The measure problem
The measure problem is an issue that happens with infinite sets, where the ratio of different categories in the infinite set is measured differently depending on which way you order the set. Take for example the task of measuring the ratio of odd to even integers. If you order all numbers with all the odd ones first, and then set a cut-off point (say the first ten numbers) and count the ratio in that sub-set, you will get 100% odd. If you order the set in a different way, you may get a different ratio.
I've read one physicist claiming that this means that existence, time, space, everything must be finite, because infinite sets are logically contradictory, as you can apparently change their ratios by changing the order in which you look at them.
However it has occurred to me that the measure problem would apply equally well to a large finite set, say a set of a billion integers which it would take a very long time to actually count in order to determine the correct ratio of odd-to-even. So I suspect that the measure problem doesn't imply that there is something inherently illogical about infinite sets at all, I think that infinite sets have concrete ratios between their categories just like a finite set would, and the measure problem simply represents the fact that if you order the set in a way that doesn't reflect its real ratios you will get an incorrect answer. Similar to what would happen if you sent our a survey to a sample of people who don't accurately represent the population.
What do you guys think? Anything wrong with my reasoning? Anything I've missed?
I've read one physicist claiming that this means that existence, time, space, everything must be finite, because infinite sets are logically contradictory, as you can apparently change their ratios by changing the order in which you look at them.
However it has occurred to me that the measure problem would apply equally well to a large finite set, say a set of a billion integers which it would take a very long time to actually count in order to determine the correct ratio of odd-to-even. So I suspect that the measure problem doesn't imply that there is something inherently illogical about infinite sets at all, I think that infinite sets have concrete ratios between their categories just like a finite set would, and the measure problem simply represents the fact that if you order the set in a way that doesn't reflect its real ratios you will get an incorrect answer. Similar to what would happen if you sent our a survey to a sample of people who don't accurately represent the population.
What do you guys think? Anything wrong with my reasoning? Anything I've missed?
Comments (34)
I wonder who would say such a thing. Where did you read this?
Quoting Fuzzball Baggins
Yeah, you missed, or rather forgot, your own argument showing that some measures just aren't well-defined. This doesn't imply anything logically contradictory, of course, only that not every measure that you care to describe is well-defined.
The measure problem in cosmology is not that you can't come up with some well-defined measure - there is no lack of candidates. The problem is in coming up with a physical justification for a specific measure - and that's a scientific problem.
Your definition of the measure problem makes much more sense!
"Infinite set" is self-contradictory. "Infinite" implies unbounded, and set implies "bounded". To say that there is an infinite set is like saying that there is an infinite object, the two concept "infinite" and "object" contradict each other, such that this is impossible.
We went through this recently on a different thread. Let's say that "set" is defined as a "well-defined collection", as Wikipedia suggests. A "collection" in the sense of a noun implies having been collected, so an infinite collection is impossible because the act of collecting cannot be complete, and such a collection cannot exist. "Collection" in the sense of a verb, meaning the act of collecting, cannot be construed as an object, a "set", because this would be a category mistake. So an "infinite set", as an infinite collection in the sense of an object, is impossible by contradiction, and it is impossible as a "well-defined" activity because it is an incomplete activity.
Perhaps with numbers and mathematics one should stick to the logic of math itself and not bother about physical time and physical doing, of what kind of numbers our present day computers or computers of the future can handle. Even a atural number that is one hundred thousand digits long can be problematic for us to handle and our Computers to handle, yet the logic of the number is totally similar to a natural number that is two digits long, basically one between 0 and 99. Otherwise you will start looking for the quite illogical "first too big number that cannot be handled by a computer".
It takes a lot longer if you're counting backwards.
Is this a good time to point out thar sets are something we make up, so they only exist insofar as someone is imagining them?
I suppose this is really more a discussion of the definition of the word set rather than whether the universe could be infinite, so I'll agree with you that with the definition that humans have given the word set, the term 'infinite set' is illogical :P
Therefore, we have "the measure problem". Doesn't it make sense to rid our mathematics of such illogical axioms? When we realize that such things are illogical, we can apply the same principle in other areas. Consider "the universe" for example. As such, it is an identified and named object. It cannot be infinite according to a very similar contradiction as mentioned above. If it were infinite it could not be individuated and identified as an object, it's existence would be indefinite and therefore not an object which has definite existence. So to speak of 'the universe" is to speak of an object, and an object cannot be infinite. Therefore by the same reason that it is illogical to talk about an infinite set, it is also illogical to talk about an infinite universe.
