Talk:Peano axioms

Induction schema
I'm not an expert on this topic, but the presentation of the induction schema in the "Peano arithmetic as first-order theory" section seems unnecessarily complicated (though I'm not claiming it's wrong). Instead of an axiom for each formula $$\phi(x,\bar{y})$$, we could just have a simpler one for each $$\phi(x)$$, namely $$ \left( \phi(0) \wedge \forall x (\phi(x) \implies \phi(S(x)))\right) \implies \forall x \phi(x) $$

The current thing for $$\phi(x,\bar{y})$$ seems like it can be realized by (infinitely many) instances of the above given by replacing $$\bar{y}$$ in the formula $$\phi(x,\bar{y})$$ by all choices of the variables (i.e. 0, S(0), S(S(0)), etc for each y_i). This seems a lot simpler, though maybe there is some subtle difference that I am not appreciating, which is why I am posting this here rather than editing. — Preceding unsigned comment added by 68.175.128.126 (talk) 02:35, 16 September 2023 (UTC)

Nominate for GA
Could this article be nominated for GA? Or maybe even FA? You can do discussion on the talk page.155.186.86.239 (talk) 00:50, 28 March 2020 (UTC)

Aside: I don't think the article should be nominated for either a GA or FA until the significance of the Peano axioms is addressed (in the article). Rhkramer (talk) 21:39, 1 August 2020 (UTC)
 * But I'm nominating it for GA anyway.--Q28 (talk) 05:50, 9 January 2022 (UTC)
 * The nomination has been reverted. Q28, please note that a nominator at GA should be a significant contributor to the article or to have consulted in advance on the talk page (which isn't what you've done). Even more important in this case, the article is A-class, a level that is higher than GA-class (between GA and FA, though not used by all WikiProjects), so you're effectively nominating it for a demotion, which is not what the GAN process is for. If you don't believe the article meets the A-class criteria for WP:WikiProject Mathematics, you should be following that WikiProject's procedures for reconsidering the status of A-class articles. Thank you. BlueMoonset (talk) 16:16, 9 January 2022 (UTC)
 * However, the project is no longer processing the revocation of A-level entries. This seems to have resulted in how we cannot revoke the A-level status of the entry.--28 (talk) 07:37, 10 January 2022 (UTC)


 * Well, however that may be, on 18 February 2022 someone did demote it from A to B. 67.198.37.16 (talk) 23:30, 28 November 2023 (UTC)

Substitution Axiom
The article currently defines equality; there was discussion above of getting rid of that, but it didnʼt go anywhere. However, if we do include the axioms for equality, weʼre missing the one for substitution. We use this implicitly in the examples of addition and in the proof that $$S(0)$$ is the multiplicative left identity; we should either include it in the main list of nine axioms, or do as we do elsewhere in the article and refer to “the usual axioms of equality”. --65.36.124.105 (talk) 06:26, 27 October 2021 (UTC)


 * At least when only 0 and S are considered as function symbols, substitutivity follows from the "only if" part of axiom 7. When also + and * are admitted, and they are defined recursively as usual, substitutivity might be provable by induction (I'm not sure about this). On the other hand, e.g. Hermes (Introduction to Mathematical Logic, London 1973, ISBN 3540058192) defines a 2nd-order logic with built-in equality, and thus avoids these problems. - Jochen Burghardt (talk) 10:18, 27 October 2021 (UTC)
 * 65.36.124.105: Your link above is to a discussion that I understood as being about including a version of the axioms that axiomatized exponentiation.  I was against that &mdash; exponentiation, unlike multiplication, does not need to be axiomatized in PA.  You can simply define it, and then prove the statements you wanted to add as axioms.  Probably more to the point, I don't think it's standard to include exponentiation as part of the language of PA.
 * As regards considering equality to be part of the logic rather than the non-logical axioms, I think that is probably standard these days, and I would look favorably on a presentation incorporating that idea, assuming it can be found in good sources. The axioms that deal purely with equality can be presented for historical interest in the "history" section, perhaps. --Trovatore (talk) 19:44, 27 October 2021 (UTC)

Downgrade the quality level of the article to B
I and some of the other editors believe that the entry does not meet the GA criteria and that the A rating for the item is already inactive. Can we downgrade it to a B rating?--28 (talk) 07:44, 10 January 2022 (UTC)
 * Q28, the place to make that request is at Wikipedia talk:WikiProject Mathematics. While they seem to have abandoned the process of assigning A-class ratings more than a decade ago, it is nevertheless a WikiProject-based rating that was never rescinded, so it's up to them to do any reassessment or retraction. You and the other editors here can certainly chime in there to explain why you feel the rating is not merited at this time. Best of luck. BlueMoonset (talk) 17:19, 10 January 2022 (UTC)


