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Might be controversial: Is addtive commutative an axiom?

Hello community,

while non-logical axioms (e.g., a + b = b + a) are substantive assertions about the elements of the domain of a specific mathematical theory, such as arithmetic.

I think additive commutative law are not considered as an axiom but an theory derived from the Peano Axiom? I did found some of the people call it an "axiom" in arithmetic. However, in early undergraduate analysis courses, it's often used as an example of basic reasoning to derive some laws in natural numbers from Peano Axiom. I doubt if it's a good example here. Alexliyihao (talk) 02:03, 30 January 2024 (UTC)[reply]

The only appearance of commutative is in #Non-logical axioms, where the context is group theory. The Peano postulates are not relevant in that context. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:43, 30 January 2024 (UTC) theory.[reply]

Commutativity of addition is a theorem in Peano theory, see these course notes. Paradoctor (talk) 12:38, 21 April 2024 (UTC)[reply]

How is that relevant to group theory? The article is about axioms, not about theorems.

As I said, The Peano postulates are not relevant in that context.

As an illustration of the importance of context, arithmetic#Axiomatic foundations discusses two approaches, in one of which the Peano Postulates are themselves theorems. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:13, 22 April 2024 (UTC)[reply]

Are we reading the same encyclopedia? It says nowhere that the Peano axiom are theorems in some set theory!

You also appear to misunderstand the point Alexliyihao made: the lead uses commutativity of addition as an example of a non-logical axiom, which is misleading. Arithemic is normally axiomatized using the Peano axioms, or some set-theoretic model thereof. There, commutativity of addition is a theorem. Surely we ought to do better? Paradoctor (talk) 13:37, 22 April 2024 (UTC)[reply]

Yes, and I never claimed that the wiki article said that. Many elementary course in Set theory derive the Peano postulate from the definition of

\mathbb {N}

from the construction mentioned in arithmetic#Axiomatic foundations.

the point Alexliyihao made is bogus, because it ignored the context and the wording. The text For example, in some groups, the group operation is commutative, is clearly talking about Group theory, not about the Peano postulates. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:21, 22 April 2024 (UTC)[reply]

Seriously, what are you reading?!? Alexlihiyao did not talk about group theory, never mentioned it. That's something you imported here. All he did is criticize the use of commutativity of addition as an example of a non-logical axiom in the lead, where it is connected to arithmetic. That is the context we're talking about. Paradoctor (talk) 16:38, 22 April 2024 (UTC)[reply]

Arithmetic is not the same as Peano arithmetic. It's perfectly possible to consider commutativity of addition as an axiom of arithmetic; it would be a different set of axioms, but the same subject matter. I think you're over-focusing on a particular axiomatization, which isn't mentioned at the point in question in the text. --Trovatore (talk) 19:02, 22 April 2024 (UTC)[reply]

I focus on the most likey interpretation, given that we're not a specialty encyclopedia. Anyway, I finally found out what tripped up Chatul: The lead does not properly represent the article. I took the liberty of fixing that. Unless someone has a better idea, that should conclude this discussion. — Preceding unsigned comment added by Paradoctor (talk • contribs) 19:35, 22 April 2024 (UTC)[reply]

"The most likely interpretation" of arithmetic is hardly the Peano axioms; that's a much more "specialty" notion than arithmetic per se.

In any case, your new text is problematic because commutativity is not an axiom of group theory, and also because we shouldn't be assuming that people know about group theory at this point in the article. --Trovatore (talk) 19:42, 22 April 2024 (UTC)[reply]

We could replace it with a + 0 = a, maybe? --Trovatore (talk) 19:47, 22 April 2024 (UTC)[reply]

MOS:LEAD: The lead should [...] summarize the body of the article

Commutativity for groups is the first example mentioned in § Non-logical axioms, and the only one mentioned in its introduction. Neutral element of addition is not mentioned at all.

not an axiom of group theory It is in the theory of commutative groups. So we add a word. Paradoctor (talk) 20:50, 22 April 2024 (UTC)[reply]

OK, the lead should generally summarize the body, but we don't have to be ultra-rigid about it. If it's useful to put an example in the lead that doesn't appear in the body, I think that's fine. And I don't think we should mention groups in the lead; too specific for the general article about axioms. --Trovatore (talk) 20:59, 22 April 2024 (UTC)[reply]

🤦 Paradoctor (talk) 21:14, 22 April 2024 (UTC)[reply]

Seriously, what are you reading?!? Obviously, the comments relating to Axiom#Non-logical axioms, NOT TO THE LEAD.

Alexlihiyao did not talk about group theory,. The text in contention, For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom, Which part of The only appearance of commutative is in #Non-logical axioms, where the context is group theory. did you not understand? And, yes, there were subsequent updates to the lead, but that has nothing to do with the validity of comments posted before them.

All he did is criticize the use of commutativity of addition as an example of a non-logical axiom in the lead,. Patently false: the lead, AT THAT TIME, did not contain any such text. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:16, 25 April 2024 (UTC)[reply]

The page as it was when Alexliyihao started this discussion. Or you could just look at the passage they quoted.

the comments relating to Axiom#Non-logical axioms, NOT TO THE LEAD Then you're missing the point, because the discussion is about a passage from the lead. Paradoctor (talk) 12:54, 25 April 2024 (UTC)[reply]

Repeating a bogus claim won't make it magically come true. The word commutative does not appear in the lead of permalink/1213388710, nor does it appear in the lead of permalink/1197258286. The first appearance is in Axiom##Non-logical axioms. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 20:01, 25 April 2024 (UTC)[reply]

paraphrase 🤦 Paradoctor (talk) 20:52, 25 April 2024 (UTC)[reply]

Article is outdated

Since the 20th century (or maybe earlier?), "axioms" can also refer to the "properties" that define a mathematical object (for example, see the definition of vector space). Student314 (talk) 13:34, 28 September 2024 (UTC)[reply]

See Axiom#Non-logical axioms Tule-hog (talk) 22:44, 15 December 2024 (UTC)[reply]