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I am just going to give an example of what I mean using Skolem's Paradox. I don't want to get into Skolem;s Paradox itself or its "resolution."

Skolem's showed that in first-order formulations of ZFC, whether some set A is countable depends on what is in the model. For example, take a model M of ZFC (assuming there is one). Let M satisfy the statement "S | S is countable." This means there exists a bijection from S to {naturals} in M (i.e. there is a particular set of ordered pairs in M). Now, remove all and only those bijections from M and call this new model M'. Assume M' is still a model of ZFC. Is S still countable? No. Countability is relative to the elements of M's domain.

How how does a mathematical object, e.g. a countable set, relate to the real world? Is it in the real world? Is is instantiated in the real world? What is meant when mathematicians say (e.g. as Cantor's Theorem says) that uncountable sets exist?

That is, what is the relation of math-objects to the real world?

Asaf Karagila
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pichael
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  • It seems that your enquiry (I said enquiry not question wording) is about existence of infinity in the real world. –  May 24 '12 at 04:10
  • youre probably familiar with the following aphorism of Reubem Hersh and Philip Davis: "the typical working mathematician is a Platonist on weekdays and a formalist on Sundays." – Tim kinsella May 24 '12 at 04:17
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    I think before you worry about infinities you should decide about 2. How does the number 2 relate to the real world? Is it in the real world? What is meant when mathematicians say 2 exists? This is already a hard problem about which reasonable people may differ, so perhaps it's hopeless to ask such questions about infinity. Just let the mathematicians get on with the job. – Gerry Myerson May 24 '12 at 04:29
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    This isn’t really a mathematical question; it’s a philosophical question. – Brian M. Scott May 24 '12 at 04:41
  • Your example is not about mathematical objects, it is about provable statements of a first-order theory and its intended model of theory. You don't have to go to real world, different mathematical theories might prove different statements about the same kind of objects. – Kaveh May 24 '12 at 04:53
  • (J.D.) Yes, but simple answers are telling: e.g. "There is a Number Realm in the real world where they exist" or as Cantor believed "All consistent mathematical results have real-world instantiations." (Tim) Haha...haven't heard that (Gerry) That does help, but some people may be very averse to actual infinitudes (e.g. how can a proper subset of something in the real world be the "same size" as that something?). Finitudes don't have quite those problems. (Brian) Hence, "a little philosophy" in the title. – pichael May 24 '12 at 04:54
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    This doesn't seem to be a real question in mathematics (since it doesn't have an answer and will lead to discussions). Might be suitable for [philosophy.se] though. There are various views towards mathematical objects and you can read about them on wikipedia or on SEP. – Kaveh May 24 '12 at 04:57
  • (Kaveh) Thanks for the links. I'm just curious to know what mathematicians have to say as opposed to philosophers. You distinguish two questions: (1) about the existence of math-objects and (2) how would you parse this other one? – pichael May 24 '12 at 05:02
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    It's a good question for philosophy of mathematics but not mathematics of philosophy. – Sniper Clown May 24 '12 at 06:08
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    In the real world, there is nothing of size $10^{1000000}$. Shall we be averse to sufficiently large finitudes, too? We are doing mathematics, not physics. – Gerry Myerson May 24 '12 at 06:58
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    @Gerry, nothing that we know of yet! The universe may be larger than we'd think. – Asaf Karagila May 24 '12 at 07:27
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    In case anyone wants to see the philosophical answer, please check the link for the crosspost. – Sniper Clown May 24 '12 at 07:29
  • You might want to take a look at http://en.wikipedia.org/wiki/Ultrafinitism –  May 24 '12 at 18:12
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    You should be wary of cross-posting a question over two different sites. When you do, you should at least explicitly link the cross-posted questions. It may give the readers some idea what you were already told. – Asaf Karagila May 24 '12 at 18:32
  • @AsafKaragila: I don't know how, but I think Mahmud did it. Thanks. – pichael May 24 '12 at 21:14

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Once mathematics began dealing properly with infinite objects it was no longer about the reality, but rather about abstract ideas.

