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Has anyone read this article? Set theory

This accomplished mathematician gives his opinion on why he doesn't think infinite sets exist, and claims that axioms are nonsense. I don't disagree with his arguments, but with my limited knowledge of axiomatic set theory and logic, I am unable to take sides. Would someone be so kind as to enlighten me on why his arguments are/aren't correct? Thanks

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I really like how he emphasizes on page 7 that the "existence of an infinite set" is basically a wrong postulate, and not in any way related to the real mathematics he does, namely representations of Lie Groups...I really wonder how much of his work is about representations of FINITE Lie Groups, and if it is, if anyone cares... –  N. S. Apr 9 '13 at 19:44
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"These grammatical constructions do not create concepts, except perhaps in a literary or poetic sense" As if a concept should correspond to something that exists. –  leonbloy Apr 9 '13 at 19:51
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A paper about any mathematical subject without any list of sources ? I wouldn't, and I won't, waste my time with it. –  DonAntonio Apr 9 '13 at 20:15
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In my opinion an infinite set doesn't need to "exist" any more than the number 3 needs to "exist", they are all imaginary concepts and there is nothing wrong with that for the purposes of doing useful/interesting mathematics –  wim Apr 10 '13 at 3:19
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@N.S. Dear N.S., The author is not going to claim that the Lie group $G_2$ (as one example, which I am bringing it up because it appears in the text) is finite. His objection is rather to a certain axiomatic treatment of mathematics, including the axiomatic treatment of infinite sets. Regards, –  Matt E Apr 18 '13 at 14:59

8 Answers 8

I stopped reading the article at this point:

(6. Axiom of Infinity: There exists an infinite set.

....

And Axiom 6: There is an infinite set!? How in heavens did this one sneak in here? One of the whole points of Russell’s critique is that one must be extremely careful about what the words ‘infinite set’ denote. One might as well declare that: There is an all-seeing Leprechaun! or There is an unstoppable mouse!

Quite frankly, he is using an layperson's interpretation of the axiom and then critiquing this interpretation for being imprecise, when the entire point having these interpretations is to give the gist without being too technical. The common form of the Axiom of Infinity used today is the following (put into words instead of logical symbols):

There is a set $X$ having the property that $\varnothing$ is an element of $X$, and whenever $x$ is an element of $X$, then $x \cup \{ x \}$ is also an element of $X$.

This is a very precise formulation which one can show yields a set which is not finite (hence infinite):

  • As $\varnothing$ is in $X$, then $\varnothing \cup \{ \varnothing \} = \{ \varnothing \}$ is an element of $X$.
  • As $\{ \varnothing \}$ is in $X$, then $\{ \varnothing \} \cup \{ \{ \varnothing \} \}= \{ \varnothing , \{ \varnothing \} \}$ is in $X$.
  • As $\{ \varnothing , \{ \varnothing \} \}$ is in $X$, then $\{ \varnothing , \{ \varnothing \} \} \cup \{ \{ \varnothing , \{ \varnothing \} \} \} = \{ \varnothing , \{ \varnothing \} , \{ \varnothing , \{ \varnothing \} \} \}$ is in $X$.
  • ...

You see that these elements of $X$ get larger and larger without (finite) bound, and so it stands to reason that such an $X$ must be infinite.

