Timeline for What's the definition of a mouse in Mitchell's handbook article "the covering lemma"?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Dec 4, 2023 at 21:58 | comment | added | Reflecting_Ordinal | @FarmerS Thank you for your good explanation. It's painful that the definition of mouse is always changing. Does it have an official version now? | |
| Dec 3, 2023 at 22:27 | comment | added | Farmer S | $\gamma=(\nu^+)^{\mathrm{Ult}(M|\gamma,E)}$ where $\nu$ is the "natural length" of $E$; the natural length is the least ordinal $\geq(\kappa^{+})^{M|\gamma}$ which contains all the generators of $E$, where $\kappa$ is the critical point of $E$. This is Mitchell-Steel indexing. Another natural alternative is $\lambda$-indexing, where $\gamma=(\lambda^{+})^{\mathrm{Ult}(M|\gamma,E)}$ where $\lambda$ is the image of $\kappa$ under the $E$-ultrapower embedding. | |
| Dec 3, 2023 at 12:52 | comment | added | Farmer S | Using $\kappa^{++}$ is more generalizable than is $\kappa^+$. For example for models $M$ with two distinct total normal measures $E,F$ with the same critical point $\kappa$, we would need to set them to be $\mathcal{U}_\gamma$ and $\mathcal{U}_\delta$ for some distinct ordinals $\gamma,\delta$. So we couldn't use $\gamma=\kappa^{+M}=\delta$. A very natural indexing is to instead use $\gamma=\kappa^{++\mathrm{Ult}(M,E)}$ and $\delta=\kappa^{++\mathrm{Ult}(M,F)}$. This also generalizes well to models with extenders, not just measures, on their sequence, indexing with... | |
| Dec 3, 2023 at 0:50 | comment | added | Reflecting_Ordinal | @FarmerS In 3.24 the author did noticed the $\mathcal{U}_\gamma$ predicate. However, in "inner models and large cardinals", definition 4.1(b), the index of the measure is required to be $\kappa^{+}$, not $\kappa^{++}$. Why there is such a difference? | |
| Nov 29, 2023 at 18:56 | comment | added | Joel David Hamkins | That makes sense, but I haven't seen that distinction before, whereas I do often see $f\upharpoonright A$ for the domain restriction case $A\subseteq X$. | |
| Nov 29, 2023 at 18:46 | comment | added | Alec Rhea | @JoelDavidHamkins Ah, I suppose in accordance with the notation $f:X\to Y$ for a function I’ve been using $\upharpoonleft$ for domain restriction and $\upharpoonright$ for codomain restriction, so the little harpoon hook points to the side being restricted. | |
| Nov 29, 2023 at 17:27 | comment | added | Hanul Jeon | Have you checked the Chapter "Beginning Inner Model Theory" also written by Mitchell? | |
| Nov 29, 2023 at 14:43 | comment | added | Farmer S | (However, I think there is a condition missing. There would usually be the requirement that if $\mathcal{U}_\gamma\neq\emptyset$ then $\gamma=\kappa^{++\mathrm{Ult}(J_\gamma[\mathcal{U}],\mathcal{U}_\gamma)}$, whereas in 3.24 it just seems to say that $\gamma=\kappa^{++J_\gamma[\mathcal{U}]}$. Of course the latter literally means that $J_\gamma[\mathcal{U}]\models$"$\kappa^+$ is the largest cardinal".) | |
| Nov 29, 2023 at 14:40 | comment | added | Farmer S | The kind of thing that $\mathcal{U}_\gamma$ is, is discussed in 3.24.1.(a),(b)... Are you looking at that? | |
| Nov 29, 2023 at 13:47 | comment | added | Reflecting_Ordinal | @GabeGoldberg I also guess this, but he didn't give any definition about $\mathcal{U}$... | |
| Nov 29, 2023 at 13:34 | comment | added | Gabe Goldberg | I guess it is implicit that $\mathcal U$ is a sequence of objects indexed by ordinals. | |
| Nov 29, 2023 at 13:08 | comment | added | François G. Dorais | Perhaps the op is actually asking about $\mathcal{U}$? | |
| Nov 29, 2023 at 11:06 | comment | added | Joel David Hamkins | @AlecRhea In my experience, $\upharpoonright$ is most often used for domain restriction, not codomain restriction. But more generally, it is often used for restriction of any kind. | |
| Nov 29, 2023 at 10:53 | history | edited | Sam Sanders | CC BY-SA 4.0 |
Small typos
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| Nov 29, 2023 at 3:59 | comment | added | Alec Rhea | $\upharpoonright$ usually denotes codomain restriction for functions, but I’m not sure about the other. | |
| Nov 29, 2023 at 1:57 | history | asked | Reflecting_Ordinal | CC BY-SA 4.0 |