## The measure problem

• 14
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?
• 688
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.

I wonder who would say such a thing. Where did you read this?

What do you guys think? Anything wrong with my reasoning? Anything I've missed?

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.
• 14
Hi SophistiCat, the physicist who said it was Max Tegmark. The first half of his book Our Mathematical Universe is very good, but in the second half he starts extrapolating a bit too much from the measure problem, and seems to think that any set that has the measure problem can't actually exist.

Your definition of the measure problem makes much more sense!
• 5.2k
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.

"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.
• 14
Why do sets have to be bounded? What about the set of all integers - is that not a proper set?
• 5.2k

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.
• 893
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.
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".
• 2.1k
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.

It takes a lot longer if you're counting backwards.
• 6.7k
in the second half he starts extrapolating a bit too much from the measure problem, and seems to think that any set that has the measure problem can't actually exist.

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?
• 14
Yeah, it probably is. Mathematics represents reality so well that I think sometimes people get carried away.

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
• 5.2k
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.
• 688
Be ware that MU's "definitions" are his own. If for some reason you want to know what MU thinks about mathematics (or anything else for that matter), then by all means read what MU has to say about it. If you want to know something about the subject as such, look elsewhere.
• 14
I'm not sure I agree that an infinite thing is ill-defined, or that the multiverse is an object, or that objects have to be bounded. It might be difficult to imagine an infinite thing but I don't think it breaks the laws of physics. Not that I'm saying you are wrong, just that I personally am unconvinced.
• 5.2k

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.
• 14
Okay, I agree with you that an object should be bounded. But I don't consider the multiverse to be an object. In fact, I think that the multiverse is a mathematical construct and that all the objects that we perceive around us are just emergent behaviours of the mathematical laws governing our reality.
• 5.2k

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.
• 14
I think you're taking the English language a little too far, using its structure to decide what can and cannot exist in reality. Think about your reasoning: I chose to say 'the' multiverse, which leads you to think that the multiverse is an object, and therefore bounded, and therefore an infinite multiverse is a self-contradictory concept. So you're deciding that an infinite multiverse doesn't exist simply because I used the word 'the'?
• 5.2k

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.. .
• 1.8k
A "collection" in the sense of a noun implies having been collected, so an infinite collection is impossible

How so impossible? You and I can each think about all the positive numbers. That well-defines a set, collected in thought. If you mean collectible in hand or basement or warehouse somehow, then you needn't resort to concepts of infinity to have uncollectible sets.
• 5.2k

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.
• 1.8k
I haven't collected them all yet in my mind.
I have them collected in my mind. But I do not understand your claim. A large number of grains of sand of sand is certainly collectible. But in terms of your apparently empirical criteria, they're uncountable - I guess you can't have a large pile of sand, yes?

Furthermore, "all the positive numbers" does not qualify as "well-defined"
Says who besides you? Your opportunity to educate.
• 5.2k
I have them collected in my mind.

You have all the positive numbers collected in your mind!? Can you list them then?

A large number of grains of sand of sand is certainly collectible.

My claim is quite simple. A large number of grains of sand is collectible. An infinite number is not.

I guess you can't have a large pile of sand, yes?

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.

If you will listen, I will oblige.
• 1.8k
You have all the positive numbers collected in your mind!? Can you list them then?
Which ones would you like me to list?

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.

— tim wood

If you will listen, I will oblige.
I know the difference - I'm by no means sure you do. And your "if you listen" in response to my question to you as to who besides you says so - your answer in response to that question - is simply an example of what I've experienced as your toxic style of discussion. You made a claim,
My claim is quite simple. A large number of grains of sand is collectible. An infinite number is not.

I ask you who besides you says so. And you do not answer. Answer or Hitchen's razor awaits you.
• 14
Your claims seem a little arbitrary. Especially your claim that the multiverse being seen on the one hand as a multitude and on the other as a single object makes it self-contradictory. A bunch of bananas is both a single object and a multitude of bananas.

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.
• 5.2k
Which ones would you like me to list?

All of them of course. I want you to prove to me what you claimed. "I have them collected in my mind."

I know the difference - I'm by no means sure you do. And your "if you listen" in response to my question to you as to who besides you says so - your answer in response to that question - is simply an example of what I've experienced as your toxic style of discussion. You made a claim,

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?

I ask you who besides you says so. And you do not answer. Answer or Hitchen's razor awaits you.

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

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..

Your claims seem a little arbitrary. Especially your claim that the multiverse being seen on the one hand as a multitude and on the other as a single object makes it self-contradictory. A bunch of bananas is both a single object and a multitude of bananas.

Yes, I went through this, I reposted it above in case you didn't read it through.

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 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.

I don't think the concept of a set having to be 'collected' quite applies to what can and cannot exist in reality.

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.

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.

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.
• 1.8k
All of them of course. I want you to prove to me what you claimed. "I have them collected in my mind."

all positive numbers of the form 2n, n being any integer.

I ask you who besides you says so. And you do not answer. Answer or Hitchen's razor awaits you.
— 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

The razor, then. "What is averred without evidence can be dismissed without evidence." Btw, a request for a respectable source is perfectly reasonable.
• 5.2k
all positive numbers of the form 2n, n being any integer.

I asked for the list, not a description of it.
The razor, then. "What is averred without evidence can be dismissed without evidence." Btw, a request for a respectable source is perfectly reasonable.

Did you read my proof? I do believe that a logical proof qualifies as "evidence".
• 14
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.

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.

I asked for the list, not a description of it.
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.
• 122
It is the constructivist/finitistic approach to maths MO is supporting. But because there is not universal concensus in mathematical and philosophical communities alike, in maths we have axioms for this stuff, like the axiom of infinity or the axiom of choice.
• 5.2k
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.

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?

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.

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.
• 122
If mathematicians were like philosophers, trying to fully resolve an issue before advancing, then there would be no progress in mathematics. So, wherever there is doubt, they create an axiom and tell their colleagues to invoke it in their work, if and when they like. In this way, they avoid the search for truth, so that to focus solely on mathematical work, numbers and proof: they just need to say which axioms were used, and leave the 'what really is the case' to philosophers. Mathematical axioms, in the way they are used by mathematicians, do not have anything to do with 'truth' in the real sense, they are just there to help them in their games. After all, if something was obviously true and accepted by everyone, we wouldn't have an axiom for it, would we? :)
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