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The author doesn't seem to argue that mathematicians can construct numbers like this, but takes issue that this construction requires more materials than exist in the universe, even on an atomic level. I don't need infinite sets to do that, just let $x = $ the number of atoms in the universe $+ 1$. –  Joshua Shane Liberman Apr 9 '13 at 20:39
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@Igor: I'll be generous and dignify your insult of me with a response; don't expect another reply from me. One does not need to read every word of a screed to determine that it is without merit. Just as I don't need to listen to Justin Bieber's entire œuvre to determine that he's a horrible songwriter. The author of the article in question was clearly using a caricature treatment of axiomatic set theory to attack axiomatic set theory. He wasn't willing to attack the practice as it is done, but resorted to what is a straw man argument to conclude that set theory is a vacuous enterprise. –  Arthur Fischer Apr 18 '13 at 17:08
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@zyx: After stating the "Axioms of Zermelo-Fraenkel" Wildberger goes on to single out his statement of the Axiom of Infinity as being imprecise (this is his appeal to Russell which I have quoted above). Perhaps I should proclaim algebraic topology to be stupid because homology groups supposedly "count the number of holes" in a space, but how can there be a hole in something that has no physical existence?! and, besides, a "hole" is a pretty imprecise concept to make mathematical reference to! [cont...] –  Arthur Fischer Apr 21 '13 at 5:35
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@Thomas: I may be a professional barista, but if I will suddenly claim tea leaves to make excellent coffee - certainly this claim will be met with ridicule. Wildberger may be a professional mathematician, but claiming that axioms are imprecise and that mathematical existence must be related to physical existence and therefore "existence of an infinite set" is equivalent to "existence of leprechauns" will be met with the same ridicule as the coffee-brewing tea leaves claim. –  Asaf Karagila Apr 21 '13 at 16:53
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@Pete: Wildberger makes demonstrably false statements about the enterprise of set theory (not even as a foundation). As most of these falsehoods are about matters that are covered in any standard graduate-level text in the area, I can only assume that he does so intentionally. My answer, if nothing else, points out exactly one of his untruths, and provides a more honest account of the actual state of affairs within set theory. –  Arthur Fischer Apr 22 '13 at 4:13

Mathematics is a mind game. It doesn't have to do with the physical world. Much like there is no number which is $\frac12$, and there is no number which is $2^{2^{10000}}$, and there is certainly no $\Bbb R^{666}$.

But mathematics is a mind game, where we pretend that for the sake of argument certain objects exists and the axioms are used to describe their properties. In our mind game we agree on certain inference rules, and we try to deduce more properties of these objects using our inference rules and our initial assumptions which we called axioms.

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But many times we get extra interesting properties about the objects which exist in reality by studing those objects which don't exists. I wonder often how many physicist/chemists realize that when they find the best line fitting a data of 666 points (or best polynomial) they are actually doing a projection in a very artificial $\mathbb R^{666}$... Probably not even all statisticians realize the abstract algebra behind regression analysis... –  N. S. Apr 9 '13 at 19:55
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Blasphemy, $\frac{1}{2}$ exists! Call me a platonist. –  Jeppe Stig Nielsen Apr 9 '13 at 22:54
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@Jeppe: It may or may not exist as an ideal mathematical object; but it certainly does not exist in our physical reality. We can never tell with accuracy whether or not you cut the cake exactly in half. –  Asaf Karagila Apr 9 '13 at 22:55
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What is 1 cake? What are 2 cakes? You cannot say either 2 cakes unless they are physically identical. Nor 1 cake, as you call it cake as an approximation to your personal ideal cake. So counting cakes is as 'real' as counting fractions of cakes, just approximations. No portal reference though... –  Francisco Presencia Apr 10 '13 at 10:05
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@Igor: Good thing that I know more than set theory. Is there a point to your thinly veiled insults? –  Asaf Karagila Apr 18 '13 at 17:09

If we take it graciously, the paper is intended to be a tongue-in-cheek essay. There are numerous claims that, if taken at face value, are extremely difficult to defend. Some examples:

  • On page 6, the author asks, "Do modern texts on set theory bend over backwards to say precisely what is and what is not an infinite set?". Of course they do, it is a simple definition in every text: a set is finite if it can be put in bijection with a natural number, and is infinite otherwise.

  • At the bottom of page 7, the author claims that the choice of postulates does not arise in his field, which is possible. But, for example, the Whitehead problem in group theory is known to be independent of ZFC, so that proving or disproving it requires more axioms than are generally accepted in mathematics. The Whitehead problem arose first in the context of group theory - not foundations - and only later was proved independent of ZFC.

  • Near the top of page 9, the author (intentionally?) confuses the property of a mathematical statement being true or false with our ability to prove it is true or false.

