Timeline for Is there a second-order expression of '$\kappa$ is Reinhardt' in $NGB$, where $\kappa$ is a cardinal?
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| Sep 29, 2020 at 20:35 | comment | added | Thomas Benjamin | Why the downvote? | |
| Feb 28, 2020 at 21:19 | comment | added | Thomas Benjamin | (cont.) If one uses Prof. Goldberg's assertion that "$NGB$ proves that for all classes $C$ [including $j$--my comment], the first-order replacement schema holds relative to $C$ [and $j$ as well--my comment also], one can see that the last clause of the Wholeness Axiom, i.e., "that for all sets $A$, $j$ $\upharpoonright$ $A$ is a set" holds as well. | |
| Feb 28, 2020 at 21:02 | comment | added | Thomas Benjamin | (cont.) If one substitutes $V$ for $M$ in the definition and $j$ for $X$ and one assumes $j$ satisfies the Wholeness Axiom, one can infer that $j$ is not definable in $V$ so $j$ is not first-order definable and therefore must be a proper class. | |
| Feb 28, 2020 at 20:44 | comment | added | Thomas Benjamin | For reference, I have decided to include the definition of 'weakly definable' (the definition is essentially footnote 6 on pg. 246 of Corazza's paper, "The Wholeness Axiom", found under title on the Web). The definition is as follows: "Suppose ($M$, $\in$) is a model of $ZF$ and $X$ $\subseteq$ $M$. Then $X$ is weakly definable in $M$ if the expanded model ($M$, $\in$, $X$) satisfies all instances of Replacement for formulas in the expanded language. It is straightforward to show that whenever $X$ is definable in $M$, it also must be weakly definable in $M$." | |
| Feb 28, 2020 at 16:35 | comment | added | Thomas Benjamin | (cont.) all imply "that for all classes $C$, the first-order replacement schema holds relative to $C$" have any relavence for $j$ when $WA_{\infty}$ is a class axiom of any of the three theories, except to show "that for all sets $A$, $j$ $\upharpoonright$ $A$ is a set"? | |
| Feb 28, 2020 at 16:24 | comment | added | Thomas Benjamin | (cont.) holds relative to $C$", consider the following formulation of the Wholeness axiom, "There is a nontrivial elementary embedding $j$: $V$ $\rightarrow$ $V$ that is not weakly definable in $V$, such that for all sets $A$, $j$ $\upharpoonright$ $A$ is a set" (it should be understood that this axiom holds for the full Separation axiom and that treating $WA_{\infty}$ as a class axiom in $NGB$, $NGB$ + $AC$, and $NGBC$ causes each of the three theories to treat $j$ as a proper class). Given this formulation (and its proper understanding), how does the the fact that $NGB$, $NGB$ + $AC$, | |
| Feb 28, 2020 at 15:59 | comment | added | Thomas Benjamin | @GabeGoldberg: It isn't any different. The beauty of having $WA_{\infty}$ as a class axiom in $NGB$ (or in $NGB$ + $AC$ or in $NGBC$--the previous comment in square brackets should read, "[through which $ZFC$($j$) is an intermediate subtheory of $NGB$ +$AC$--my comment]") is that $NGB$, $NGB$ + $AC$, and $NGBC$ treats $j$ as a proper class (that way it is not weakly definable in $V$) and so avoids the problem of why $ZFC$ + $I_3$ is of greater consistency strength than $ZFC$ + $WA_{\infty}$. As regards your comment, "$NGB$ proves that for all classes $C$, the first-order replacement schema | |
| Feb 21, 2020 at 5:00 | comment | added | Gabe Goldberg | I don't see why the statement of the Wholeness Axiom, when formulated in second order set theory, is any different from the statement that there is a Reinhardt cardinal... NBG proves that for all classes $C$, the first-order replacement schema holds relative to $C$. | |
| Feb 20, 2020 at 22:41 | comment | added | Thomas Benjamin | @GabeGoldberg: It is the Wholeness Axiom holding for the full Separation axiom in the language {$\in$, $j$} [which is an intermediate subtheory of $NGB$ = $AC$--my comment]. See Prof Hamkins' paper "The Wholeness axioms and $V$ = $HOD$", arXiv:math/9902079v1 [math.LO] 13 Feb 1999. | |
| Feb 20, 2020 at 2:45 | comment | added | Gabe Goldberg | What's $\text{WA}_\infty$? | |
| Feb 19, 2020 at 23:58 | comment | added | Thomas Benjamin | @GabeGoldberg: The arXiv preprint link was part of David Robert's fine edit (thanks, Prof. Roberts). Thanks for pointing it out as well, Prof. Goldberg. The trick is, I presume, is to use Gaifman's Lemma (Lemma 2 in both papers) to show that "a class $j$ from $V$ to $V$ that is an elementary embedding is a fully elementary embedding? Can one then use Gaifman's Lemma to show the following relative consistency results: "If $Con$($NGB$) then $Con$($NGB$ + $WA_{\infty}$)", "If $Con$($NGBC$) then $Con$($NGBC$+ $WA_{\infty}$)"? | |
| Feb 19, 2020 at 2:58 | comment | added | Gabe Goldberg | I'm referring to the preprint on arXiv, which you linked to in the question. The only problem in formulating the axiom is to assert that a class $j$ is an elementary embedding from $V$ to $V$, since it seems to require a truth predicate. Once you know the trick for doing this, you just say "there is a class $j$ that is an elementary embedding from $V$ to $V$." The trick is explained in great detail in "Generalizations of the Kunen Inconsistency." | |
| Feb 18, 2020 at 22:44 | comment | added | Thomas Benjamin | (cont.) quantifying only over sets, allowing finitely many class parameters" (this is also on the third page of the published version), it may be able to escape the fact that in ($ZFC$, $j$), the wholeness axiom is below $I_{3)$ in consistency strength because ($ZFC$, $j$) is not a class theory.. | |
| Feb 18, 2020 at 22:38 | comment | added | Thomas Benjamin | @GabeGoldberg: Page 5? In the version published in Annals of Pure and Applied Logic, the quote that got rid of the downvote was on the third page of the published version. Are you referring to page 5 in some preprint version of the paper? If so, then could you supply a link to the preprint in question? I would be much obliged if you would. Thanks. I am also interested in what would happen if one added the Wholeness axiom as a class axiom of $NGB$ ($NGBC$ as well). Because "$NGB$ includes the replacement...axiom only for formulas having only first-order quantifiers, that is, | |
| Feb 16, 2020 at 23:27 | comment | added | Gabe Goldberg | The answer is essentially contained in "Generalizations of the Kunen Inconsistency" starting at the last paragraph of page 5 and continuing until the end of the metamathematical preliminaries. | |
| Feb 13, 2020 at 3:08 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
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| Feb 13, 2020 at 0:23 | history | edited | Thomas Benjamin | CC BY-SA 4.0 |
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| Feb 13, 2020 at 0:13 | history | edited | Thomas Benjamin | CC BY-SA 4.0 |
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| Feb 6, 2020 at 22:46 | comment | added | Thomas Benjamin | Why the downvote? | |
| Feb 6, 2020 at 3:40 | review | Close votes | |||
| Feb 13, 2020 at 3:05 | |||||
| Feb 6, 2020 at 0:43 | history | asked | Thomas Benjamin | CC BY-SA 4.0 |