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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?

LATER:

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?

Bill Dubuque
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M. Winter
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    A relevant discussion here – MathematicsStudent1122 Apr 25 '17 at 08:23
  • The question has a big philosophical impact and math (and scientific) theories are not able to answer phil questions. Since Socrates and Plato philosphers are struggling with defintions (the essence, τὸ τί ἐστι): the Good, the Beautiful and also "the number". – Mauro ALLEGRANZA Apr 25 '17 at 08:29
  • @MathematicsStudent1122 I have no problem with accepting that there is a meta language in which we already know about some stuff, e.g. an intuitive understanding of $\mathbb N$. But then there is serious mathematical literature which refers to standard natural numbers and I got the feeling there must be a definition out there which is beyond simple intuition. – M. Winter Apr 25 '17 at 08:31
  • But there is a well-established and well-know (since school days) usage of numbers to count and this fact is perfectly described by the characterization of the number progression provided by the Peano-Dedekind axioms – Mauro ALLEGRANZA Apr 25 '17 at 08:32
  • Note that "second order logic is categorical" is basically a set-theoretic statement in disguise, since it refers specifically to the properties of set-theoretic models in which "the set of propositions on $X$" is interpreted precisely as the power set of the interpretation of $X$. If you remove the set-theoretic attachment, second-order logic is just the first-order theory of first-order logic. –  Apr 25 '17 at 08:34
  • @MauroALLEGRANZA So even the academic usage of the term standard natural number trusts in our intuitive understanding from preschool? – M. Winter Apr 25 '17 at 08:35
  • @Hurkyl I always wondered how second-order logic can talk about sets if there is no set theory lurking in the background that suffers from the same first-order problems as everything else. But what exactly do you mean by "the set of propositions on X"? Are the subsets accessible by quantifiers in second-order logic just the sets definable by predicates? Why is it then, that second-order logic is more expressible than first-order logic with an axiom scheme (one axiom per predicate)? Maybe I should ask this in a separate question. – M. Winter Apr 25 '17 at 08:37
  • @M.Winter: I probably should have said "type" there. But really, set, class, type, collection, and such are all basically synonyms. E.g. the sets of set theory and the types of type theory are basically the same thing, just studied in different ways. –  Apr 25 '17 at 08:47
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    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. – Mauro ALLEGRANZA Apr 25 '17 at 08:47
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    How do you know anything? – Asaf Karagila Apr 25 '17 at 08:56
  • @Asaf Of course ;) More like this: how do the mathematicians that write standard natural numbers have formal consensus on what they are talking about. But probably to long for a title. – M. Winter Apr 25 '17 at 10:16
  • It seems that all modern number theory and classical analysis can be derived from Peano's Axioms using the ordinary rules of logic and set theory. That in itself makes them useful as a working definition of the set of natural numbers. – Dan Christensen Apr 25 '17 at 12:36
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    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! =) – user21820 Apr 25 '17 at 14:10
  • @DanChristensen, I am not sure what you mean. If you are limited to the Peano Axioms then you are in a strictly weaker system than set theory, so how can you apply "the usual rules" as you put it of set theory? There are certainly many important results in modern mathematics that are inaccessible if one is limited to PA. – Mikhail Katz Apr 25 '17 at 14:11
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    @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! – M. Winter Apr 25 '17 at 14:14
  • @goblin I agree that my question might be a duplicate of this and that some of the answers over there could be answers to mine (but actually, I don't know, because I can only look at them with the knowledge I already got from the answers to my own question). Further, I think I prefer the answers here, there are really good ones, and it seems they are still coming in. So what should I do? Closing my question for now seems to early, not because of reputation, but because of good answers still coming. But also, I cannot explain why my question should be significantly different from the other one. – M. Winter Apr 25 '17 at 14:56
  • @M.Winter, yes, this is a flaw in the system; questions that get closed as duplicates may have attracted new answers, which sucks. But I've already cast my "mark as duplicate" vote. Fwiw, I voted to close mainly because I think the question is poorly written (no offense). For instance, this sentence: "First order logic fails to be categorical, i.e. there are always non-standard models." I mean, for starters, this isn't true for finite models, of course. $\forall x(x=0 \vee x=1) \wedge 0 \neq 1$ pins down sets with $2$ elements, for example. – goblin GONE Apr 25 '17 at 15:01
  • And, that sentence is just very vaguely worded... I just feel like you could have tried hard to be precise. If you're precise, but wrong, then at least we can clear that up and everyone goes home satisfied. But if you're wrong in imprecise ways... it's just harder, you know? Honestly, I'm not sure I really buy the whole "some questions should be closed" thing. But if there's ever a situation where this is appropriate, and I'm not sure there is, but if there is, surely really vaguely worded questions is that situation. Anyway, I feel less bad about it because marking as duplicate... – goblin GONE Apr 25 '17 at 15:04
  • is much less problematic, from a moral standpoint, than closing a genuinely new question. Also, please take a look at my new answer under my old question; I think it will help you a lot. – goblin GONE Apr 25 '17 at 15:04
  • @goblin: More than half of my answer is not found in the suggested duplicate. And I actually don't think your just-posted answer resolves anything philosophically, because you cannot refer to $\mathbb{N}$ without having defined it, which you cannot actually do... for reasons described in my answer. =) Also, I took the asker's "there are always non-standard models" to be in the context of models of PA, so it's not that bad. I prefer to interpret inquisitive questions leniently; they are much preferable to homework questions. =) – user21820 Apr 25 '17 at 15:21
