Talk:Peano axioms

message: 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)
thread_title: Induction schema

message: 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)
replies: text: But I'm nominating it for GA anyway.--Q28 (talk) 05:50, 9 January 2022 (UTC)
replies: text: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Nominate for GA 

message: 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)
replies: text: 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)

text: 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)
replies: text: 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)
thread_title:  Merge proposal 

message: 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)
replies: text: 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)
thread_title:  Zero 

message: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
replies: text: I, too, engage in a wide variety of unwise activities. 67.198.37.16 (talk) 07:13, 29 November 2023 (UTC)
thread_title:  Zero is not a natural number