Timeline for Ultrainfinitism, or a step beyond the transfinite
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| May 31, 2023 at 23:07 | comment | added | cody | I guess I took a too strong version of the pill and now I only see tiny "Peano universes" (finite sets of size the smallest proof of inconsistency of $\mathrm{PA}$). | |
| Jun 30, 2012 at 18:31 | comment | added | Mirco A. Mannucci | hyperinfinities. There is though a deep seated limitation, namely ZF itself. After all, ZF was concoted precisely to eliminate the inconsistencies of the original naive Cantor's theory, and it was very successful at that. But in the process set-theory was a bit, let us say, straight-jacketed. The multiverse, insofar as it is made of models of ZF, shares these limitations. Continues in my comment on Joel's answer... | |
| Jun 30, 2012 at 18:22 | comment | added | Mirco A. Mannucci | Why not, I say, creating a full fledged class theory whose task is to bring us up the ladder of infinity? Of course, there is another option, which is precisely the one that you suggest, and that Joel and the other set theorists follow, namely to SIMULATE this higher class hierarchy INSIDE ZF (via an enrichment of ZF with suitable large cardinals axiomns, or other axioms guaranteeing some reflexivity that lets in genuine models (ie set models) to the desired effect. That is in fact a v options, and, once the mythological V is banished, leads to multiverses with an abundance of -simulated- | |
| Jun 30, 2012 at 18:18 | comment | added | Mirco A. Mannucci | To begin with, I have to say that I, unlike Cantor and many many others, do not believe in such a beast as V (to me V is just as unreal as N inn PA). To me, talking of V when one talks about sets (ie entities well described by the theory ZF) is a bit disingenuous: ZF has no name for V, it simply does not exists as far as ZF goes. Of course, V does exist, when one extends conservatively ZF in an ambient class theory, such as GB. But then, I ask, why limiting ourselves to a tame class theory, who only job is to give a language for talking about classes which are made of objects living in ZF? | |
| Jun 30, 2012 at 1:15 | comment | added | Mirco A. Mannucci | Andreas, you are not a megalomaniac, but a very very very smart man, and it is always a honor and a privilege to hear your voice. I will comment to your answer tomorrow, and likewise with Noah's and Joel's (both packed with excellent material). To all of you my great thanks! | |
| Jun 30, 2012 at 0:25 | history | answered | Andreas Blass | CC BY-SA 3.0 |