  • @MikhailKatz I don't know about every aspect of modern mathematics, but I understand that what we call modern number theory and classical analysis is derivable from Peano's Axioms. Not sure what you mean by "weaker system than set theory." IMHO, it seems highly unlikely that any of Peano's Axioms is not true of the natural numbers or that some new axiom will ever be required. It is a well known result that every structure that satisfies Peano's Axioms is order isomorphic to every other such structure. – Dan Christensen Apr 25 '17 at 20:11
  • @DanChristensen, would you include the Lebesgue measure as part of what you call "classical analysis"? Can you tell me if the countable additivity of the Lebesgue measure, routinely taken for granted in analysis textbooks, is provable from PA? – Mikhail Katz Apr 26 '17 at 07:19
  • @MikhailKatz I would think that the formal development from scratch of any theory that refers the natural numbers would have to start with a statement of Peano's Axioms. – Dan Christensen Apr 26 '17 at 21:24
  • @DanChristensen, I didn't say anything to the contrary but your claim seems to be that one doesn't need any further foundational material. The latter claim is dubious. – Mikhail Katz Apr 27 '17 at 09:10
  • @MikhailKatz AFAIK, to formally develop modern number theory and classical analysis you need nothing more that Peano's Axioms and the ordinary rules of logic and set theory as a starting point. Depending on your set theory, you may even be able to derive Peano's Axioms (e.g. ZFC). In this case, such a derivation would probably be a necessary first step to formally develop number theory and classical analysis from scratch. – Dan Christensen Apr 27 '17 at 12:28
  • "Peano's Axioms (e.g. ZFC)" What do you mean? @Dan – Mikhail Katz Apr 27 '17 at 12:52
  • @MikhailKatz Apparently it is possible to actually derive Peano's Axioms using the ZF (Zermeo-Fraenkel) axiom of infinity. Alternatively, if you admit the existence of any Dedekind-infinite set, you can select a subset of it that satisfies Peano's Axioms. – Dan Christensen Apr 27 '17 at 19:49
  • In other words, the axioms of logic, set theory and Peano's Axioms are necessary and sufficient to be able to formally derive modern number theory and classical analysis. IMHO. – Dan Christensen Apr 28 '17 at 01:58
  • @DanChristensen, ZFC certainly entails PA, as I mentioned earlier. The problem is the opposite implication. You haven't answered by question about Lebesgue measure. – Mikhail Katz Apr 28 '17 at 06:49
  • @MikhailKatz Nowhere have I suggested that Peano's Axioms entails ZFC. Again, AFAIK classical analysis (including measure theory) and modern number theory can be formally derived from Peano's Axiom using nothing more more than the rules and axioms of ordinary logic and set theory. No other rules or axioms are required. Are you really suggesting otherwise or is this all just a misunderstanding? – Dan Christensen Apr 28 '17 at 13:24
  • @MikhailKatz You may be right. I'm not sure, but transfinite arithmetic may require additional axioms, at least in the way it is usually presented in analysis textbooks. – Dan Christensen Apr 28 '17 at 13:38
  • So, apart from perhaps considerations of transfinite arithmetic, classical analysis, too, can be formally derived from Peano's Axioms. – Dan Christensen Apr 28 '17 at 14:47
  • @DanChristensen, I am indeed suggesting otherwise. Incidentally what is the source of your claim? Intense faith? You still haven't answered my question concerning Lebesgue measure. – Mikhail Katz Apr 30 '17 at 12:11
  • @MikhailKatz Not a claim so much as an opinion based on my experience. As for the countability theorem you cite, it refers the set of natural numbers, so I would think that it is formally derivable starting from Peano's Axioms using only the ordinary rules and axioms of logic and set theory. I would be interested to know what other rules and axioms you think might be required. I don't think it requires any transfinite arithmetic. – Dan Christensen Apr 30 '17 at 15:18
  • @DanChristensen, the countable additivity of the Lebesgue measure is not even provable in ZF, let alone PA. If you give me an email address I can send you a recent publication of ours on this subject. – Mikhail Katz Apr 30 '17 at 15:20
  • @MikhailKatz If you can't prove it in ZFC, how do you know it is true? Do you have some other set theory (or foundational system) in mind with which you can prove it? – Dan Christensen Apr 30 '17 at 15:26
  • @DanChristensen, I didn't say you can't prove it in ZFC. You certainly can. You can't prove it in ZF. Nuance! – Mikhail Katz Apr 30 '17 at 15:27
  • @MikhailKatz So, you need AC, a part of what I think of as "ordinary set theory." – Dan Christensen Apr 30 '17 at 15:32
  • @DanChristensen, I agree about AC, but the point is ZF implies PA without any need for AC. – Mikhail Katz Apr 30 '17 at 15:34
  • @MikhailKatz Agreed, but even if PA is derivable from your set theory, a first step in formally deriving most if not all classical analysis would have to be actually deriving PA.Otherwise, you would need to state PA as an initial assumption. – Dan Christensen Apr 30 '17 at 15:46
  • @DanChristensen, I have no quarrel with the idea that PA is fine as an initial assumption. Incidentally, I feel that this question should be reopened as it contains answers that are more interesting than the ones at goblin's post. – Mikhail Katz Apr 30 '17 at 16:48
  • @MikhailKatz I love to here that, but are the answers a reason for reopening despite the fact, that the actual question is indeed a duplicate? I am not so familiar with the correct behavior for such situations. – M. Winter Apr 30 '17 at 17:32
  • @M.Winter, it seems to me that your question is rather different, since it concerns the role of Goedel's theorem in the story. This was not addressed at all in the other question. I think you can legitimately vote to reopen your question. – Mikhail Katz Apr 30 '17 at 17:35
  • Goedel once stated from his incompleteness results there're only 2 possible disjunctive philosophical conclusion: either math including natural numbers exist independently as Platonism, or the idealized human mind having the ability to conceive natural numbers cannot be formalized as any Turing machine like deterministic mechanism compatible with constructive intuitionism. Regarding the coexistence of non-isomorphic or even contrary standard/non-standard models of a same first order theory like PA/ZFC is actually not that surprising, there was ancient book named like "Infinite meaning sutta". – cinch Sep 27 '21 at 05:42

