Timeline for Large cardinals without replacement
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Feb 24, 2021 at 10:55 | comment | added | Gro-Tsen | I'm afraid I easily get lost in the twisty maze of theories all alike, but the reference here is Mathias's paper “The Strength of Mac Lane Set Theory”. Let's just say I was mostly suggesting a look at the theory $\mathsf{KP}^{\mathcal{P}}$ (in the terminology of this paper), which is somewhat better behaved than $\mathsf{Z}$ and presumably admits an interesting hierarchy of “small large” cardinals (whose existence in $\mathsf{ZFC}$ would be probable). | |
| Feb 24, 2021 at 0:27 | comment | added | Tim Campion | @Gro-Tsen Thanks, that's good to know. As is probably apparent, I don't really know enough set theory to say with confidence that ZC is the base theory I'm "really" interested in. Just to be sure I have it straight, we have that ZC interprets KP+Powerset and KP+Powerset interprets BZC=Mac Lane, but between KP and Mac Lane neither one "directly" interprets the other, right? | |
| Feb 24, 2021 at 0:22 | comment | added | Gro-Tsen | (contd.) … So I am tempted to believe that there should be notions like “power-recursively inaccessible”, “power-recursively Mahlo”, etc., waiting to be studied. AFAICT this hasn't received any attention, but it could be an interesting hierarchy to look at. | |
| Feb 24, 2021 at 0:19 | comment | added | Gro-Tsen | This is probably not what you had in mind, but just in case: if we weaken ZC even more and restrict Separation to Δ₀ formulae, but add Δ₀ replacement, we get Kripke-Platek (KP) + powerset. Now KP has its own hierarchy of “recursively large ordinals” (recursively inaccessible, recursively Mahlo, nonprojectible, etc.), which reflect the “small” large cardinals. The theory Power-KP obtained by adding a powerset function (rather than axiom) to KP has been studied, and seems to have much in common with KP (its transitive models are called “power-admissible” sets). … (contd.) | |
| Feb 24, 2021 at 0:02 | vote | accept | Tim Campion | ||
| Feb 23, 2021 at 23:13 | answer | added | Dmytro Taranovsky | timeline score: 6 | |
| Feb 15, 2021 at 3:26 | history | became hot network question | |||
| Feb 14, 2021 at 20:24 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Feb 14, 2021 at 20:00 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Feb 14, 2021 at 19:47 | answer | added | Zuhair Al-Johar | timeline score: 5 | |
| Feb 14, 2021 at 19:26 | history | asked | Tim Campion | CC BY-SA 4.0 |