Wikipedia:WikiProject Mathematics/A-class rating/Peano axioms

The following discussion is preserved as an archive. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this page.

Peano axioms

Support. I don't see much wrong with it. It would be nice if a logician had a look at it, especially since I have a couple of issues which I can't resolve myself.
1. Is there a subtle reason why the articles uses both N and N; specially, in the "Binary operations and ordering" section.
2. "The main source of difficulty is the second-order induction axiom." — is there another source of difficulty?
3. "This is one reason that the first-order axioms of PA are generally considered to be weaker than the second-order Peano axioms." — in my happy world as an applied mathematician who never has to bother about foundational issues, the fact that there is more than one model proves that PA is weaker. So why the weaselish "generally considered to be weaker"? Any references to contrary opinions?
Jitse Niesen (talk) 12:10, 26 March 2007 (UTC)[reply]

I went through the article and made an effort to address these. Except for the section "Existence and uniqueness", which is working with things that clearly are not natural numbers, the bold N should be used. The second two bullets have been addressed by rewording the sentences. I will go through and reformat the references this week. CMummert · talk 13:13, 26 March 2007 (UTC)[reply]

I'm not comfortable that no specialists have commented, so I asked a couple of editors who know more about logic than me to comment. I hope that this discussion can be kept open for a few more days to give them a chance to have a look. -- Jitse Niesen (talk) 08:20, 29 March 2007 (UTC)[reply]

Support. This article seems to be well written, and provides a good overview of the Peano axioms. My only reservation is that the first-order vs. second-order discussion seems to involve a fair amount of hand waving. it's also worth noting that the Peano axioms are generally (sop far as I, a nonspecialist, know) considered in a first order context, and they're not really expected to be categorical. Greg Woodhouse 15:50, 26 March 2007 (UTC)[reply]

Support. I only have two concerns (aside from N vs. N).
1. It is not obvious, and possibly needs to be stated, that induction does imply recursion or inductive definition.^[1] (The reason I have that book is left as an exercise for the reader.)
2. I also don't recall reading that "there is no countable nonstandard model of PA in which either the addition or multiplication operation is computable." I assume it's in one of the references....
Everything else falls within a logician's "common knowledge" or can easily be derived from such. (Well, my knowledge, anyway.) — Arthur Rubin | (talk) 13:48, 29 March 2007 (UTC)[reply]

Re number 2, this is called "Tennenbaum's theorem" and is proved in Kaye's book. I added a reference and the point. I am not sure what you are proposing for #1. CMummert · talk 14:08, 29 March 2007 (UTC)[reply]

In Peano axioms#Binary operations and ordering, the text reads: "To define the addition operation + recursively in terms of successor and 0,...." What I'm saying is that this definition requires set theory or second-order logic. Once addition and multiplication are defined, it's possible to formalize definitions of exponentiation and sequences (or finite set theory) within first-order PA, which then allows the formalization of recursive definitions. — Arthur Rubin | (talk) 01:45, 30 March 2007 (UTC)[reply]

I added a few sentences, which I hope will resolve your concerns. CMummert · talk 02:20, 30 March 2007 (UTC)[reply]

Weak oppose.
- (+) I didn't spot any glowing errors, and I think the coverage is pretty good.
- (+) I am glad the authors have taken pains to inline short definitions of all jargon. It is a style that I wish would be adopted more generally in WP articles.
- (–) For an article on logic, there is a general feeling of imprecision in the text. To start with, I would greatly encourage a syntactic separation of object and meta levels. There is a lot of repetition and redundancy.
- (–) I don't see any reason to present the axioms based off 1. Peano's reasons for leaving 0 out are obscure, and it would be better to state that Peano's original formulation deviates from the standard formulation off 0 rather than adopt his idiosyncracies.
- (–) I am not happy with the Construction of the natural numbers in set theory section. It spends too much time on the definition and not enought time discussing the relation of Peano#9 to the axiom of infinity from ZF.
- (–) I wish the article would use the <math> tags and depend on texvc for the formatting, instead of homebrewing with wiki syntax.
- Overall, I think the article is almost, but not quite feature quality. Congratulations to the authors. — Kaustuv Chaudhuri 15:01, 30 March 2007 (UTC)[reply]
I'm not sure how to resolve these comments. Would you be willing to make some changes yourself? There isn't a lot of metamathematical study here, so I don't see how to mark "meta" syntax. The axiom of infinity is mentioned; how would you like to see its description expanded? The issue of texvc versus wiki formatting surely isn't a serious issue. In general, the standard for an A-class article should not be "perfection". CMummert · talk 17:02, 30 March 2007 (UTC)[reply]
Sorry for the lack of specificty in my evaluation. There are several changes I intend to make. It is a matter of finding an hour of uninterrupted editing time. Perhaps over the weekend. — Kaustuv Chaudhuri 17:08, 30 March 2007 (UTC)[reply]
One of the specific points is whether to start at 0 or 1. If the axioms are nowadays usually based off 0, then that would seem to be a good reason to do the same here, especially as this is already done later in the article (sections on addition/multiplication and Kaye's formulation).

The homebrewed wiki syntax is perfectly fine for me. It's widely used here and it often gives better results that the <math> tag. -- Jitse Niesen (talk) 14:59, 31 March 2007 (UTC)[reply]

I started making small changes today and ended up rewriting significant portions of it. My goal was to be precise, avoid needless repetition, and adhere to the WP:MOS as best I can. I won't alter the main article until this nomination finishes, but here is my proposal: User:Kaustuv/ip/Peano axioms. — Kaustuv Chaudhuri 02:44, 1 April 2007 (UTC)[reply]

The rewriting seems very nice, although I have a few quibbles that could be discussed on the talk page. The content is essentially identical to that in the current article in terms of scope and depth. The reading level is slightly higher than the current version. I hope you will implement your changes in the main article. CMummert · talk 03:18, 1 April 2007 (UTC)[reply]

Comments I don't think I'd ever seen the distinction "Peano axioms" = 2nd order, "Peano arithmetic" = 1st order, prior to Wikipedia. Is this really standard? (Confusing matters even further is that "2nd order Peano arithmetic" is a first-order (two-sorted) theory, at least in my usage; do we not have an article on PA2 somewhere?) --Trovatore 08:17, 31 March 2007 (UTC)[reply]

I think that when nonlogicians say "Peano's axioms" they usually mean the unformalized ones, which correspond most closely to the second order way of thinking (like all of informal mathematics). On the other hand, the article needs to have something to call the second order axioms, and "Peano arithmetic" is right out.

There is indeed an article on Second order arithmetic - I'll make sure it is in the see also section. The article on second order arithmetic also has to explain that there are both first and second order semantics and that the first order ones are more commonly used. CMummert · talk 14:14, 31 March 2007 (UTC)[reply]

References

^ Rubin, Jean E. (1990). Mathematical Logic: Applications and Theory. Saunders College Publishing. p. 311. ISBN 0-03-012808-0. In an inductive definition, the proof that the function F is unique is an easy proof by mathematical induction and will be left as an exercise. However, the proof that the function F exists uses set theoretical techniques that we shall not discuss here.

The above discussion is preserved as an archive. Please do not modify it. Subsequent comments should be made on the appropriate discussion page, such as the current discussion page. No further edits should be made to this page.

[1] Rubin, Jean E. (1990). Mathematical Logic: Applications and Theory. Saunders College Publishing. p. 311. ISBN 0-03-012808-0. In an inductive definition, the proof that the function F is unique is an easy proof by mathematical induction and will be left as an exercise. However, the proof that the function F exists uses set theoretical techniques that we shall not discuss here.

[1]