Talk:Set has Rank/Proof 1

This proof seems incomplete to me. (Caveat: novice here)

Early on, we say: Suppose that all the elements $a \in S$ have a rank.

Ok, but what about the alternative case? What if all of the elements $a \in S$ don't have a rank?

If that is impossible, then we don't need to merely suppose - it just is axiomatically.

Also, we state:

That is, $a \in \map V x$ for some $x$.

We should immediately clarify what $\map V x$ is (Definition:Von Neumann Hierarchy).

Finally - a concrete example would be extremely useful/helpful.

As an example: What if the set had $2$ elements: the words "red" and "blue".

So $\map V x = 4$: $\O$, $\set {red}$, $\set {blue}$, $\set {red, blue}$

How does this fact lead us to our conclusion that our set S has a rank? --Robkahn131 (talk) 19:35, 15 January 2023 (UTC)

This whole proof relies heavily on Epsilon Induction, which only holds because of the Axiom of Foundation. It's true that, in your example, the set $\set {red, blue}$ doesn't have a rank, but in the Universe of Discourse, $red$ and $blue$ simply don't exist.

The only base case, for the induction is the Empty Set, for which all of none of its elements have a rank. Therefore, all of them are in $\map V 0$, so $\empty \in \map V 1$. This proof is trying to show that foundation implies that every set can be built up from just power sets and unions, as seen in Von Neumann Hierarchy.

Note that there is a variation of set theory called ZFA (which I could not find on this site) where other objects like $red$ and $blue$ exist, and this proof would be incorrect (and in fact impossible, since it is untrue). -- CircuitCraft (talk) 20:19, 15 January 2023 (UTC)

I had reservations about merging that page with this page, mainly because for both pages it is unclear exactly what axiom schema each was dependent upon. While there has been some tentative exploration at putting class theory onto a sound axiomatic footing via the presentation in Smullyan and Fitting a short while back, which appears (although has not been explicitly stated) to be ZFC or a variant, little progress has been made with alternative schemata.

It is intended that all such axiomatic approaches be explored, and their differences explained (and their similarities likewise) so as to be able to prove the common foundational basis from any which one, but unfortunately it has not been possible to find anyone who is prepared to go into the development with quite the level of attention to detail that this requires. My own skills are fading because of age. --prime mover (talk) 22:31, 15 January 2023 (UTC)

I understand your concerns with including theorems that rely heavily on particular axiomatic systems. However, in this case I have attempted to sidestep the issue by avoiding invoking any class theory at all. I wrote out the alternative proof for Epsilon Induction so that this page would be consistent.

Nevertheless, it is probably worth putting a note on the top-level page about the requirement of Foundation. Most theorems in set theory don't invoke it, but it's essential to this one. We might need to copy that onto other pages that invoke this one as well. -- CircuitCraft (talk) 02:13, 16 January 2023 (UTC)

Feels like a makeshift solution to a deeper problem. I'd prefer that the job be done properly. --prime mover (talk) 06:19, 16 January 2023 (UTC)