Do I get this right? Gödel's incompleteness theorem applies to first order logic as it applies to second order and any higher order logic. So there is essentially no way pinning down the natural numbers that we think of in everyday life?

  • First order logic fails to be categorical, i.e. there are always non-standard models.
  • Second order logic is categorical here, but does not allow us to prove all its true statements?
  • Defining the numbers from set theory (e.g. ZFC) suffers from the same problem as all first order theories, i.e. there are non-standard models of ZFC which induce non-standard natural numbers?

How do we even know what natural numbera are, if we have no way to pin down a definition. What are the standard natural numbers? Or do we accept that second-order logic gives us this definition and we just cannot prove all there is?


I think I can summarize my question as follows:

How do the mathematicians that write standard natural numbers have formal consensus on what they are talking about?

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    $\begingroup$ A relevant discussion here $\endgroup$ Apr 25, 2017 at 8:23
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    $\begingroup$ Exactly; the fact that we have difficulties in picking up in a $\mathsf {ZFC}$-based meta-tehory the unique structure satisfying the first-order P-D axioms, and that we still lack (from Plato until today) a philosophical satisfying def of "number" does not mean that we have to consider the "usual" natural numbers: $0,1,2,\ldots$ we are accustomed to use to count with their well-known properties as some sort of unknowable phantoms like unicorns or the philosoper's stone. $\endgroup$ Apr 25, 2017 at 8:47
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    $\begingroup$ How do you know anything? $\endgroup$
    – Asaf Karagila
    Apr 25, 2017 at 8:56
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    $\begingroup$ I added a bit at the top of my answer to address your "LATER" question. However, I recommend you don't do that in subsequent questions on Math SE because answerers are not notified of changes to the question and besides cannot keep editing their answer to match the question. Anyway I hope my answer has satisfied you! =) $\endgroup$
    – user21820
    Apr 25, 2017 at 14:10
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    $\begingroup$ @user21820 I'm sorry about the change. I felt it was a good idea to compactify my question into one sentence. My hope was to not change the meaning, and this sentence was also already contained as a comment under my question. Anyway, I will stay away from doing this again ;) And your answer was amazing! $\endgroup$
    – M. Winter
    Apr 25, 2017 at 14:14

5 Answers 5


How do the mathematicians that write standard natural numbers have formal consensus on what they are talking about?

Mathematicians work in a meta-system (which is usually ZFC unless otherwise stated). ZFC has a collection of natural numbers that is automagically provided for by the axiom of infinity. One can easily define over ZFC the language of arithmetic (as in any standard logic textbook), and also that the standard natural numbers are terms of the form "$0$" or "$1+\cdots+1$" where the number of "$1$"s is a natural number. To disambiguate these two 'kinds' of natural numbers, some authors call the standard natural numbers "standard numerals".

Do I get this right? Gödel's incompleteness theorem applies to first order logic as it applies to second order and any higher order logic. So there is essentially no way pinning down the natural numbers that we think of in everyday life?

Yes. See this post for the generalization and proof of the incompleteness theorems that applies to every conceivable formal system, even if it is totally different from first-order or higher-order logic.

First order logic fails to be categorical, i.e. there are always non-standard models.

Yes, so first-order PA does not pin down the natural numbers.

Second order logic is categorical here, but does not allow us to prove all its true statements?

Yes; there is no (computably) effective deductive system for second-order logic, so we cannot use second-order PA as a practical formal system. In the first place the second-order induction axiom is useless if you do not add some set-existence axioms. In any case, any effective formal system that describes the natural numbers will be incomplete, by the incompleteness theorem.

So although second-order PA is categorical (from the perspective of a strong enough meta-system), the categoricity does not solve the philosophical problem at all since such a meta-system is itself necessarily incomplete and hence the categoricity of second-order PA only ensures uniqueness of the natural numbers within each model of the meta-system, and cannot establish any kind of absolute categoricity.

Defining the numbers from set theory (e.g. ZFC) suffers from the same problem as all first order theories, i.e. there are non-standard models of ZFC which induce non-standard natural numbers?

Exactly; see previous point.

How do we even know what natural numbers are, if we have no way to pin down a definition.

We can only describe what we would like them to be like, and our description must be incomplete because we cannot convey any non-effective description. PA is one (incomplete) characterization. ACA is another. ZFC's axiom of infinity is a much stronger characterization. But there will never be an absolute categorical characterization.

