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I've been looking around for a long time about how to found mathematics on a solid base. This led me to a long and painful journey of avoiding circular loops.

It led me to do a bit of elementary logic and learning what are first-order formal systems, second-order logic, third-order logic, and so on. The notion of a predicate and so on. When that was done, I started having a look at ZF and ZFC, and realized that defining the set $\Bbb{N} $ correctly seemed highly non-trivial.

An issue that I have is that the axiom schemas of set theory (such as the axiom schema of Specification or Replacement) use natural numbers in them to admit an arbitrary number of finitely many logical variables in the formula defining the various sets involved. This pre-supposes that there should be a model where natural numbers are defined so that they can be used to construct this theory.

So for me, this meant that there should be a model of the natural numbers that doesn't use set theory, otherwise the axioms of set theory cannot be properly laid down.

Under what theory would such a model be built? The Peano axioms use strongly the notion of a "successor function", which seems to assume the notion of a set already.

dmtri
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Patrick Da Silva
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    What is your definition of "model"? – Mark S. Jan 03 '19 at 18:33
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    Your use of letters, words, sentences..., presupposes the existence of a coherent contextually appropriate interpretation for them.... Just like an axiom schema's use of dummy symbols presupposes a contextually appropriate interpretation for them. – Not Mike Jan 03 '19 at 19:09
  • @MarkS. : An interpretation of a theory that satisfies its axioms. Essentially, maybe I'm misunderstanding how logic is normally done because I'm just beginning to understand formal systems, models, theories... but it seems circular to me that some axioms of set theory look like $\forall w_1,\cdots,\forall w_n$ without even discussing integers. How do we deal with that? Is there some human understanding of the axiom schema involved or is it still mathematical and I'm missing something to work with natural numbers? – Patrick Da Silva Jan 04 '19 at 16:06
  • Usually the definition of a model of a first order system is a set, or at least includes a set in a key way, which is why I asked. For instance, see the Wikipedia pages for "interpretation" in logic or model theory. "Some axioms of set theory [involve something like subscripts up to $n$]" Not in ZFC they don't. The closest thing I can think of that comes up would be the concept of a first order formula, but you don't need to be able to count (say, with natural numbers) to build that. – Mark S. Jan 04 '19 at 16:16
  • @Mark S. : Then maybe I misunderstood how axioms need to be read. How do I formulate the axiom schema of specification without having the notion of a natural number? It refers to "having an arbitrary but finite number of variables". That is a quite hard to define axiom if you don't know what "finite" means. Where am I wrong? – Patrick Da Silva Jan 04 '19 at 23:26
  • [looks at wikipedia] Oh, I see; there are different ways to do things. The "finite" business is sort of incidental. The key points are that each formula $\varphi$ is built in a standard way (which necessarily leads to it being a finite string), and that all the extra variables are universally quantified. You don't need to use an internal-to-math definition of finite to understand how to build logical formulas. For instance, if $\varphi$ and $\psi$ are formulas, then so is something like, say, $(\varphi\lor\psi)$. – Mark S. Jan 05 '19 at 00:30
  • @Mark S. : I see. So I don't really need a formal setting to give meaning to a positive integer to write formulas, but I need one to define what it means for a set to be finite. That's probably because logical formulas are meant to be written whereas finite sets are meant to be discussed, I guess? – Patrick Da Silva Jan 06 '19 at 11:38
  • @PatrickDaSilva I think most of your questions are about the role of the metalanguage. I don't know the best source, but you can read about it at these philosophy notes and how it can be a formal language at this MSE question. Basically, when you use a word like "finite" or "natural number", you could be using their everyday English sense ("'abc' has finite length"), or as shorthand for formulas (about sets, say). As long as we're clear when we do which, there's no problem. – Mark S. Jan 06 '19 at 14:05
  • @MarkS. : Yes, I agree with you. At some point something has to be meta because we need to start somewhere. Logic would be a good way to go. So you're claiming that all those results in logic come from first using "meta-logic" to build set theory, use that to build formal logic and then deduce the logic results within set theory?.. Feel free to convert your last comment into an answer. – Patrick Da Silva Jan 07 '19 at 10:43
  • See Does mathematics become circular at the bottom? What is at the bottom of mathematics? and Are sets and symbols the building blocks of mathematics?. – user21820 Apr 25 '20 at 10:36