There is surely an indefinite aspect of infinite, which is not so commonly developed in talk of "infinite". One definition of indefinite is limitless, and, something which continues indefinitely is infinite.
An object must be bounded, because it is an individual, a unity, a whole. Without these conditions it is indefinite. It's not an issue of what can be imagined, or the laws of physics, but it's an issue with the laws of logic, specifically the law of identity. When we identify an object, we point it out, then proceed to describe it by assigning properties or attributes. "Indefinite" refers to what we cannot grasp, what is beyond our apprehension. So, when we assign to an object, the property or attribute of "indefinite", we are saying that there is something about that object which is impossible to apprehend.
This is an act of judgement which is made, the object is judged as incomprehensible. It does not mean that the object really is incomprehensible, it has just been judged as incomprehensible. This is a self-defeating judgement. It impairs the will to understand the object, by identifying it as not understandable. Further, if any aspect of the object appears to be incomprehensible, illogical, or logically inconsistent with another aspect of the object, we can accept these logical inconsistencies of the object, by concluding that they are due to the indefiniteness of the object.
Therefore it is completely unreasonable to identify an object as indefinite, or to in any way assign "indefiniteness" to an object. We must assume that the appearance of indefiniteness is due to our inability to understand, and not part of the object itself. This will inspire us to continue to try and understand the object, to develop our minds rather than just assuming it is impossible to understand. And, from the other perspective, if the object really is indefinite, and therefore impossible to understand, we would never be able to know this with certainty, because this would require knowing the object which cannot be known. So it is completely unreasonable to assume that any object is indefinite, or infinite, no matter how you look at it.
Actually I don't really know what a multiverse is. In one way, "multi" implies a multiplicity of objects, but also in another way "the" implies a single object. It's probably self-contradictory like infinite set.
No it's not quit like that. What I said is that "the" implies a single object, while "multi" implies a multitude. I could have just as easily said that the name "multiverse" identifies a single object, so it's not really "the" which is the problem, "the" was just an indicator. It indicated that "multiverse" is a name which identifies an object.
So it appears, at first glance, that there may be an issue with self-contradiction, because it is suggested that a multitude of objects is a single object. However, we do commonly speak of a multitude of objects as a single object, that's what happens in arithmetic; 2, 3, 4, etc., are each representative of a single object, a number, but each number defines a multitude as well. What happens with "infinite" is that the multitude is undefined, and even specified as undefinable. But the object, the particular number, 10, 15, 25, or whatever, only has existence because it defines a multitude. Its very existence, as a number, is completely dependent on its capacity to define a multitude. If any such number which is signified by a numeral, "6", "7", "8" etc.,, did not define a multitude, it would not exist as an object. "Infinite" signifies an undefined multitude. So by the very fact of what it signifies, the possibility of it being an object is denied. What "infinite" signifies is "it is impossible that I am an object like a number", because a number necessarily defines a multitude while "infinite" necessarily does not.. .
I haven't collected them all yet in my mind. So how could they be collected in my thought? Furthermore, "all the positive numbers" does not qualify as "well-defined" in a mathematical sense, because how many positive numbers that there are is indefinite.
You have all the positive numbers collected in your mind!? Can you list them then?
Quoting tim wood
My claim is quite simple. A large number of grains of sand is collectible. An infinite number is not.
Quoting tim wood
Do you understand that there is a significant difference between "a large pile of sand", which obviously has a finite number of grains of sand, (as any pile of sand does), and "an infinite number of grains of sand"? The latter is not a pile of sand.
Quoting tim wood
If you will listen, I will oblige.
I don't think it's a good idea to rely on human intuition when it comes to physics. The human brain has evolved to cope with our everyday experiences, not with the laws of physics. Think about relativity and quantum mechanics - very unintuitive!
Do you have any logical reasoning (not involving human intuition, but based on the laws of physics or mathematics) for why an infinite thing could not exist in reality?
I don't think the concept of a set having to be 'collected' quite applies to what can and cannot exist in reality. I may not be able to create an infinite collection, or even imagine all the members of an infinite set, but reality doesn't have to 'collect' anything - infinite things can exist simultaneously without having to be created one by one.
All of them of course. I want you to prove to me what you claimed. "I have them collected in my mind."
Quoting tim wood
If what I say is true, then what difference does it make, whom, besides me, says so? Can you not read, and judge what I say, for yourself, without requesting an appeal to authority?
Quoting tim wood
I'll take the razor, and here's my proof. I'll reproduce from above, as it appears like you haven't read the thread. Tell me which part you dislike
Quoting Metaphysician Undercover
Quoting Fuzzball Baggins
Yes, I went through this, I reposted it above in case you didn't read it through.
