Infinite sets don't exist!?

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.

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.

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.

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.

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.