Talk:Axiom of Foundation (Strong Form)

What's the relationship between this and Axiom:Axiom of Foundation (Classes)?

They both seem to assert the existence of a minimal element (sorry, possibly obvious, I haven't kept up with all this class-theoretic stuff you've all been adding).

For that matter, is Axiom:Axiom of Foundation the same statement for sets?

And lastly, does it bother anyone else to have a proof of something labeled an Axiom (that is in the main/proof namespace)? --Alec (talk) 00:53, 26 August 2012 (UTC)


 * We're proving this within ZF set theory. We can't assume Axiom:Axiom of Foundation (Classes), which is not part of ZFC (it's part of NBG).  We can only assume Axiom:Axiom of Foundation, which is the analogous statement for sets.  The proof of the statement for classes is not obvious. --Andrew Salmon (talk) 01:09, 26 August 2012 (UTC)


 * I have added the category "Zermelo-Fraenkel Class Theory" to show that this is ZF specific. --Andrew Salmon (talk) 01:16, 26 August 2012 (UTC)


 * So we're going with the strategy that the two branches of set theory are completely divorced from one another? Fair enough, as long as I know what's been decided. --prime mover (talk) 09:02, 26 August 2012 (UTC)


 * There's not 'two' branches of set theory (well, I note the distinction between naive and axiomatic set theory). There are numerous axiomatisations. A satisfactory system for interlinking and interweaving them has not been developed on PW. I have no ideas towards such an encompassing device just yet. It will be a project of the long breath; I imagine a page containing the naive set theory result, and one comprising the same result in the various axiomatic approaches; these will in some way be interlinked. Because naive set theory is not consistent, there is little hope for any rigorous approach at a higher than result-by-result level. --Lord_Farin (talk) 11:27, 26 August 2012 (UTC)


 * The issue I have, as always, is that we are developing two separate parallel threads of stuff which is "basically the same" except for the detail that in one branch we say "sets" and in the other we say "class". I hear and hear again the argument that they are not the same, and I appreciate that.
 * But there is necessarily overlap and common ground. For example, the Axiom:Axiom of Extension is indisputably part of every analysis, as it determines what a set (and class) actually is. We should, therefore avoid branching this one in any way at all. The case for the others grows fuzzier as we progress through to more sophisticated axioms.
 * This page, for example, makes no reference to the Axiom:Axiom of Foundation page, which IMO is wrong. It's not even in the same namespace. In fact, there is no reference on any of these pages to any other of them. At the very least there ought to be an also-see, but really they ought to be linked by transclusion.
 * I would also suggest that rather than refer to "Zermelo-Fraenkel Class Theory", would it be better to refer to it as "Bernays-Von Neumann-Godel Class Theory" and refer to the ZF name under an "also known as" section? This should, I believe, reduce confusion.
 * Unless, of course, ZF $\ne$ BNG (I don't know, I'm guessing not) in which case further overlaps are needed. --prime mover (talk) 12:44, 26 August 2012 (UTC)


 * ZF $\ne$ NBG. However, it is provable that NBG is a conservative extension of ZF, so everything that is expressible in ZFC and true in NBG is true in ZFC as well.  Therefore, whatever we prove in NBG, we could also prove in ZFC and vice versa.  ZFC cannot create statements that quantify over classes, but NBG can.  However, we may develop "class theory" within ZFC, which is what we have done.  It's not that these are different branches of set theory.  They just have different axiomatics, and thus, some systems can prove things that others cannot.  There are relationships between these different axiom systems, which would allow all these theorems proven for ZFC to translate to theorems for NBG and vice versa. --Andrew Salmon (talk) 17:04, 26 August 2012 (UTC)

Minimality
$\in$ is not a strict ordering, except of strange things like the ordinals. It's not transitive. --Dfeuer (talk) 20:45, 29 March 2013 (UTC)