  • The existence of uncomputable reals, which the author discusses on page 11, is well known by results in computability theory to be necessary for statements such as "every bounded increasing sequence of rational numbers converges" to be true - even when we require the sequences themselves to be computable. In particular, the claim on page 12 that the computable real numbers are complete is not constructively provable, as it is disprovable in ZFC.

There are well-written and cogent explanations of different philosophies of mathematics, such as finitism and intuitionism, which the author describes only obliquely. This paper might be better as something to read after you are familiar with those philosophies, so that you get the jokes that the author is making.

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Carl, if you look the author's page up and check his youtube videos, you may start to think this was not a tongue-in-cheek essay. –  Asaf Karagila Apr 10 '13 at 8:17
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Wildberger could presumably have made those arguments if he wanted to, but he did not. Instead, he claimed that set theory books don't define "infinite set" and that the computable real numbers are complete, with no further comment. This is why the essay should be treated as tongue in cheek. The plain appearance is that Wildberger went out of his way to avoid making any point clearly. Those who already understand finitism can read his essay in that light, at which point they will understand the jokes. But in evaluating the essay I look at what is literally said, not at what could have been. –  Carl Mummert Apr 21 '13 at 12:27
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But Carl, did you "think clearly about the subject for a few days"? –  Quinn Culver Apr 21 '13 at 14:48
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As hinted above, it is an interesting and controversial question whether the Whitehead problem (or any other independent statement) is an example of set theory "appearing in practice" in mathematics. Harvey Friedman, who has devoted much of his career to trying to demonstrate the relevance of set theory, has often articulated (to set theorists) what I believe to be the overwhelming consensus among mathematicians, which is that none of the famous examples are convincing. Citing Whitehead as a self-evident refutation of Wildberger's argument is restating your assumptions as a conclusion. –  zyx Apr 22 '13 at 5:18
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"Wildberger could presumably have made those arguments if he wanted to, but he did not." - a reading of the whole essay, and not only the one sentence in isolation, shows that Wildberger did make those arguments and others about lack of definition of infinite sets in textbooks. A search for the string SPECIF might help, it finds some of the passages in which he says that the standard presentations do not discuss how (or what it means) to specify a set, how to compute with sets as a data type, and other things he takes as a requirement for being well-defined. –  zyx Apr 22 '13 at 5:25

Recently I was reminded of the following gem of an aphorism: "The most annoying thing about an incorrect proof of a correct theorem is that it is very difficult to give a counterexample." It is certainly true that infinite sets do not necessarily "exist" in most uses of the word other than the mathematical one. It is not, however, true that accepting set theory as foundations foces one to believe in such existence in any sense beyond the mathematical. Furthermore, the existence of "infinite sets" is no more contentious than the existence of "finite sets", in my opinion.

I am not a logician (yet), but the picture in my head is as follows. Mathematicians at the end of the day deal with certain systems of rules on how to manipulate symbols on a piece of paper. Such systems are composed of two parts: a language which consists of the rules that say which strings of symbols are valid (i.e. are sentences or formulas), and the transformation (inference) rules which say how to transform certain (collections of) sentences and formulas into other sentences and formulas. Formally, this is all we do as mathematicians: we come up with languages and inference rules, pick some sentences or formulas in the language that seem interesting and then we go on and try to obtain certain other interesting sentences and formulas (you get at mathematical logic if you ask yourself whether you can obtain certain interesting sentences and formulas at all).

From this formal perspective, the relation to the real world is that occasionally a more scientifically inclined mathematician (or more commonly, a mathematically inclined scientist) would use or create a language in which to describe the things in the world he or she observe, and the relationships between the things he or she hypothesizes. Then, they apply whatever set of inference rules they use (usually basic logic) to their initial conditions and laws, and thus arrive at a new sentence or formula, which they label a prediction about the real world. Then they go and see if the prediction is true. If yes, they say that the formal system they came up with describes the real world, which is never true: the formal system only models the real world, i.e. functions to predict rather than describe things about the real world.

