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When one writes down the axioms of ZFC, or any other axiomatic theory for that matter, and making statements like "let x, y ..." doesn't this assume an understanding (and thus existence) of natural numbers implicitly? (Q1)

How is the reader to interpret statements such as existence of separate symbols, nevermind sets, without an intuitive notion of numbers?

Bourbaki talks about this within the framework of metamathematics, but then declares that the reader can read words, differentiate between different words etc. and that to assume otherwise is idiotic.

Is there an introduction to these circle of ideas & debates somewhere you'd recommend? (Q2)

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    $\begingroup$ This kind of issue is raised by Poincare in "Science and Method", and (if I both remember and understand correctly) forms part of his criticism of impredicative definitions (such as the definition of the number two as the equivalence class of all two elements sets under the equivalence relation of bijection). $\endgroup$
    – Emerton
    Commented Nov 26, 2010 at 6:05
  • $\begingroup$ Related question: mathoverflow.net/questions/40296/… $\endgroup$
    – Martin Brandenburg
    Commented Nov 26, 2010 at 9:02

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This issue came up both in the logic course that I taught last semester and in the set theory course that I am teaching this semester.

What I told the students is that in order to do mathematical logic, we need a basic understanding of words over a finite alphabet. We cannot build a theory from less than that.
But if we understand finite strings, we basically have the natural numbers.
Some parts of mathematical logic assume some basic set theory, such as the completeness theorem of first order languages over uncountable alphabets (or just alphabets that are not recursively enumerable). But this can be avoided if you stick to sufficiently simple alphabets (or even finite alphabets).

Similarly, you cannot do axiomatic set theory without a basic understanding of logic, which in turn requires a basic understanding of strings.

On the other hand, once you have built a sufficient theory of logic and set theory, you can use that in order to analyse mathematics. This is somewhat similar to the way that we learn mathematics: You learn to add natural numbers first, and then (usually something like 12 or more years after that) you learn about Peano Axioms that put everything on a solid foundation. I believe that this sort of circle cannot be avoided.

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  • $\begingroup$ In some sense, we formalise the things we want to study (sets, numbers, whatever), and that formalisation necessarily is built on the (observed) properties of the things we want to study. I.e. we observe that natural numbers have certain properties, and invent a formalisation (e.g. Peano arithmetic) that fits. So in this sense, the circularity is by design. $\endgroup$
    – Ketil Tveiten
    Commented Nov 26, 2010 at 9:02
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    $\begingroup$ @Ketil: I agree that the circularity is by design, but I also don't see a way it could be avoided. You have to built on something in order to built a foundation of mathematics. There is one striking example where we seem to built something from nothing, and that is the von Neumann hierarchy of sets. Starting from the empty set, we construct all sets by just iterating the power sets operation. But in order to carry out this approach, we need to be able to define things using formulas. And hence we need formulas. And hence we need strings. And then we are back to the natural numbers. $\endgroup$
    – Stefan Geschke
    Commented Nov 26, 2010 at 9:45
  • $\begingroup$ @Stefan -- exactly! We suspend our disbelief for a bit, and try to 'simulate' set theory, and then construct numbers and so on in this make-believe universe. We already assumed numbers but that was "in the real world". Two ways to think about this: If we forego any notion of "meaning", then we are not constructing numbers, just manipulating strings to get other strings. This dissolves the question. But if we do imbue our axioms with the intended "meaning", it is unclear to me why assuming ZFC and building something that corresponds to our intuitive notion of numbers is the natural foundation $\endgroup$
    – Deniz
    Commented Nov 26, 2010 at 9:58
  • $\begingroup$ @Deniz: Well, it is one foundation, and it is one that turned out to be extremely successful and one that was developed after quite a bit of struggle with other attempts. I agree that ZFC might actually be stronger than what is realistically needed to carry out most of mathematics. But I believe that the most striking argument for some theory like ZFC is that practically all mathematician work within the system without necessarily being familiar with it on a formal level. This indicates that ZFC is natural after all. (To be continued.) $\endgroup$
    – Stefan Geschke
    Commented Nov 26, 2010 at 10:10
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    $\begingroup$ @Stefan: I would say that practically all mathematicians work with sets and functions (with the category of sets and functions, if you like). The further reduction of functions to sets (thus founding mathematics on a single membership predicate, leading to ZFC if taken seriously) is an optional move which has pluses and minuses, but it leads to issues most mathematicians have little practical interest in. $\endgroup$
    – Todd Trimble
    Commented Nov 26, 2010 at 10:59
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It seems that Deniz is raising a slightly uncomfortable question of whether some circularity is built into mathematical foundations. A similar common-sense circularity is what might be called the "paradox of the dictionary": since all words are defined in terms of other words, either dictionaries are hopelessly circular, or some words need to be left undefined in order to break out of the impasse.