 * Q28 I agree with you that the A-rating seems a bit too high. I'm not an expert in logic, but the article seems to have many deeper flaws. For instance, the induction is expressed in second-order logic, but then the addition (defined immediately after the axioms) is given like the induction is a first-order schema. No examples of interesting predicates are talked about which differentiates this system from simpler axiomatic schemes (e.g., primality, divisibility). Decidability isn't discussed at any length. TheZuza777 (talk) 00:46, 29 January 2022 (UTC)
 * By the WikiProject maths assessment rating guidelines anyone can change the assessment, and disputes should be settled on the talk page.
 * Since no one so far seems to come forward to defend the A-Rating, I would suggest downgrading it and seeing if anyone reinstates it. Then we could take it to Wikipedia talk:WikiProject Mathematics. Felix QW (talk) 12:03, 31 January 2022 (UTC)

Vkuncak (talk) 10:01, 29 January 2022 (UTC) I suggest as the main way towards improving the article to have more balance between 1) the historical second-order formulation that seems to be the way the article started and 2) more modern and more widely used first-order variant (called Peano arithmetic) here. Many people would be more than happy to take FOL variant as the starting point for an entire article.

I went ahead and split the article into the 1) historical+2nd-order part and 2) FOL part. I feel that this was the right thing to do because the historical+2nd-order part one is meant to describe natural numbers in a fairly general sense as witnessed by categoricity, whereas the second part has clear recursive axiomatization(s), exhibits non-standard models, and was the subject of Goedel's incompleteness theorem. Vkuncak (talk) 18:25, 31 January 2022 (UTC)


 * [ Maybe you are the right person to explain to me why Gödel's incompleteness theorem is considered to refer to first-order logic in Wikipedia, while his original 1931 article (e.g. ) refers to Principia Mathematica, and, in particular, higher-order logic (in sect.2, p.176, variable symbols of  arbitrary high order are introduced before fn.17; on p.177, axiom I.3 is the classical 2nd-order induction axiom, with x2 a predicate variable, ".", "Π", and "⊃" meaning "and", "for all", and "implies", respectively). I guess I'm unaware of some modern treatment of Gödel's result which transcribed it to FOL. Maybe you could even devise a clarifying remark for the page Gödel's incompleteness theorems? Many thanks in advance - Jochen Burghardt (talk) 19:06, 31 January 2022 (UTC) ]
 * Jochen: I'm not Vkuncak, but if I could mix in, I think that may be more of a Principia Mathematica question than a general math-logic question.  The notation and terminology used by Russell turned out to be pretty much of a dead end; hardly anyone learns it anymore for mathematical reasons (though historians may still be interested).  And yes, I suppose it's probably good to know about if you want to read Gödel's original paper, but the same remarks apply &mdash; Gödel was doing something for the first time, and putting it in the context of the day, but there are much more efficient ways to learn the content now.
 * I'm not competent to answer the question directly because I've never bothered to put in the effort needed to understand Principia Mathematica, but I could offer a guess. It likely has to do with two different senses of the phrase "second-order logic".  In the context of this article, for example, second-order logic with "full semantics", plus the original Peano axioms, is enough to make the theory categorical &mdash; there is only one model up to isomorphism.  "Full semantics" means that you have first-order variables that range over natural numbers and second-order variables that range over sets of natural numbers, and the latter really range over all sets of natural numbers.
 * On the other hand, you can take exactly the same theory but interpret it with Henkin semantics, which means that you give it an interpretation by specifying what both the first-order and second-order variables range over. That is, you limit the second-order variables to ranging over a pre-specified class of sets of naturals, rather than all of them.  Now the theory is no longer categorical.  It's actually a theory of first-order logic, though it has second-order variables.
 * Does that help? It might be a nice thing to bring up at the refdesk if you want more details.  We could use some good questions like that. --Trovatore (talk) 19:42, 31 January 2022 (UTC)


 * Jochen: Thank you for pointing out to this historical development, of which I was not aware of! I was merely following the expositions in several textbooks, including Mendelson and the handbook by Samuel Buss. These all refer to FOL formulation of the problem. What matters for the incompleteness theorem is to have a system where provability is recursively enumerable. Strong enough FOL theory with decidable axiom schemas is one typical way to get recursively enumerable definition of provability. Second order logic with full semantics is not such an effective definition: there is no way to enumerate all true facts according to such semantics. Vkuncak (talk) 20:25, 31 January 2022 (UTC)


 * Trovatore: Henking semantics for second order (and higher-order) logic could be applied to the question of models of Peano arithmetic and provability, but do we actually have interesting facts to mention in that direction (other than it is a valid thing to consider)? If yes, this line of thinking could be used to expand and clarify analogously the paragraph on categoricity proof carried out in first-order axiomatization of ZFC. I thought how to clarify that paragraph, but I am not entirely sure about the intention was with it originally, so I left it as it is. Vkuncak (talk) 20:25, 31 January 2022 (UTC)

 (Please reply at Talk:Gödel's_incompleteness_theorems.)