You might hear a common attempt at defining natural numbers as those that can be obtained from 0 by adding 1 repeatedly. This is circular, because "repeatedly" cannot be defined without essentially knowing natural numbers. We are stuck; we must already know what are natural numbers before we can talk about iteration. This is why every useful foundational system for mathematics already has something inbuilt to provide such a collection. In the case of ZFC it is the axiom of infinity.

What are the standard natural numbers?

Good question. But this is highly philosophical, so I'll answer it later.

Or do we accept that second-order logic gives us this definition and we just cannot prove all there is?

No, second-order PA does not actually help us define natural numbers. The second-order induction axiom asserts "For every set of natural numbers, ...", but leaves undefined what "set" means. And it cannot possibly define "set" because it is circular as usual, and it does not help that the circularity is tied up with natural numbers...

Now for the philosophical part.

We have seen that mathematically we cannot uniquely define the natural numbers. Worse still, there does not seem to be ontological reason for believing in the existence of a perfect physical representation of any collection that satisfies PA under a suitable interpretation.

Even if we discard the arithmetical properties of natural numbers, there is not even a complete theory of finite strings, in the sense that TC (the theory of concatenation) is essentially incomplete, despite having just the concatenation operation and no arithmetic operations, so we cannot pin down even the finite strings!

So we do not even have hope of giving a description that uniquely identifies the collection of finite strings, which naturally precludes doing the same for natural numbers. This fact holds under very weak assumptions, such as those required to prove Godel's incompleteness theorems. If one rejects those... Well one reason to reject them is that there is no apparent physical model of PA...

As far as we know in modern physics, one cannot store finite strings in any physical medium with high fidelity beyond a certain length, for which I can safely give an upper bound of $2^{10000}$ bits. This is not only because the observable universe is finite, but also because a physical storage device with extremely large capacity (on the order of the size of the observable universe) will degrade faster than you can use it.

So description aside, we do not have any reason to even believe that finite strings have actual physical representation in the real world. This problem cannot be escaped by using conceptual strings, such as iterations of some particular process, because we have no basis to assume the existence of a process that can be iterated indefinitely, pretty much due to the finiteness of the observable universe, again.

Therefore we are stuck with the physical inability to even generate all finite strings, or to generate all natural numbers in a physical representation, even if we define them using circular natural-language definitions!

Now I am not saying that there is absolutely no real-world relevance of arithmetical facts.

Despite the fact that PA (Peano arithmetic) is based on the assumption of an infinite collection of natural numbers, which as explained above cannot have a perfect physical representation, PA still generates theorems that seem to be true at least at human scales. My favourite example is HTTPS, whose decryption process relies crucially on the correctness of Fermat's little theorem applied to natural numbers with length on the order of thousands of bits. So there is some truth in PA at human scales.

This may even suggest one way to escape the incompleteness theorems, because they only apply to deterministic formal systems that roughly speaking have certain unbounded closure properties (see this paper about self-verifying theories for sharp results about the incompleteness phenomenon). Perhaps the real world may even be governed by some kind of system that does is syntactically complete, since it has physical 'fuzziness' due to quantum mechanics or the spacetime limitations, but anyway such systems will not have arithmetic in full as we know it!

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    $\begingroup$ Brilliant+1.... $\endgroup$ Apr 25, 2017 at 13:47
  • $\begingroup$ I am a bit confused about the objection concerning the usage of "repeatedly". The axiom of infinity states $\exists N \, (\emptyset \in N \land \forall x \, (x \in N \to S x \in N))$, where $Sx := x \cup \{ x \}$ or smth. similar. I think it is safe to informally read this as "$0$ is in $N$ and starting from it, repeated usage of $S$ produces further elements of $N$". $\endgroup$
    – Léreau
    Aug 21, 2021 at 8:00
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    $\begingroup$ @Lereau: Well do you realize that no one would ever even think of, much less propose, the Infinity axiom in Z set theory unless they already know about naturals? Just think about it; the very notion of "closure" under an operation is inherently based on the natural numbers. Axioms that claim the existence of some kind of closure obviously do not justify the existence; they do nothing more than assert it without justification. And you cannot justify it non-circularly. $\endgroup$
    – user21820
    Aug 21, 2021 at 13:37
  • $\begingroup$ It is noteworthy to add that at least in the case of the first-order logic, the problem is not just so much Goedel's theorems as it is the fact that first-order logic does not have the ability to unambiguously represent any infinite structure, even those where Goedel does not apply, e.g. elementary Euclidean geometry. The interesting question, then, is whether these philosophical problems run deeper than just Goedel's theorem. $\endgroup$ Sep 27, 2021 at 3:47
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    $\begingroup$ @The_Sympathizer: You're not wrong that FOL cannot pin down any infinite structure uniquely, even up to isomorphism, due to Lowenheim-Skolem. But it's also worth noting that this is philosophically problematic only if we assume the existence of uncountable structures. For instance, there is an essentially unique countable DLO without endpoints. So a very natural question is, can there be a consistent computable extension T of PA that has an essentially unique countable model? No, because the incompleteness theorems lets us construct uncountably many countable non-isomorphic models of T! $\endgroup$
    – user21820
    Sep 27, 2021 at 6:54