Our "natural" intuitions (i.e. those we have from a pre-mathematical education time) are often very wrong about the infinite, to list some examples:

  1. The rationals are countable;
  2. The real numbers are uncountable;
  3. There are uncountably many ways (up to isomorphism) to well-order a countable set;
  4. Hilbert's Grand Hotel.

The list itself is infinite. It gets even larger if you wish to consider it in early 1900's eyes where the axiom of choice were still researched thoroughly.

However mathematics no longer deals solely with describing the real world, it deals with deductions from assumptions. Once accepting that it seems that a lot of the problems with infinities dissipate, as they follow from definition.

There comes a new problem with foundations of mathematics, the independence of claims, in particular the set theoretical ones. How can a set be countable in one model and uncountable in another? Let me use, once again, my usual analogies from field theory.

Suppose $F$ is a field (of characteristics $0$ if you prefer). What is the size of $\{x\in F\mid\exists n\in\mathbb N^+: x^n=1\}$

In the rational numbers the answer is $2$, in the rational numbers adjoined by a complex unit root of order $3$ the answer is $4$; in the algebraic closure of the rationals the answer is countably infinite. In the complex numbers you don't increase the size of this set, but you find a lot more transcendental elements on the unit circle which you can't even describe so nicely.

Note that field theory cannot express in a single formula the notion of being a unit root; but it can express the notion of being a unit root of order, say, $72$ or less. This should give us enough examples ($\mathbb Q$ still has only two; different extensions have four, five, etc.) of a specific definable set which changes in size between the models.

Why does no one complain when they are told that "in this field there are more unit roots than in that field"? My guess is that we are being educated to accept that "all numbers live in $\mathbb C$", so some are rationals, some are algebraic, etc. and thus different fields would have different amount of unit roots.

But set theory deals with sets, is this a surprise that different models of set theory would have different sets and if we pass from one model to a smaller model we may lose some of the information? No. If you study some axiomatic set theory you find out it's not surprising at all. It's what you'd expect, much like the way you may lose some unit roots in passing to a smaller field.

Now you are probably thinking, "he must be cheating me somewhere, because I feel completely fine with the unit root example, but it's impossible for sets to be countable here and uncountable there!". Well, sticking to first-order logic, you have to ask yourself what is the language that you use to describe the axioms and the model. In field theory you essentially describe the operations and the polynomials which have a solution in the field. In set theory you only have $\in$, but you describe a more complicated creature.

Is it a surprise that we have computers and an amoeba have only one cell? No, we are a far more complicated creature. Set theory is far more complicated, as a theory, than field theory. It should not be a surprising understanding that some of the things it can say about objects in the universe are more complicated. Since those are complicated it often seems that there should be some "canonical answer", but so far there is none. Whether it is good or bad, I can't tell. I hope there won't be a canonical answer because I enjoy the plethora of models, much like (I suppose) people studying measure theory enjoy the plethora of measures and spaces attached to those.

I will finish with one last point, Skolem tried to show in his paradox not that there is an inherent problem with set theory describing the world but rather that there is an inherent problem with using first-order logic to describe set theory. As it happens to be, he actually made clear the distinction between "internal" and "external" points of view in logic.

