Definition talk:Zermelo-Fraenkel Axioms

Do the standard postulates of geometry (i.e. the line postulate, the parallel postulate, etc.) follow from these Axioms, or should we make pages for them too? Sorry to keep asking this kind of question, I've never seen a complete derivation of math from Axioms before, so I don't know what's necessary. --cynic 20:41, 3 October 2008 (UTC)

(Exciting, isn't it?) Good question. IMO we should present whatever pages necessary of Euclid's axioms, postulates, etc. etc., the full nine yards, as they're interesting enough in their own right, whether they can be proved from set theoretical axioms or not.

Geometry's a particular challenge that I won't be up to (my skills aren't up to it, I'm an abstract algebraist myself) but I'd love to see someone documenting the whole of Euclid. (Then Lobachevsky of course.)

I believe (but I've never seen it proven) that one can directly derive the theorems of geometry from coordinate geometry which in turn can be derived from theorems proved on $\R \times \R$ etc. but first you need the complete rounding off of modules, vectors etc. which haven't been started on yet.

In any case, there will definitely be connecting links between Euclid and Descartes as and where necessary, I'd have thought.

In any case, there's no harm in specifying whatever axiom systems we like in the "Axioms" section, e.g. the axioms of naive set theory, the axioms of number theory, the axioms of real analysis, etc. with notes and links to the effect that certain axiom systems can be derived from other more "basic" systems. It would be a shame to lumber all the poor students of (for example) calculus with all the hard slog of hacking their way through the undergrowth of mappings, rings, naturally ordered semigroups, all that, just so as to find out what a "real number" is when all they need to know is where to start to prove continuity. Or whatever. --prime mover (talk) 21:02, 3 October 2008 (UTC)

I would consider it bad practice to quantify over the propositional functions. The more usual practice is to define an 'axiom' while in fact stating many axioms: one for each propositional function we may think of (and that is expressible in the language $\{\in\}$). --Lord_Farin 17:07, 23 October 2011 (CDT)


 * I lack a decent text in this area - so my understanding is intuitive. Do what you think works. --prime mover 15:34, 24 October 2011 (CDT)

Axiom namespace?
Shouldn't this be in the axiom namespace? --barto (talk) 02:30, 21 October 2017 (EDT)

done. --barto (talk) 11:45, 21 October 2017 (EDT)