1 Answers1

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The Question

Here is a rewriting of what I think your main question is, and it's what I'll try to address directly. If I've missed something, just let me know:

I would like to know how formal mathematics can be built-up from first principles without seeming circular. For example, Wikipedia's outline of the Peano axioms for the natural numbers mentions a successor "function", which would seem to require set theory (to define the meaning of "function"). But Wikipedia's statement of the axiom schema of specification involves notation like $\forall w_1,\ldots,w_n$, and the use of the index $n$ would seem to require already having the naturals.

Too-specific Answer

For the specific concerns you mentioned, I think the Wikipedia pages are just a little misleading because of their goal of being accessible, and the preferences of various authors.

It's true that the Peano page links the general function page, and functions are usually defined in terms of sets (at least having a set as a domain). But the word "function" is overloaded, and in Logic it has a meaning separate from set theory. The key phrase on the Peano page is "The non-logical symbols for the axioms consist of a constant symbol $0$ and a unary function symbol $S$.". Briefly, in logic, a function (symbol) is something that acts syntactically like a function; for more details, see the non-logical symbols section of the page on First-order logic(s).

In many books, the axiom/theorem schema of specification is not written with subscripts like on Wikipedia. For example,

More-important General Answer

No matter how we set up formal mathematics, whether or not we use some form of bootstrapping, we'll have to rely on something informal at the bottom of things. For example, a standard logic treatment might take the concept of "strings of symbols" for granted. Or maybe we could define "strings of symbols" in terms of some formal construction of the natural numbers, but then we need some other informal concept to set up those natural numbers.

A key general idea is that of a "metalanguage" that we use to talk about some particular formal logic/language. I don't know the best source, but you can read about it at these philosophy notes and how it can be a formal language at this MSE question. For example, when we use a word like "finite" or "natural number", you could be using their everyday English sense ("'abc' has finite length" or "$w_1,\ldots,w_n$ is a finite bunch of variables"), or as shorthand for logical formulas (about sets, say). As long as we're clear when we do which, there's no real problem.

Side Comment

all those results in logic come from first using "meta-logic" to build set theory, use that to build formal logic and then deduce the logic results within set theory?

We don't need set theory to build a logical system (syntactically). We could just describe the axioms and the allowed rules for writing a proof to go from axioms/theorems to new theorems. (If you have questions about a particular source's treatment of how this is done, that could be a good separate question on MSE.)

But if you want to give semantics to a first-order logic system, the most common approach is to assume you have some version of set theory in your pocket (in the meta-language?), and then use a set as a model (this is described a bit here on the wikipedia page for model theory).

For example, if we say the axioms of the first-order theory of groups are something like:

  • $\forall x, e*x=x\land x*e=x$
  • $\forall x, x^{-1}*x=e\land x*x^{-1}=e$
  • $\forall x, \forall y, \forall z, (x*y)*z=x*(y*z)$

Then a model could be some set $G$ with a distinguished element to take the role of $e$, a function (in the sense of sets) $G\to G$ to take the role of $\cdot^{-1}$, and a function $(G\times G)\to G$ to take the role of $*$.

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Mark S.
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  • I will eventually have a look at that answer in detail, but I definitely want to have a discussion with you! Are you knowledgeable about second-order logic? It has been a very painful journey trying to understand what that actually is, no matter how many books I've read and people I've spoken to. – Patrick Da Silva Mar 27 '19 at 15:47
  • @Patrick "a discussion" Feel free to contact me through the email link in my profile or to invite me to an MSE chatroom. I wouldn't call myself knowledgeable about second order logic, per se, but I feel like I understand the basic definitions from wikipedia and a couple books that mentioned it briefly. – Mark S. Mar 31 '19 at 12:14