Talk:List of first-order theories

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In the definition of real closed fields, in the item that 0 is not a non-trivial sum of squares, shouldn't "a_1=0 or a_2=0 or...or a_n=0" read "a_1=0 and a_2=0 and ... and a_n = 0"? In this case, the "and" statement follows from the "or" statement, by virtue of the quantification "for every positive n" and the method of descent, but this seems like a pointless obfuscation, and I think "and" was meant. Do any experts want to weigh in? --OinkOink 00:19, 8 July 2006 (UTC)[reply]

It's funny, but "and" and "or" are actually equivalent in this case. The version with "or" looks weaker, but applying it n times you can get the version with "and". Nevertheless what you propose sounds better, the only reason I am not changing it is because I don't know which of the two versions of the axiom is standard. --Hans Adler 21:19, 13 November 2007 (UTC)[reply]

Linkification

{{Unanswered}} In the section on second-order arithmetic it states "The axioms are those of Robinson arithmetic, together with axioms schemes of induction and comprehension." ; upon reading that I immediately wondered what the heck might be meant by 'Robinson arithmetic', so went looking and found there does exist a page linkable as Robinson arithmetic. I have not the expert knowledge to know whether the token 'Robinson arithmetic' in this article corresponds to the notion that the article Robinson arithmetic is about. Thus I did not feel qualified to edit the page to make that token into a link, and resorted to this talk-page approach to the problem. I also noticed that neither the token 'induction' nor the token 'comprehension' were links, and a similar thought process ensued: do they refer to notions Wikipedia has pages on, if so do they correspond to the pages linked to by putting double-squarebrackets around those tokens? For all I know the reason none of these three tokens are links is because none of them actually refers to the notion one would find by putting square-brackets around the token! Ouch. Thus, this request for someone more explicitly qualified to look into this. Finally, I notice the string 'axioms schemes' and wonder is this grammatical? Can it be re-written as 'axiom schemes' and/or as 'axioms scheme'??? (Or even 'the axioms schemes' or 'the axiom schemes' or 'the axioms scheme'??? The end result of all this is my naive urge was initially to replace the quoted material with the following: "The axioms are those of Robinson arithmetic, together with axiom schemes of Mathematical induction and comprehension."; but comprehension is a disambiguation page that does not yet contain any definition that says it is applicable to or used in arithmetic theory nor in first-order theories... So please, someone who knows this stuff, please can clear up these possibly trivial but for all I know extremely nontrivial matters for me? :)

Knotwork (talk) 14:21, 18 April 2008 (UTC)[reply]

I have also now noticed (dunno why I didnt earlier) that the token 'axioms' and the token 'schemes' might well be intended in some rigourous and/or formal sense thus that this same linkification issue also applies to them. :)

Knotwork (talk) 18:13, 18 April 2008 (UTC)[reply]

Blank and Redirect at First-order arithmetic

First-order arithmetic was blanked and redirected to Peano axioms in November, and I now changed the redirect target to List of first order theories#Arithmetic, which covers first-order arithmetic more holistically. There is some content in the page history there, so if anyone here sees anything that merits inclusion here, please feel free to (selectively) merge to here instead. Felix QW (talk) 10:41, 8 January 2022 (UTC)[reply]

Pure theory of equality

This article seems to be playing an important role in improving coherence of presenting the foundations of mathematics on wikipedia and it seems likely that it should eventually become a list of summaries of main articles on each of the individual topics. I suppose those should be explicitly identified as main. I just found that there was a stub article Theory of pure equality and now made connection from and to here, having it expanded somewhat before. I think the material in Theory of pure equality can be strengthened with what is already provided here. Optionally the material here can be shortened as this may be viewed as a somewhat degenerate case compared to other theories. Vkuncak (talk) 10:28, 5 February 2022 (UTC)[reply]