5 Answers5

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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!

user21820
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    Brilliant+1.... – Mikhail Katz Apr 25 '17 at 13:47
  • 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 $N$". – Léreau Aug 21 '21 at 08:00
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    @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. – user21820 Aug 21 '21 at 13:37
  • 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. – The_Sympathizer Sep 27 '21 at 03:47
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    @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! – user21820 Sep 27 '21 at 06:54
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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!

hmakholm left over Monica
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    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... – user21820 Apr 25 '17 at 13:31
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    "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! – Yakk Apr 25 '17 at 14:03
  • @user21820 The word "circular" is silly. Use the word "repetition" instead. –  May 03 '17 at 07:22
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    @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. – user21820 May 04 '17 at 13:25
  • 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. – Tony Jun 23 '19 at 00:12
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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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    Hurkyl, it doesn't look like the OP is willing to swallow the ZFC package just to save "the standard numbers" :-) – Mikhail Katz Apr 25 '17 at 14:25
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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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    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. – Mauro ALLEGRANZA Apr 25 '17 at 14:21
  • @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. – Mikhail Katz Apr 25 '17 at 14:23
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    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. – Mauro ALLEGRANZA Apr 25 '17 at 14:25
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    @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 $\mathbb N$. – Mikhail Katz Apr 25 '17 at 14:28
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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.

The_Sympathizer
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