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In thinking about the years circa 1900 when people were first seriously attempting the formalization of mathematics (e.g. via a formal logic like Frege’s or Whitehead & Russell’s), it seems like it would have been far from obvious at that time that all of mathematics actually could be formalized in this way.

But nowadays it seems that, although only a fraction of mathematical results have actually been formalized in this way, people seem to generally be confident that all mathematical concepts, proofs, etc., can in fact be formalized in first-order logic within some sufficiently strong and “consistent” foundational theory (I use “consistent” in scare-quotes as shorthand for “widely believed by the community of mathematicians to be consistent”). One example of this confidence can be seen in the comment thread (unfortunately moved to chat) of this earlier question of mine.

My question is: what exactly is the basis for this confidence?

  • Is it purely “empirical”? Namely, the fact that all attempts made so far to formalize mathematics in this way have been successful.
  • Or is there some sense in which one can “prove” that all mathematical reasoning can be formalized in first-order logic within some sufficiently strong and “consistent” foundational theory? (obviously any attempt at such a proof would be complicated by the difficulty of defining what exactly is meant by “all mathematical reasoning”)

Side Note: I’m assuming in this question what seems to be a fairly mainstream philosophical stance, namely that mathematics needn’t necessarily be formalized in order to be “legitimate” (see for example this answer for a fuller articulation).

Update, to address the (hopefully temporary) closure of this question as “opinion-based”:

To clarify, I’m specifically not looking for answers consisting of an individual SE user’s personal opinion on the first-order formalizability of all mathematics.

If indeed there is a general consensus in favor of confidence that all mathematics is first-order formalizable in the sense described above, then this consensus surely is based on reasoned arguments, rationales that have been expressed and communicated in papers, books, lectures, etc. I’m interested in knowing what is/are the “common” or “mainstream” expressed rationale(s) justifying this confidence in first-order formalizability. This could be answered by reference to, say, a frequently-cited journal article. It could also be answered on the basis of someone’s experience as a research mathematician (discussions with colleagues, etc.) That is, the question is answerable in a fact-based way that answers the question “what are the reasons that people commonly give for thinking this?”

Of course, if I am wrong in my impression that there is a general consensus in favor of all mathematics being first-order formalizable in the sense described above, then the wrongness of this impression is also a fact-based matter answerable in a fact-based way (“actually, most mathematicians don’t take this view”)