Merge proposal
I believe that including an example of how the Peano axioms prove associativity of addition, say, would be helpful for an understanding of the Peano axioms. On the other hand, Proofs involving the addition of natural numbers is of little use as an independent page without the context from here. Therefore, I suggest selectively to merge Proofs involving the addition of natural numbers into Peano axioms, and would love to hear some input on what content to retain. Felix QW (talk) 15:16, 31 January 2022 (UTC)


 * I suggest to reuse just the image (File:Inductive proofs of properties of add, mult from recursive definitions svg.svg). However, I'm definitely biased. - Jochen Burghardt (talk) 17:14, 31 January 2022 (UTC)


 * I think that textual rendering would make the proofs more accessible. If this is to be merged, it should go into the Arithmetic section because they are one part of the proof of the claim that the axioms imply the properties of an ordered semiring. The proofs are easy exercises in induction, so I would say that one of them is likely sufficient to state as an example? Vkuncak (talk) 17:31, 31 January 2022 (UTC)
 * My personal preference would be only to add the single proof of associativity. I think it aids the understanding of how the Peano axioms can be used to infer well-known properties of arithmetic, and reflects on the axioms themselves. Felix QW (talk) 18:16, 31 January 2022 (UTC)


 * Oppose merge. The material at the "proofs" article is too detailed to include here.  Honestly I'm not sure the "proofs" article belongs in Wikipedia at all, but that's not really a road I want to go down; it isn't hurting anyone where it is. --Trovatore (talk) 17:40, 31 January 2022 (UTC)

Zero
ISO 80000-2:2009 apparently requires 0 to be an element of N. The obvious problem with that is that zero can't be put into a one-to-one correspondence with anything. There is also the just plain false claim that zero is intuitive. Well, no: it must be taught; it is not a natural concept. (Everyone who knows its history knows this.) I don't argue for a revolution in the Abstract Algebra main-stream here. I DO argue that more discussion should be included with an (alternative) N that does not include 0. Even Peano did it initially, (before he understood its value in Abstract Algebra and how it allowed N to have lots of connections that an N-without-0 would lack.) Note that other than the silly ahistorical claim that 0 is intuitive (for certain values of 'intuitive', there's nothing that isn't 'intuitive'), justification of its inclusion is omitted here. So, it seems to me that we have the "inside baseball" people and the peanut gallery (general public). It's the latter that I believe would greatly benefit from some mention of the "with or without 0" issue. (other than just an aside). Oh, speaking of asides, it's claimed here that N is necessarily infinite. OK, but that should be changed to "countably infinite", imho.174.131.48.89 (talk) 06:40, 19 August 2022 (UTC)
 * While there are nontrivial points to be made about the conception of zero and the mental steps needed to make it intuitive, I respectfully suggest that this is not the right article to go into detail on that. This is an article about a particular axiomatic framework, not about zero and not even about the natural numbers.  The "zero" issue should be touched on in the history section (I assume it already is; haven't checked) but beyond that it's a distraction from the main concerns of the article. --Trovatore (talk) 19:27, 19 August 2022 (UTC)

Zero is not a natural number
This article is historically mistaken. Zero is not a natural number. The first natural number is one. The Ancient Greeks and Romans had no zero in their arithmetic. In fact, zero isn't even a positive integer.

Bertrand Russell observed that any other natural number may be given as a number with no predecessor, for example, the natural number two. But then 1 would have to be a variable, rather than a constant.

Accordingly, this article needs to be edited and revised. — Preceding unsigned comment added by 108.41.98.105 (talk) 00:22, 23 May 2023 (UTC)
 * It is true that the older notion of natural number did not include zero. That was a historical mistake, which has largely been corrected.  The more modern and useful conception of natural number does in fact include zero.  However terminology has not converged to a standard; there are still mathematicians who do not include zero. --Trovatore (talk) 18:36, 23 May 2023 (UTC)


 * Eh? Many natural formulas become awkward if you let zero be a natural number. They have to be marked up by changing x to x+1 which often makes the formula twice as large to display. That, or requires some qualification that all of the $$x_n>0$$ everywhere, which also adds text and lengthens formulas. The term "non-negative integer" works, if you absolutely have to have zero. Stick to the conventional, historical usage taught in grammar school. It works cleanly. 67.198.37.16 (talk) 23:44, 28 November 2023 (UTC)
 * I take it your edit summary, argument for the sake of argument?, was meant to describe your own edit? Anyway this is not the place to discuss it. I unwisely responded in that key half a year ago, but the right answer is the one I gave in the previous section, at 19:27, 19 August 2022 (UTC). --Trovatore (talk) 00:45, 29 November 2023 (UTC)
 * I, too, engage in a wide variety of unwise activities. 67.198.37.16 (talk) 07:13, 29 November 2023 (UTC)