As it happens, I am preparing an article for eventual exportation to the nLab which at the outset deals with precisely this question. In the present draft, I have this passage:

Logical foundations avoids this paradox ultimately by being concrete. We may put it this way: logic at the primary level consists of instructions for dealing with formal linguistic items, but the concrete actions for executing those instructions (electrons moving through logic gates, a person unconsciously making an inference in response to a situation) are not themselves linguistic items, not of the language. They are nevertheless as precise as one could wish. $$ $$ We emphasize this point because in our descriptions below, we obviously must use language to describe logic, and some of this language will look just like the formal mathematics that logic is supposed to be prior to. Nevertheless, the apparent circularity should be considered spurious: it is assumed that the programmer who reads an instruction such as "concatenate a list of lists into a master list" does not need to have mathematical language to formalize this, but will be able to translate this directly into actions performed in the real world. However, at the same time, the mathematically literate reader may like having a mathematical meta-layer in which to understand the instructions. The presence of this meta-level should not be a source of confusion, leading one to think we are pulling a circular "fast one".

In other words, to break out of the circularity, it is enough to observe that computers can be programmed to recognize certain strings as well-formed terms or formulas (of a given axiomatic theory), and how to recognize inferences as valid. It's not as if there needs to be some background theory, or the prior existence of a completed or actual infinity of all expressions which might come up, sitting inside the computer. The computer is programmed to handle finite parts of the theory correctly, and the same applies to human users of a theory (although we say "taught", not "programmed").

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    $\begingroup$ @Deniz: I do not agree with Todd's proposed escape from circularity. His proposal implicitly assumes that there is a purely deterministic procedures to check a proof encoded as a string and stored in some physical medium. That is in fact assuming more than the consistency of PA. How do you know that the computer actually does what is claimed? You don't, unless you assume that the computer is in a world governed by some underlying system that already contains a manifestation of the natural numbers. $\endgroup$
    – user21820
    Commented Dec 29, 2015 at 6:56
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    $\begingroup$ @Deniz: Worse still, if the universe has finitely many particles, then no such physical procedure for checking proofs over first-order logic can even exist! The conventional response is that we are talking about an ideal computer, such as a Turing machine, but that is self-defeating because any idealization can only be done in yet another logic system, and precisely because it is once again estranged from the real world, the same reliance on the abstract notion of natural numbers and their properties is an obviously unjustifiable one. $\endgroup$
    – user21820
    Commented Dec 29, 2015 at 7:02
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    $\begingroup$ Regarding meaning in the real world, that is not the requirement of theorems: what is the meaning of Fermat's last theorem? $\endgroup$
    – Fan Zheng
    Commented Oct 15, 2016 at 1:35
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    $\begingroup$ @user21820, I don't know enough about Python 3 to know whether the most trivial objection to that meaning is valid: does Python support bigints natively? Even if this trivial objection isn't satisfied, I'm not sure that I want the meaning of Fermat's theorem to be tied to a particular piece of software, necessarily (not anything to do with Python, just with software) with its own bugs. If it comes down to that, what if I compile Python with a version of Ken Thompson's compiler designed specifically to make Python programs report true when they detect attempts to probe FLT? $\endgroup$
    – LSpice
    Commented Oct 6, 2017 at 17:22
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    $\begingroup$ @user21820, I am not claiming that theorems have no real-world consequences, only that I am not prepared to judge the theorems' meanings by those real-world consequences—especially if they are tied to highly fallible things like a specific piece of software. (I would be equally sceptical if the mathematical meaning of a theorem were to be judged completely by one person's proof of it.) I don't even know what it means to say that a piece of software would work if it had no bugs, and that that verifies the mathematics; that seems too circular an argument to be falsifiable. $\endgroup$
    – LSpice
    Commented Oct 7, 2017 at 17:01
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Note that there are different ways of thinking about set theory and more generally about Logic. They can be thought as a foundation for mathematics or they can be thought as a part of mathematics. If you are thinking of them as a foundation, at the end you have to accept some intuitive concepts, the point of foundation is not that it does not assume anything and builds on nothing, the point is that it is based on accepted theories. Almost all of mathematicians accept the very weak theories about natural numbers and they are sufficient for building the needed metamathematics for set theory. (Primitive Recursive Arithmetic would suffice but even weaker theories are sufficient).

I would suggest the introduction of K. Kunen's "Set Theory" book (the part he discusses the formalist viewpoint) and S.C. Kleene's "Metamathematics".

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If i may add (although this post is kind of old)

This circularity "problem" (if one wishes to see it as such), appears in (what is refered as) classical mathematics.

Now one should bear in mind that even these classical mathematics (continuing along the lines of aristotle, euclid, archimedes, leibniz, cantor, hilbert, russel, goedel, etc..), have been cutoff and formalised (or sterilised if you like) to a greater extened that originaly meant.

In any case this is not the main argument.

But i would like to draw attention to the intuitionistic flavor of mathematics (and especially of the LEJ Brouwer path) (see for example LEJ Brouwer, Cambridge Lectures on Intuitionism, most of first lecture plus the appendix on marxists.org).

There Brouwer, aware of the problem, explicitly takes on the issue of mathematics over language or syntax.

Excerpt:

FIRST ACT OF INTUITIONISM

Completely separating mathematics from mathematical language and hence from the phenomena of language described by theoretical logic, recognising that intuitionistic mathematics is an essentially languageless activity of the mind having its origin in the perception of a move of time. This perception of a move of time may be described as the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory. If the twoity thus born is divested of all quality, it passes into the empty form of the common substratum of all twoities. And it is this common substratum, this empty form, which is the basic intuition of mathematics.

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