Yes, as you phrased it in a comment:

So even the academic usage of the term standard natural number trusts in our intuitive understanding from preschool?

That's exactly how it is.

We believe, based on experience, that the intuitive concept we learned in preschool has meaning, and mathematics is an endeavor to explore that concept with more powerful tools than we had available in preschool -- not to construct it from scratch.

There are the Peano axioms either in their first- or second-order guise, of course. However, even if Gödel hadn't sunk the hope that they would tell us all truth about our intuitive natural numbers, they would still just be ink. The fundamental reason why we care about those axioms is that we believe they capture truth (only some of the truth, but truth nonetheless) about our intuitive conception of number.

Indeed it is hard to imagine how one could construct the natural numbers from scratch. To demand that, one would need to build on something else that we already know -- but it can't be turtles all the way down, and somewhere the buck has to stop. We can kick it around a bit and say, for example, that our fundamental concept is not numbers, but the finite strings of symbols that make up formal proofs -- but that's not really progress, because the natural numbers are implicit even there: If we can speak of strings, then we can speak of tally marks, and there's the natural numbers already!

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    $\begingroup$ Yup! Many people fail to realize the fundamental circularity as you have put it: "If we can speak of strings, then we can speak of tally marks, and there's the natural numbers already!" And (as described in my answer) even if we agree on our present circular definition of natural numbers, there is no indication that it even describes anything in the real world, from which we originally drew inspiration for the axioms of arithmetic... $\endgroup$
    – user21820
    Apr 25, 2017 at 13:31
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    $\begingroup$ "It cannot be turtles all the way down, somewhere the buck has to stop" already assumes the natural numbers; even making analogies about the natural numbers being standard is hard without them! $\endgroup$
    – Yakk
    Apr 25, 2017 at 14:03
  • $\begingroup$ @user21820 The word "circular" is silly. Use the word "repetition" instead. $\endgroup$
    – user132079
    May 3, 2017 at 7:22
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    $\begingroup$ @WaqarAhmad: I do not know what you mean at all. A definition is said to be circular if it relies on prior understanding of the notion that it is purported to define. It is necessary to have some background in logic before one can understand that there is a circularity in the definition of the natural numbers, not to say describe it accurately. I suggest you look very very carefully at the meaning of "repetition"; you cannot define repetition without already understanding the counting numbers. $\endgroup$
    – user21820
    May 4, 2017 at 13:25
  • $\begingroup$ Just my own philosophical tint, but some of the answers here seem to presuppose formal system before numbers, while I do think the better view is to say the logics are used to study numbers. The formal systems in question already use more complicated string manipulations than numbers, so saying a formal system is more fundamental than number puts the cart before the horse. Numbers themselves are given by physical reality, they don't need any assumptions or formalism to exist. $\endgroup$
    – Tony
    Jun 23, 2019 at 0:12

Any set-theoretic universe (or similar sort of object) will define a unique (up to isomorphism) set of natural numbers.

In that universe, that set of natural numbers is the standard one.

In some sense, natural numbers, manipulating strings, and logic are all synonymous subjects. So if you are working with some form of logic, one could consider the associated natural numbers to be the standard natural numbers.

(e.g. formal logic incorporates string manipulations, string manipulations let you compute arithmetic on numerals, and Godel's work shows how to encode logic in natural numbers)

I get the impression that this adequately describes the usage of the term by mathematicians who believe the notion has some real meaning.

However! We can conceive of logic as a mathematical construction rather than an attempt to capture the behavior of real-world mathematicians — that is, formal logic is something you develop within a mathematical universe.

In this setting, both of the above definitions of "standard natural numbers" are consistent; e.g. if you develop formal logic in ZFC, then the natural numbers associated to that development of logic are precisely the natural numbers of set theory.