Things like the natural numbers, basic rules of arithmetic, or the finite set theory Wildberger prefers, are simply formal systems which have always given correct (when testable) predictions about the real world. What people actually mean when they say that 1+1=2 is a self-evident statement is that in almost all contexts, the statement "one thing and another thing give us two things" has proven true. But this is of course tautological, since the idea of 1+1=2, i.e. the language of arithmetic and its basic properties are considered interesting exactly because of the fact that they model so many phenomena that we observe extremely well. It is absurd, however, to claim that the number 1 "exists" in any sense other than the mathematical, which is that there is a certain practice we engage in, which has always accurately predicted certain situations in the real world (i.e. if I take one apple, and another apple, I now have two apples).

What about "infinite sets" and ZF(C)? What aspect of reality do they model? Well, ZFC models the very real practice of doing mathematics in the above sense. It gives symbols and rules with which to express strings of symbols (the set of all finite strings), the language (the subset of all valid strings), and inference rules (functions on sets of valid formulas). We even have for certain kinds of formal systems Godel's completeness theorem which states that if a theory is consistent (its set of theorems/formulas derived from axioms does not include "P and not P" for any P), then ZFC can model that theory in a standard way. Assuming that ZFC is consistent, the implication goes the other way as well, i.e. ZFC models only consistent theories if it is itself consistent.

For this reason, almost all mathematicians of an object (in a theory) have agreed to understand mathematical existence to mean that any way in which ZFC models that theory, the object is represented in the model. This is why defining, say, the rational numbers or the real numbers as equivalence classes of whatever is not as insane as it might seem: it is actually showing that the rationals and the reals exist in the sense that their theories pass the test of consistency relative to ZFC. This is important if we want to have some standard by which to be confident that these formal systems (of the rational numbers, of the real numbers) are free of contradictions, i.e. would not simultaneously predict "P and not P". Otherwise, because of how our inference rules are set-up, their theorems are trivial (every formula is a theorem), and thus their utility as models of the real world is null.

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Woodin has a nice paper that makes some of these points as well: The Tower of Hano. In Truth in mathematics (Mussomeli, 1995), 329–351, Oxford Univ. Press, New York, 1998. He sometimes describes himself as a "conditional platonist". Let me quote from the paper: "By a routine Gödel sentence construction we shall produce a formula $\Omega(x_1)$ in the language of set theory which implicitly defines a property for finite sequences." "If there exists a sequence of length $n$ with this property, then $Exp_{2011}(n)$ does not exist". (Cont.) –  Andres Caicedo Apr 24 '13 at 23:22
    
Here, $Exp_1(n)=2^n$ and $Exp_{k+1}(n)=2^{Exp_k(n)}$ for all $k$. "However this sentence has the feature that, if arbitrarily large sets can exist, then, for each suitable $n$, there is no proof of length less than $n$ that no such sequence of length $n$ can have this property." "We shall argue that there are limitations to the extent our experience in mathematics to date refutes the existence of such sequences." (Cont.) –  Andres Caicedo Apr 24 '13 at 23:26
    
"In fact we shall argue that a consistent philosophical view must in effect acknowledge the possibility that the sequences of length $10^{24}$ could exist, just as those who study large cardinals must admit the possibility that the notions are not consistent." –  Andres Caicedo Apr 24 '13 at 23:28

This joker is just playing to the gallery. "Maths $-$ who needs it? Ha ha ha!"

To take a specific example, on page 10 he ridicules the standard definition of a rational number as an equivalence class of ordered pairs of integers. As I hope you know, this is perfectly standard, and no "accomplished mathematician" should have any problem with it at all.

