B. [math]= 1 + 1 + 1 + 1 + ... [/math]
C. 1 is 0% of the way to counting all numbers / reaching infinity (because 1/?=0%)
D. If n is 0% of the way to counting all numbers / reaching infinity, then so is n+1 (n+1/?=0%)
E. (eg 100^100/?=0% of the way to counting all numbers / reaching infinity)
F. So by induction on [C] and [D], for no natural n are we greater than 0% of the way to counting all numbers / reaching infinity
G. ?/?=UNDEFINED, so infinity is never in this sense 'completed'. In fact there is no number n such that n/?=1%, n/?=10%, n/?=50%, so the sum of the series never gets anywhere near infinity / counting all numbers - there seems to be a massive discontinuous 'train wreck' at the end of the series associated with 'infinity' - it is unattainable?
Comments (30)
I think my doubts are around how infinity is defined. Limits seem to do a good job of defining potential infinity but how is actual infinity defined? Set theory defines 'countable actual infinity' as the the 'cardinality of the set of natural numbers' - which is about as meaningless a definition as could be imagined.
Actual infinity is not computable and not (it seems) definable, leading to a suspicion that it cannot be a real concept - it may exist only in our minds (along with talking trees and square circles) and is therefore not realisable.
Maybe @JohnDoe, @softwhere or @John Gill have opinions?
To tackle it from another angle, my cousin, who has a degree in mathematics, explained to me, there are infinities of different sizes. There is an infinity between the numbers one and two because numbers are infinitely divisible. So there's no need for conversion between finite and infinite: the two are part of the same system.
This point in a slightly different guise, is discussed at length on another thread:
https://thephilosophyforum.com/discussion/comment/362707
In short (and my opinion only) - a number has zero width, so the number of numbers between 1 and 2 is 1/0=UNDEFINED and not infinity.
If a number has non-zero width, that leads to a finite number of numbers between 1 and 2.
If you were to try to represent numbers in the reality external to your mind, you would find that any line used to represent the real number line would be composed of molecules so there would be a finite number of numbers on a real line segment.
I have already given my opinions of infinities as being limit concepts, with virtually no mention of "the point at infinity". Infinitesimals, on the other hand, have been placed in a proper mathematical model and can be used to generate calculus. I would think these tiny little objects might generate more controversy than infinity, but apparently not. As part of the hyperreal number system they are intimately connected with infinities.
I never went beyond naive set theory, so these mathematical notions are merely mildly amusing. Keep in mind what I said before: the foundations of mathematics is a relatively recent subject that attempts to place all the math that has worked so well over millennia in "proper" logical frameworks. I suspect there are a few analytic philosophers on this forum who know far more than me.
1. A zero width
2. A non-zero width
3. A undefined width
So how many reals between 0 and 1?
1. 1/0=UNDEFINED
2. 1/(some finite small number)=(some finite large number)
3. 1/UNDEFINED = UNDEFINED
I believe that a number is a purely imaginary concept so we can imagine it to have zero width so definition 1 above is what we do in our minds. This is also consistent with the definition of a point in maths as having zero extent - so we can imagine a point corresponding to a number on the real number line.
Now you could claim that a number has an infinitesimal width. But that is 1/? and in my opinion ? and its inverse are not well defined - the point of the whole OP.
As you can imagine I am not a believer in the possibility of continua occurring in nature and I am not alone in this belief (eg loop quantum gravity). If spacetime is actually discrete, then case 2 above will have a particular, assignable, finite width. Time will tell. I'd point out that we will never, ever be able to empirically prove spacetime is continuous, but we may be able to prove it is discrete.
Depends on the hardware/software you are using I guess. Computers use discrete binary representations of numbers. Computers are real. Reality is real. Continua maybe just a figment of our imagination as far as reality goes. Time will tell.
There would be no “when” because there is no beginning or end to infinity. I’m prone to think a symbolic representation of infinity suggests finitude, but then again I am a layman in math.
Yes you have a good point. It seems that most people do not understand infinity.
Infinity just means 'without end'. It is not a number and cannot be used in any calculation.
You can count the integers as far as you like and you will never reach 'infinity'. So to say that the integers are infinite is just to say that you can count them as far as you like without reaching the end.
Similarly no physical measurement can ever be 'infinite'. So to claim that time or space are 'infinite' is just to hypothesize that the end of time or space can never be reached.
To hypothesize that some things are infinite, such as hotel rooms, is to enter a fantasy world, where nothing is real.
This forum has many folk like @tim wood who accept the received 'wisdom' of Cantor and co without question - it is refreshing to talk to someone with an open mind!
BTW Did you know the reason actual infinity is enshrined within maths as a number is that Cantor was a devout Lutheran, believed that God was infinite and believed that God was talking to him telling him to put infinity into maths!
Fast forward to today and ironically, it is mainly the atheists who believe in infinity - theists and deists often question its existence and rightly so.
The correct terminology is that the series [math]\sum_{i = 0}^\infty 1[/math] diverges. In other words, the limit of the partial sums [math]\sum_{i = 0}^n 1[/math] does not converge to a number as [math]n[/math] tends to infinity. The notation [math]\sum_{i = 0}^\infty 1 = \infty[/math] does not mean that the series equals some number called 'infinity' and denoted '[math]\infty[/math]'.
Accepting the non-numeric / purely imaginary / unrealisable status of infinity implies:
- The commonly given definition of infinity is wrong: 'a number greater than any assignable quantity or countable number'
- Transfinite arithmetic is a work of pure fiction
- Ideas about space and time that assume the existence of actual infinity are not mathematically sound
You are misinterpreting my response. I was responding to your argument by clarifying what people mean when they say things like "the series equals infinity." This is simply a shorthand for "the series diverges" or "the limit of the partial sums does not converge to a real (complex) number." It doesn't follows that cardinal and ordinal arithmetic is "false" (I'm not even sure what that means) or that space and time cannot be unbounded.
By definition, [math]\aleph_0[/math] is the cardinality of the natural numbers. Your argument does not establish that the natural numbers have finite cardinality. I thought that clarifying what people mean when they use the series notation might disabuse you of the notion that there must be some "transition" from "finite numbers" to "infinite numbers" happening.
Just a definition. Sort of like defining ?=?. Meaningless IMO.
Quoting quickly
Why?