NikS
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    Nobody has found a case where it hasn't worked. Certainly, there are some things like homotopy type theory, which avoids the restriction in FOL that there is one "type" of entity. But you can encode it as FOL, I think. This is a sort of Church-Turing thesis for logic, rather than computing. – Thomas Andrews May 26 '25 at 05:23
  • Probably relevant: https://math.stackexchange.com/questions/4891060/is-first-orderizability-a-requirement-for-legitimate-mathematical-reasoning and https://math.stackexchange.com/questions/4912787/on-the-limitations-of-first-order-logic-in-mathematical-reasoning – Eric Towers May 26 '25 at 07:23
  • https://mathoverflow.net/questions/24874/what-is-the-reverse-mathematics-of-first-order-logic-and-propositional-logic may provide an alternative perspective on the problem. – Eric Towers May 26 '25 at 07:28
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    See Metamath. – Mauro ALLEGRANZA May 27 '25 at 07:13
  • @ThomasAndrews I don't know your definition of FOL, but multisorted first order logic is also a form of FOL in my opinion. That contains multiple types of objects. NBG for example can be formulated using a "set" sort and a "class" sort. – Trebor May 28 '25 at 09:20
  • NikS, Can you clarify what you mean by "first-order logic" exactly? For example, would you consider "subsystems of second order arithmetic" as "formalized in first-order logic"? – Mikhail Katz May 28 '25 at 09:33
  • @Trebor Just as "exponential Diophantine equations" are not (all) Diophantine equations, this is not a first order logic. It is an extension of first order logic. First Order Logic has a particular definition, and it doesn't include types. You can usually fudge types in first order logic, however, with predicates. – Thomas Andrews May 28 '25 at 14:15
  • @MikhailKatz — According to my understanding of the notion that “everything is formalizable in first-order logic”, the argument would be: Second-order arithmetic can be modeled within a first-order theory like ZFC, thus anything formalizable in second-order arithmetic can be regarded as “ultimately” formalizable in a first-order theory — i.e. whatever you prove in second order arithmetic can also be formalized in a sufficiently strong first-order theory (ZFC). – NikS May 28 '25 at 20:56
  • @ThomasAndrews : I think the single-sorted vs many-sorted distinction probably(?) isn’t significant for the purposes of this question since (as you mentioned) a many-sorted theory is often (always?) equivalent to a single-sorted theory with “type predicates” – NikS May 28 '25 at 21:16
  • @Trebor : See my comment just posted @-ing Thomas Andrews. As I recall, Godel’s original paper describing NBG is technically formulated as single-sorted but has the predicate $\mathfrak{M()}$ to distinguish classes which are sets (“menge” in German). Interestingly, this SEP article uses the phrase “many-sorted first-order logic”, on the other hand I think many of the “classic” logic theorems assume single-sorted. But as mentioned in my earlier comment, the many vs single sorted distinction probably doesn’t matter for this question. – NikS May 28 '25 at 21:29
  • I don't think this question is precise enough. I don't mean this as a criticism, it is of course a very complicated question, but I think in order to remedy this, you should begin by asking something more basic: Can all of mathematics be formalised in propositional logic? I will only accept that your question makes sense when you can convince me that the answer to this is "no", with at least some sort of definition of "formalise" – Carlyle Jun 07 '25 at 20:13

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Far from obvious or not, Frege gets credit for discovering that there is a means of formalizing the foundations of mathematics into FOL. Obviousness, so far as I understand the history of logic and mathematics, isn't really a factor in truth or proof, and we often take things in these subjects as obvious after other people figured them out for us.

Now, whether we can formalize all mathematical reasoning in FOL is something in need of proof, and we may have some evidence that we can't. The go-to counterexample that I know in this philosophical domain is the fact that, just to be able to catch a ball, human brains (or, heck, even dog brains) have to be doing some kind of calculus. However, it's clear that we're not consciously working out those equations in our heads, with all of the variables at play, to react within a range of acceptable accuracy. The question is: Is all of what our brains do formalizable, or are there mathematical intuitions that we cannot formalize?

Another viewpoint would attack the question, itself. That view, known as mathematical formalism, states that mathematics just is the symbol systems that we make up, each with varying utility, so there's nothing to know about whether all of mathematics is formalizable, since they're one and the same thing.

Joshua Harwood
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  • My aim with the “side note” in my question is to explicitly assume we’re not adopting the formalist philosophy. Although, it seems to me that even given a certain flavor of formalism, my question survives in the following form: how do we know someone won’t invent some better formal language capable of expressing mathematics that is beyond the reach of FOL in its current form (not formalizable within any consistent theory based on contemporary FOL)? – NikS May 29 '25 at 05:20
  • There's active work all the time in working out formal languages that expand the expressiveness (just like how FOL expands the expressiveness of PL), and thus the inferential scope, of logic. Some logicians have proposed expanding the expressiveness of FOL with generalized quantifiers, for example. – Joshua Harwood May 30 '25 at 17:25
  • If you believe that GPT is an algorithm which is very complex, but a computation none the less, and not a person typing an answer to your query, then surely you can appreciate that our brains do some sort of computation, complex as it may be, in some kind of a way. Just because it's in some "deep level" doesn't mean it's not a computation. – Asaf Karagila May 30 '25 at 18:30
  • @JoshuaHarwood : OK, but do any of those other languages actually go beyond what FOL can do? That is, is there some other language $\mathfrak{L}$ which can formalize some mathematical reasoning which is not formalizable in a sufficiently strong first-order theory? My understanding is that there is not (currently) any such language — i.e. the capabilities of those other languages can always be replicated in some sufficiently strong first-order theory (albeit perhaps less elegantly). – NikS May 31 '25 at 05:39
  • Yes, there are obviously SOL and HOLs, which go beyond what FOL can do and, because they still contain all of FOL, formalize all of foundational mathematics in the same way. For example, you can't formalize the strong axiom of induction in FOL, but you can in SOL. SOL is also irreducible to FOL. – Joshua Harwood Jun 02 '25 at 07:49
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As you acknowledge in the comments, Zermelo-Fraenkel set theory (ZFC) is formulated in first-order logic. Therefore the issue is not so much first-order logic but whether or not mathematical entities of interest can be expressed in ZFC. For instance, the natural numbers are built following von Neumann starting with the empty set, then the rationals and the reals (for instance, as Dedekind cuts on the rationals).