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Yes, he's trying to impose a sort of realism on the construction of numbers that no one ever claimed existed. To say that "$1 = \{ 0 \}$ is ridiculous" is to beg the question of what $1$ is at all. –  Andrew Salmon Apr 9 '13 at 21:39
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@Kaz: it depends on which $2$ you mean, but in $\mathbb{Z}$ constructed as the extension of the cancelative semigroup $\mathbb{N}$ to a group, $2$ is indeed the equivalence class of all pairs of naturals $(a,b)$ such that $a = b+2$. This is the standard construction of $\mathbb{Z}$ from $\mathbb{N}$. –  Carl Mummert Apr 10 '13 at 2:06
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I'm not denying that there is such a class. It's just not what comes to mind as 2. –  Kaz Apr 10 '13 at 2:11
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@Kaz: That's how I like to define $\mathbb Q$. If it makes me a moron in your eyes, I can live with that. –  TonyK Apr 10 '13 at 15:56
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@Kaz: You said "pretty moronic". Which was a pretty moronic thing to say, in my opinion. Are you retreating from that stance now? It's difficult to tell from your latest comment. –  TonyK Apr 10 '13 at 19:18

I totally agree with Asaf Karagila here. Mathematics is a mind game, but Professor Wildberger assumes that mathematics should relate to the real world:

Elementary mathematics needs to be understood in the right way, and the entire subject needs to be rebuilt so that it makes complete sense right from the beginning, without any use of dubious philosophical assumptions about infinite sets or procedures. Show me one fact about the real world (i.e. applied maths, physics, chemistry, biology, economics etc.) that truly requires mathematics involving ‘infinite sets’ ! Mathematics was always really about, and always will be about, finite collections, patterns and algorithms. All those theories, arguments and daydreams involving ‘infinite sets’ need to be recast into a precise finite framework or relegated to philosophy.

But as far as I know, mathematics is not, and should not be about reality. I would like to quote Albert Einstein here

"As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality."

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Ok, so we should not do physics. Physicists should quit their jobs. Because mathematics should not describe the real world, because it us humans who decide how the universe works. –  Igor Ultra Apr 18 '13 at 16:50
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@Igor Ultra: ??? Mathematics can be used to describe the physical world. That doesn't mean it has to be limited to such. –  mike4ty4 Apr 21 '13 at 7:40

Wildberger makes the point

clear definitions are necessary

and

People use the term ‘Axiom’ when often they really mean definition. Thus the ‘axioms’ of group theory are in fact just definitions. We say exactly what we mean by a group, that’s all. There are no assumptions anywhere.

Oddly, he seems to miss the obvious application of this idea in regards to universes of sets. I suppose it shouldn't be that surprising, due to all the effort he spends setting up a caricature of modern set theory so that he could proceed to mock the straw-man.

Nor does he seem to take the time to apply this principle to whatever philosophy he is suggesting. Read through his article; is there anything in there you could have predicted that he would say before he actually said it? A fair bit, probably, but very few (if any) of those predictions will be because due to the correct use of logic as applied to clear definitions.

Let's consider one particular point. He talks about the natural numbers. This is clearly a concept he accepts in some fashion, even if he dislikes the modern treatment.

(aside: much of the controversy would go away if he would simply do something like define "Wildberger sets" and "the Wildberger numbers" and develop their theory. Instead, he leaves them undefined, calls them "sets" and "natural numbers", and then tries to browbeat everybody to stop using the usual meaning for "set" and "natural number")

He even admits implicitly that "natural numbers" are some sort of object that can be reasoned with -- e.g. one can make statements such as "$f(n) = n^2 + 1$ defines a function that inputs natural numbers and outputs natural numbers" -- so in his treatment, "natural number" is clearly not simply some metaconcept.

Now recall Cantor's approach to set theory. One of the most basic ideas is comprehension: if $P$ is a proposition, then

$$ \{ x \mid P(x) \} $$

is a set. One may have some a priori ideas about sets as some sort of "collection" (but then, what is a "collection"?), but in Cantor's set theory, the notion of set equates to the notion of predicate. And I don't believe the underlying idea was particularly new -- philosophers had been struggling with such things for a long time -- the novel feature is that it was cleanly and precisely stated and one could reason rigorously with it, and that Cantor was willing to fully explore what could be done with it.

