Talk:Axiom of Subsets Equivalents

No universe in ZF

In ZF set theory the universal set is (deliberately) not defined. So I'm not sure whether these proofs are actually valid. --prime mover 16:42, 9 September 2011 (CDT)

In any case, the universal class is virtual and unreal, and it is the collection of all sets $x$ (not classes) such that $x = x$. Any statement involving membership to the universal set $A \in U$ is equivalent to another statement written $\exists x: x = A$ (i.e. that $A$ is a value of $x$). Membership to the universal class only signifies that it belongs to our universe of discourse, and that we can talk about the class being a member of other classes. Anyway, ZF allows the capability to talk about classes that are virtual and unreal even if they aren't part of the universe of discourse. In ZF, we are allowed to talk about classes that don't exist as long as we don't refer to them as values of variables in the same way that we are allowed to refer to the class of Natural Numbers as long as we don't make the mistake of assuming it's the value of some variable before the Axiom of Infinity. -Andrew Salmon 18:03, 9 September 2011 (CDT)

Flaw in statement

Is it that Takeuti and Zaring have this fundamental flaw in its analysis, or that it has been mispresented here? --prime mover (talk) 11:22, 26 August 2017 (EDT)

Upon close examination, it is implicit in T/Z that the $A$ in their statement is a class. And of course we can define the class $\{x|\phi(x)\}$; that's just a notational convenience. It is however not allowed to quantify over classes inside the formal language, so it seems T/Z is more accurate here. Also, no equivalence is asserted in T/Z. I think this disqualifies the content of this page in its current form. — Lord_Farin (talk) 11:37, 26 August 2017 (EDT)

Given that there's a lot of T/Z in here and it has not been addressed consistently, is it worth stepping through everything and fixing it up where it's wrong, or should we just rip it all out and start again when someone has the urge to to T/Z from scratch?

I have a Takeuti ("Proof Theory") on my shelf but I don't think I've ever opened it. I expect it follows similar ground, so there may be an opportunity to revisit this thread of development from that instead. --prime mover (talk) 13:12, 26 August 2017 (EDT)

I've been preparing for ripping it out and re-covering from scratch, more so since the ground has been covered before and a consistent integrated approach is needed. Many of the current T/Z pages are beyond repair. Starting from scratch is easier. — Lord_Farin (talk) 15:04, 26 August 2017 (EDT)

Let me add that I can imagine — given the almost impenetrable style it is written in — that there is actually quite a need for a reasonable, fleshed-out, comprehensive treatment of T/Z, of the type PW can provide. So there might be added value here. Or we agree T/Z is and has always been crap, and we don't look at it again, instead spending our time on more worthwhile publications. — Lord_Farin (talk) 15:07, 26 August 2017 (EDT)

I'm wary of dismissing a book as crap until I've completely analysed it, so I'll suspend judgment. There are few books I'd class as crap in this context, but I call to your attention 1997: David Wells: Curious and Interesting Numbers (2nd ed.) whose page-to-mistake ratio is of the order of $2 : 1$ (yes, that's right, it's that bad -- I'm up to page 156 and I've found 87 mistakes in it so far), and I was somewhat disappointed in the shallowness of 1980: D.J. O'Connor and Betty Powell: Elementary Logic, and all my Game Theory works are either too unstructured or too surface-level to properly make into the basis for a consolidated $\mathsf{Pr} \infty \mathsf{fWiki}$ approach. But I have Takeuti's Proof Theory on my shelf, which I am looking to crack open sometime, once I've finished this (flawed but fascinating) overview of Number Theory and Recreational Mathematics, after which I will alert Wells to the errata page ... --prime mover (talk) 16:56, 26 August 2017 (EDT)