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    $\begingroup$ Hurkyl, it doesn't look like the OP is willing to swallow the ZFC package just to save "the standard numbers" :-) $\endgroup$ Apr 25, 2017 at 14:25

We only know what the naive counting numbers are. According to Reeb these do not exhaust all of $\mathbb N$. The belief in "standard" natural numbers parades under impressive-sounding terms like intended interpretation. The impossibility of fixing such was argued by no less a heavywight than Hilary Putnam. The deleterious effects of such beliefs were studied in this article.

The only "formal consensus," as you put it, among such mathematicians is to act as if they know what $\mathbb N$ is.

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    $\begingroup$ But there is not only "formal consensus"... There is also the "practical consensus" that we parctice every day when we pay with our credit card, when we take a look at the clock, and many many other examples of the all-pervading language-game of counting. $\endgroup$ Apr 25, 2017 at 14:21
  • $\begingroup$ @MauroALLEGRANZA, the examples you mentioned are those of naive counting numbers that enjoy a universal consensus. The tendency to identify those with $\mathbb N$ is a tell-tale sign of a platonist. $\endgroup$ Apr 25, 2017 at 14:23
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    $\begingroup$ Exactly, and they are the "intended" interpretation of any reasonable mathematical theory of numbers. We for sure will reject every theory that disproof some of the well-known properties of "counting" numbers. $\endgroup$ Apr 25, 2017 at 14:25
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    $\begingroup$ @MauroALLEGRANZA, you remarks are unclear. If you talk about intended interpretation you must have a precise mathematical model in mind. But here you are referring to the naive counting numbers apparently. In other words, you are confusing naive counting numbers for $\mathbb N$. $\endgroup$ Apr 25, 2017 at 14:28

You're right, we don't.

We also don't even need the full power of Goedel's theorem or any other such no-go result to understand this at least on an intuitive level, because we're only talking about the natural numbers themselves here.

The key point in these discussions is the induction postulate:

Induction Postulate. Let $S$ be any arbitrary set of natural numbers. If $0 \in S$ and for every $n \in S$, $n + 1 \in S$, then $S$ contains all the natural numbers.

Intuitively, this is pretty much precisely what we mean by that "all natural numbers consist only of those we reach from $0$ by counting by $1$s." So where is the trouble?

First, let's flip it around so that instead of simply looking like a hand-wavy statement about what we wish to exist, it looks more like a process.

Induction Postulate, v2. Suppose we are given a set $S$ of natural numbers containing $0$ and such that, if $n \in S$, then $n + 1 \in S$. Then there exists no natural number $m$ which, if we removed it from the set of all natural numbers, would not also be removed from $S$.

In other words, the induction axiom establishes a sort of weeding procedure: we rip through all the subsets of the natural numbers, and we check to see if any subset containing the chain of successors of $0$, i.e. "the numbers you get from counting", doesn't happen to contain some weeds or impostors (i.e. what are called elsewhere as "non-standard natural numbers"). If it does, then we know we didn't have the natural numbers.

Or, to put it another way, suppose we are given some set that is purported to be the natural numbers, but perhaps contains some weeds. We could go through each subset, check the above, and if we find an $m$, we can toss it. We can repeat this over and over until we finally have the natural numbers.

Still sounds pretty good, right?

But it shouldn't take much to see the trouble. You see, how many subsets do we have to do that for? The answer is an infinite amount - even more, it's a continuum amount, $\beth_1$. Thus, this is a supertask of high order: a task we imagine to complete, but with an infinite number of steps.

It, then, is not something we can ever carry out physically.

Thus, of course, one might then wonder what the deal is with first order logic and Goedel and all that no-go stuff. Well, all those things basically deal with ways to try to make the task "easier". What they do is replace the idea of taking arbitrary subsets, which is something which while maybe we can imagine it in our head is not something we can ever make concrete, with something that can be made more concrete - subsets we can describe in some kind of description language or method that we can actually realize. Then we can carry out the weeding. However, because any such method necessarily excludes indescribable sets from consideration and no two such languages have any need to flag all the same sets as describable, there may still be "weeds" left over, but we will never see them so long as we don't work outside the confines of that language.

And that's how that this axiom practically "defines" the natural numbers. We never get to see what the "real" natural numbers are, or even if there are such a thing. Instead, what we do is settle for next best: throw out enough weeds that at least all the theorems we are interested in can be proven. If any weeds are still left, but don't affect those, then we just let them be. The axiom lets us do that, for whatever realm of interest we want to explore, and that is enough.


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