So if Wildberger is willing to grant that we can reason about "natural numbers" -- and even go so far as allowing them to be some sort of object -- then the natural numbers are a Cantor set. (of course, it's apparently not a Wildberger set, but I'm not talking about Wildberger sets, we're talking about Cantor sets)

By any reasonable definition of the word "finite", the natural numbers should not be a finite Cantor set. Thus, infinite Cantor sets exist, even in Wildberger's way of thinking. Assuming, of course, that Wildberger's way of thinking can even give a reasonable definition of "finite".

Well, I should be careful; Wildberger has not used clear definitions. If he is talking about the natural numbers as he claims to be, then I can conclude that even in Wildberger's mathematics, infinite Cantor sets exist. However, if he is talking about Wildberger numbers instead, I honestly don't know if they form an infinite Cantor set. e.g. I'm not really sure if there is a largest Wildberger number or not.

Now, mind you, people like to study other universes of sets. The universe of finite sets, for example. This has rather severe limitations, and it's not generally adequate to study mathematics -- e.g. the notion of a "function whose inputs are natural numbers and whose outputs are natural numbers" cannot be encoded in such a universe. We can recover a fragment of such a notion by defining things like Turing machines.

A lot of controversy would have been avoided if Wildberger simply said "I want to study constructive analysis rather than real analysis" and maybe even presented reasons why teaching students constructive analysis would be better than teaching calculus and real analysis. But then, I suppose that would have had the drawbacks of being less provocative, and actually exposing his rationale to be critiqued (if he even has one!).

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-1 On meta, you wrote things like "poor fit for this site ... to argue specifically with a well-known crankish essay, a task that is generally considered fruitless" and "Furthermore, giving serious responses to it is arguably counter-productive to begin with, by implying the original essay actually merits a direct response". I think your answer contains "debate, arguments and extended discussion", all these things why you correctly explained that such a question should be closed. Why on earth then do you write such an answer??? –  Thomas Klimpel Apr 21 '13 at 15:04
    
@Thomas: As I said there, I was trying to represent the viewpoint the questioner asked about, but wasn't sure if I actually had that view (or the contrary one). At the time, it had appeared that nobody who was involved in the closing was paying attention, and so I wanted the person to get a timely answer. While I am happy to argue the case for closing this and the other question, I haven't convinced myself, and it appears the current majority opinion is that this (and the other) question should be open. –  Hurkyl Apr 21 '13 at 15:51
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Do you really think MSE is a place for "debate, arguments and extended discussion"? The "majority opinion" is one thing, but the stack-exchange framework explicitly tries to minimize such things, probably for good reasons. Anyway, I was just explaining why I downvoted. –  Thomas Klimpel Apr 21 '13 at 16:18

If no infinite sets exist, then the natural numbers are not infinite. Consequently, the natural numbers are finite. It follows that there exists a last natural number, call it L. Since it's the last natural number, then S(L) the successor of L or (L+1) does not exist. In other words, at some point, no matter how "high" you count, you will NOT have the ability to count to another number and no one, nor anything in the universe will. Nor will you have the ability to find a natural number greater than L ever. So do you believe that there exists a last natural number? Does the set of natural numbers have a least upper bound? Does the set of natural numbers have a greatest member?

If not, then you reject the notion of infinite sets not existing.

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It might be the case that the natural numbers do not form a set, but every number still have a successor. In this case the natural numbers form a class rather than a set. –  Asaf Karagila Apr 10 '13 at 8:16
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@AsafKaragila I hadn't considered that. But is the question "do infinite sets exist?" or "do infinite collections exist?"? –  Doug Spoonwood Apr 10 '13 at 11:59
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Of course that if you are going for "nothing infinite exists" then either there are finitely many natural numbers, or you cannot really talk about the collection of natural numbers in a meaningful way. But if you consider the theory of $\sf ZF$ without the axiom of infinity, and replace it with its negation, then you have a theory which is strong enough to develop a lot of basic mathematics (it is bi-interpretable with first-order Peano), and yet there are no infinite sets there. –  Asaf Karagila Apr 10 '13 at 12:01

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