Timeline for Axiomatic strength of the cumulative hierarchy
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Jul 3 at 13:09 | comment | added | Zuhair Al-Johar | @TimButton, No, I'm not making this assumption either. I'm defining everything in the base language, then adding the axioms, then proving that ZFC would follow. I'm making no assumptions whatsoever other than those of first order logic with equality. I'm exactly starting from the base as you are doing. But, the formulation is more complex of course. | |
| Jul 3 at 11:27 | comment | added | Tim Button | I think you're assuming this all takes place within ZF(C). Again, there's no problem with that! But I wanted to work with much weaker assumptions; assumptions that can be justified by this story: Sets are formed at stages. At each stage, you find (all and only) every "possible" set whose members are among those things found earlier. Nothing seems to ensure the stages are well ordered, or to guarantee one can provide definitions by transfinite recursion. | |
| Jul 2 at 10:42 | comment | added | Zuhair Al-Johar | @TimButton, it depends on the definitions. If you define ordinals to be transitive sets of transitive sets not having more than one element with no successor, and so we'll not assume well ordering at the outset. As for the first point, I think it is circumvented by defining $x= V^F_\alpha$ in a piecemeal manner by the formula asserting the existence of a function $g$ from $\alpha+1$ with $g(\emptyset)=\emptyset$ and $g(m+1)=\mathcal P(F(m))$ and $g(\lambda)=\bigcup_{\beta <\lambda} g(\beta)$ for limit $\lambda$, and $x= g(\alpha)$. So, the symbol $V$ is eliminable totally. | |
| Jul 2 at 8:23 | comment | added | Tim Button | One could do things this way. But it makes a couple of assumptions I would prefer to avoid. (1) We have a recursion theorem which ensures the success of these definitions. (2) We explicitly want the stages of the hierarchy to be well ordered. I was—following in the footsteps of Dana Scott, and some others—considering how we might avoid building these two assumptions in at the outset. | |
| Jun 28 at 10:44 | history | answered | Zuhair Al-Johar | CC BY-SA 4.0 |