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S Jun 6, 2020 at 10:52 history bounty ended user820789
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May 31, 2020 at 4:13 answer added SSequence timeline score: 1
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May 28, 2020 at 18:30 comment added Dave L Renfro What I believe is the slowest "mode of travel" faster than Veblin hierarchy methods was developed (I think) by Heinz Bachmann in early 1950s. The best introductory survey I know of (in English) is Normal functions and constructive ordinal notations by Larry William Miller. See also, at least for the references, A survey on ordinal notations around the Bachmann-Howard ordinal by Wilfried Buchholz (2016).
May 28, 2020 at 11:42 comment added Fedor Pakhomov @meowzz I am not sure what would be the best source for this. But you could check the paper "Ordinal notations based on a hierarchy of inaccessible cardinals" by Wolfram Pohlers.
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May 27, 2020 at 21:20 answer added Simply Beautiful Art timeline score: 2
May 27, 2020 at 21:11 comment added user820789 @DaveLRenfro Well put! I would upvote your comment more times if I could.
May 27, 2020 at 20:35 comment added Dave L Renfro having a bit of a hard time understanding even the smallest of the large cardinals --- I've pretty much given up on getting much of an understanding of how large they are (and they're a bit outside my main areas of expertise anyway), and I now mainly view them as larger than I could ever hope to reach by the kinds of vaguely explicit or the least bit constructive (in a very loose sense) modes of travel that I know about. It's as if I only know how to walk (proceed step-by-step) and I want to reach some of the distant quasars; another mode of travel is needed.
May 27, 2020 at 19:59 comment added user820789 @DaveLRenfro That portion of Levy's work does indeed look interesting! Also, I have made that Google search numerous times. Still having a bit of a hard time understanding even the smallest of the large cardinals..
May 27, 2020 at 19:56 comment added user820789 @SSequence I have looked at ITTMs a bit & have been wanting to look into OTMs. If you have any good reccomendations for resources I'd appreciate it.
May 27, 2020 at 19:54 comment added user820789 @FedorPakhomov I would be interested in reading what you wrote, if at all possible. I haven't seen much about ordinal collapsing functions with inaccessibles (I normally see them with uncountables) - but it seems interesting & I will look into it more. If you have any resources you would reccomend, I'm open to suggestions.
May 27, 2020 at 16:05 comment added Dave L Renfro A useful google search is Mahlo + cardinal + hyperinaccessible.
May 27, 2020 at 16:05 comment added Dave L Renfro Now take the diagonal intersection of the $A^{(\beta)}$'s to obtain the class of ordinals $\gamma$ such that $\gamma$ is the $\gamma$-th ordinal $\alpha$ such that $\alpha \in A^{(\beta)}$ for all $\beta < \alpha,$ and now repeat the process with the class of ordinals in the diagonal intersection, and continue for another diagonal intersection, and another and another into the transfinite, then look at ordinals $\gamma$ such that $\gamma$ doesn't show up as a least (nonzero?) ordinal in any of these classes until you've done $\gamma$ many diagonal intersections, $\ldots$
May 27, 2020 at 16:04 comment added Dave L Renfro You might find 4.33 (p. 152) of Levy's Basic Set Theory of interest. Let $A^{(0)}$ be the class of alephs, let $A^{(1)}$ be the class of ordinals $\gamma$ such that $\aleph_{\gamma}=\gamma,$ let $A^{(2)}$ be the class of ordinals $\gamma$ such that $\gamma$ is the $\gamma$-th ordinal $\alpha$ such that $\aleph_{\alpha}=\alpha,$ let $A^{(3)}$ be the class of ordinals $\gamma$ such that $\gamma$ is the $\gamma$-th ordinal in the enumeration of $A^{(2)},\ldots$ (process continues transfinitely so that $A^{(\beta)}$ is defined for all ordinals $\beta).$
May 27, 2020 at 12:52 comment added SSequence I haven't looked at the post in enough detail, but my impression upon a cursory look is that the following should be a safe upper limit. In OTM model add an extra tape or two basically serving as an "oracle" to calculate $\omega_i$ given $i$ number of $1$'s on the "advice tape" [shouldn't be too difficult to give equally powerful additional commands in other models]. I suppose this should be a safe upper-limit to cover the kind of cases mentioned in OP? Or am I missing something here?
May 27, 2020 at 12:18 comment added Fedor Pakhomov @meowzz Nevertheless, if I understood correctly what you are doing here, this cardinals definitely have been considered before. Namely, if you look at stronger ordinal notation systems based on collapsing functions, in the same time they provide a notation system for (relatively) large cardinals. In particular consider the notation system that is used in analysis of $\Delta^1_2\text{-}\mathsf{CA}_0+\mathsf{BI}$ and $\mathsf{KPi}$ that collapses first inaccessible cardinal. In particular it has notations for your analogues of $\Gamma_0$, small Veblen ordinal, and large Veblen ordinal.
May 27, 2020 at 12:10 comment added Fedor Pakhomov @meowzz I am not sure whether I correctly understood all the definition that you proposed. Nevertheless the idea of making the analogue of Veblen functions by starting from $x\longmapsto \aleph_x$ instead of $x\longmapsto \omega^x$ seems to be pretty straightforward. Although, I am not familiar with any particular works that studied this, I personally have considered an analogue of Veblen functions based on $x\longmapsto \beth_x$ when I was studying superintuitonistic propositional logics that require frames of large cardinality for Kripke completeness (this never got to a publication).
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May 29, 2020 at 22:28
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May 27, 2020 at 10:23 comment added user820789 @Wojowu Based on your feedback & this meta question I have updated my question. [Aside: I'm a big fan of your Higher order set theory]
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May 27, 2020 at 9:53 comment added Wojowu I'm afraid MathOverflow is not the place to ask about general thoughts on some idea of yours (see this meta discussion), though if you have some specific questions it will be more on-topic. As to how far it reaches - this is hard to answer, because there aren't any common notations which reach cardinals this large, bar any variations of the one you have just presented. No notation of this kind can reach an inaccessible, nearly by definition. High level indication is that ZFC proves such notations have limits, but it can't prove inaccessibles exist.
May 27, 2020 at 9:39 comment added user820789 @Wojowu Is there any problems with this notation & how far does this reach?
May 27, 2020 at 9:34 comment added Wojowu What is your question?
May 27, 2020 at 8:30 review First posts
May 27, 2020 at 10:12
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