Ordered pairs $\langle a,b\rangle$ are introduced via $\{a,\{a,b\}\}$, etc. The superstructure of ZFC enables one to introduce functions, function spaces, topological spaces, the functionals of functional analysis, etc., using a miniscule part of the superstructucture. So this apparently enables one to set-theorize analysis, functional analysis, topology, algebra, geometry, etc.

Now mathematics, unlike mathematical entities, is a fairly ill-defined concept since it is not entirely clear what domains of inquiry it covers exactly. But if you suspect that some area of mathematical activity is not set-theorizable, you should mention specific examples of such.

Mikhail Katz
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  • In FOL, ZF has an infinite family of specification axioms. To obtain a finite description of the theory, don't we need to quantify over formulas? – MJD May 29 '25 at 10:35
  • @MJD, I am not sure how you mean your question. The OP didn't say anything about "finite descriptions"; nor was this issue raised in either of the existing answers. – Mikhail Katz May 29 '25 at 10:41
  • Well, not everything is formalizable in ZFC. Examples are discussed here where ZFC is insufficient but “upgrading” to some stronger theory (NBG, or MK, etc.) will do the job. The question is: Can we in some sense “prove” that any mathematical theory will always be formalizable in a sufficiently strong first order theory? Or do we only know that *so far* this has always been possible when it has been attempted? – NikS May 29 '25 at 20:25
  • To be clear, I don’t have an actual example where it hasn’t been possible. The question is whether such an example *could* arise. – NikS May 29 '25 at 20:29
  • Side note: @MJD, the question of finite axiomatizability (versus infinite axioms schemas) is an interesting one, but isn’t what I’m concerned with here. – NikS May 29 '25 at 20:33
  • @NikS, you wrote: "The question is whether such an example could arise" but as I already mentioned in my answer, the term "mathematics" is sufficiently ill-defined to make it impossible to rule out such a possibility. The kind of mathematics that physicists sometimes do sounds like voodoo to mathematicians, and is certainly not formalizable in either first-order logic or any kind of logic as far as the mathematicians know. – Mikhail Katz Jun 03 '25 at 07:09
  • @MikhailKatz : As to your comment that “the term ‘mathematics’ is sufficiently ill-defined to make it impossible to rule out such a possibility”, that matches my intuition as well. But I’m not sure what you had in mind re “voodoo physics math” not being formalizable in FOL. Of course there’s non-rigorous stuff like Dirac’s hand-wavy approach to delta functions. But the hand-wavy stuff can be made rigorous (e.g. defining delta functions via the theory of distributions), at which point I imagine it should be formalizable in FOL, is it not? – NikS Jun 03 '25 at 10:24
  • The Dirac delta has been formalized, but a lot of what physicists do hasn't. Perhaps the most famous example is the Feynman path integral in quantum mechanics. This has not been fully formalized. @NikS – Mikhail Katz Jun 05 '25 at 13:03