Quoting Fuzzball Baggins
I went through this already. It is unreasonable to assume that any thing is infinite because such an assumption impedes our capacity to know that thing, and it is also impossible to know that a thing is infinite. So it's not the case that it is impossible that an infinite thing exists, in reality, but it is impossible to know that any given thing is infinite, and detrimental to the understanding of that thing, to assume that any given thing is infinite. Therefore it is unreasonable to assume that there is an infinite thing in reality.
Quoting Fuzzball Baggins
Of course it applies. We create descriptive terms, and the laws of logic to reflect reality. When something is contradictory, like a square circle, we say that it is impossible because it cannot exist in reality. So, if we produce a concept of descriptive terms which contradict (a contradictory description), we say that this thing cannot exist in reality. That is the case with "infinite collection". As a noun "collection" implies having been collected, as a verb "collection" implies the act of collecting. The noun contradicts "infinite" and the verb when qualified by "infinite" signifies an indefinite act.
Quoting Fuzzball Baggins
You can say whatever you want about the "infinite thing", that's the problem with assuming an infinite thing. Because the thing is indefinite, it cannot be properly identified, and laws of logic cannot be applied. That is why assuming an "infinite thing" is detrimental.
I asked for the list, not a description of it.
Quoting tim wood
Did you read my proof? I do believe that a logical proof qualifies as "evidence".
You may not be able to observe through empirical evidence that an infinite thing exists, but that doesn't mean it's unreasonable to infer that it exists. Take numbers, for example. I cannot count all the way to infinity, but I can infer that there are infinite numbers from the fact that if there were a finite biggest number then asking 'what is that number plus one' would break that limit.
Quoting Metaphysician Undercover
I think that is a reasonable way to define an infinite set of numbers, it is used all the time in mathematics. Just because he didn't list all those numbers separately doesn't mean they don't all exist. You were complaining earlier that infinite sets are inherently undefined. '2n, with n being any integer' is an example of how to define an infinite set.
Sure, I agree that's the case, but "numbers" is not a thing. That's the whole point. So your argument is nothing other than a category mistake. 'I can infer that there are infinite numbers, therefore I can infer that there is an infinite thing', requires the undisclosed premise that numbers is a thing. Don't you think that we're just going around in circles here?
Quoting Fuzzball Baggins
Again, you're just missing the point. Tim claimed that all the positive numbers were collected in thought, "I have them collected in my mind." So I asked for proof. A description, or definition, is not proof of that. I can say that there is a circle which is square, I have it in my mind, but that does not prove that a square circle exists in my mind. The fact is that we can say things which aren't true. Likewise, mathematics can use axioms which are not true.
We call that a self-evident truth. Some axioms are self-evident truths, other axioms are not self-evident truths.
Actually I believe that numbers ARE things - that our physical reality is based on mathematics, and numbers are at the heart of everything. But we'll put that aside for the moment because you and others may not agree with it. 'I can infer that there are infinite numbers, therefore I can infer that there is an infinite thing' is NOT the argument I was making in my last post, I was arguing that just because we can't empirically observe an infinite thing doesn't mean that it's always unreasonable to assume the existence of an infinite thing. Numbers was just an example. Here's another example: something caused the big bang. In the absence of any evidence indicating that this event could only happen once, the hypothesis that are physical laws which cause big bangs to spontaneously happen at random point in time and space is more simple and relies on fewer assumptions than the hypothesis that something caused only one big bang to happen and then something else stopped that process from reoccurring. Because of this I can infer that a multitude of big bangs have always been and will always be happening, and therefore there is an infinite multiverse.
That's fine, but the issue here is whether "numbers" (note the plural) refers to a thing.
Quoting Fuzzball Baggins
I explained to you why it is always unreasonable to assume the existence of an infinite thing. If you have a reasonable rebuttal then please present it.
Quoting Fuzzball Baggins
Sorry, but I can't see any argument here. You seem to misunderstand "the big bang", representing it as something which occurred at some time, in some place.
Science makes a rule: naturalistic solutions only allowed. No magic. Why do we make an exception to this rule for infinity? If I said I had a ruler longer than any other thing it would be a magic ruler would it not? Infinity is supernatural and it should not be allowed in science.
Infinity meant to be (say) larger than anything else; but 'larger' is a property of quantities and infinity is demonstrably not a quantity; it's a concept and a logically flawed one at that.
Really the burden of proof should be on those who believe in infinity - its an irrational belief and it requires evidence to back up its existence. But there is no such evidence and plenty of evidence that infinity does not exist.