Asaf Karagila
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  • I wrote a somewhat similar answer here. – Asaf Karagila May 24 '12 at 06:33
  • "Once mathematics began dealing properly with infinite objects it was no longer about the reality, but rather about abstract ideas." I think this happened long before mathematics began dealing with infinite objects. Where is the "reality" in the square root of minus one (no pun intended)? I'm not sure there's any reality in one-half; half a piece of chalk is still a piece of chalk. – Gerry Myerson May 24 '12 at 06:54
  • @Gerry: I was told that $\sqrt{-1}$ is actually useful for describing things in electronics. Of course numbers are never a good approximation for reality, but it's an abstraction of it. Indeed it is not exactly the same, but the idea was to describe something real in an abstract way. Much like a bag with five oranges has the same number of fruits as a bag with two oranges and three apples. When we formalized transfinite numbers we no longer described anything which is an abstraction of the real world. We went "to infinity, and beyond!". – Asaf Karagila May 24 '12 at 07:03
  • @AsafKaragila: your comment here, is really the kind of answer I'm after. First you say (roughly) "numbers are an abstraction of reality." Also, you say, "But when we get into the Buzz Lightyear Realm (the transfinites) we lose the tie between reality and our mathematics being an abstraction of it. Why do we lose the abstraction-relation to the real world in the finite-to-transfinite transition? – pichael May 24 '12 at 08:02
  • @pichael: This is exactly what Gerry said. At some point you have way more objects than observable objects and you no longer describe anything. – Asaf Karagila May 24 '12 at 08:19
  • @AsafKaragila: OK. I'm trying to see where you're coming from. (I think I agree.) Is this last comment connected to there being only countably many names? So even if there were real-world infinitudes, there would be a theoretical limit saying we could only describe e.g. a countable number of reals. Or is what you're saying: the number of observable objects can be countable at most, and so when describing "things" of higher cardinalities, we aren't describing anything observable? Or perhaps you're coming from elsewhere? – pichael May 24 '12 at 11:06
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    @pichael: This has nothing to do about names. We are finite beings, we can observe only finitely many things in our lives. We cannot observe them accurately, too. We cannot really measure "one half" of something. Infinite, therefore, is completely beyond our physical reach and can only remain as an abstract idea. – Asaf Karagila May 24 '12 at 12:16
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    Yes, $\sqrt{-1}$ has applications. So do uncountable infinities; every time you do even one-variable calculus in physics, you're dealing with the real line as a model for physical reality, and (I don't have to tell you) the real line is uncountable. All numbers, finite, infinite, real, complex, positive, negative, whole, fractional, are equally abstractions, models, mathematical objects, not physical ones. pichael will not understand how infinity relates to the real world until he/she comes to grips with how two relates to the real world. – Gerry Myerson May 24 '12 at 12:39
  • @Gerry, while I strongly disagree with the finitistic approach to mathematics, I do accept it for physics. Using the real numbers is a good approximation of the reality which (as we model it now) is somewhat discrete (correct me if I'm wrong, there is some very very small number that below it we don't understand how gravity works). I also know that recently physics has somewhat of a motion towards $p$-adic fields (so I hear from my office-mate which has to do with these things). How on earth are you going to use $p$-adics to explain reality? It's as non-Euclidean as it gets! :-) – Asaf Karagila May 24 '12 at 12:44
  • Are you sure "you find a lot more transcendental unit roots" is a correct statement? Isn't any solution of $x^n=1$ an algebraic number by definition? – Thomas Klimpel May 25 '12 at 06:58
  • Are you sure your example can be expressed in first-order field theory (without any axioms related to sets or natural numbers)? Don't you need to add at least some axioms to define the meaning of $n\in\mathbb N$ for your example? I admit that it's probably also possible to find an example in first-order field theory that equally illustrates your point. However, I fear that your current example creates (or cements) more misconceptions than it clears up. – Thomas Klimpel May 25 '12 at 07:07
  • @Thomas: For the first comment, you are correct. My meaning was that we have sequences of unit roots which converge to other elements on the unit circle. I will clarify this in the post. The second comment is indeed true. You cannot define all unit roots with one formula in first-order. I could very well ask how many unit roots there are of order $72$, and the idea would be similar. Recall, however, that ZFC is a foundational theory and field theory is not. This is because ZFC allows us to express a lot more than what field theory allows. The example is not mathematically accurate (cont.) – Asaf Karagila May 25 '12 at 07:15
  • (cont.) but rather to illustrate that certain sets which are "nice" can change between the different fields. As I said, set theory and field theory are very different creatures (note that ACF is complete, after a choice of characteristics) while ZFC is far from complete (and any completion is not effectively recursive so it's hard to describe it in full). This is an example to make people think of what we are able to write or prove from a given theory. I will make some changes in this analogy to clarify my point better. If you have a better idea, let me know. – Asaf Karagila May 25 '12 at 07:19