Timeline for What are the difficulties involved in proving that the Kunen inconsistency holds in $NGB$
Current License: CC BY-SA 4.0
21 events
| when toggle format | what | by | license | comment | |
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| Oct 30, 2018 at 20:58 | comment | added | Gabe Goldberg | Let us continue this discussion in chat. | |
| Oct 30, 2018 at 17:21 | comment | added | Thomas Benjamin | But either way (as long as $V_{\lambda + 1}^{\sharp}$ exists) $L$($V_{\lambda + 1}$) is 'very far from $V$', correct? | |
| Oct 30, 2018 at 14:41 | comment | added | Gabe Goldberg | $I_0^\#$ at $\lambda$ is precisely the statement that $V^\#_{\lambda+1}$ exists and is an Icarus set. But if $\lambda$ is least such that $I_0$ holds at $\lambda$ and $V_{\lambda+1}^\#$ exists, then $I_0^\#$ fails at $\lambda$ by Cramer's results. | |
| Oct 30, 2018 at 13:27 | comment | added | Thomas Benjamin | So if $V_{\lambda + 1}^{\sharp}$ exists, it is an Icarus set (since $V_{\lambda + 1}^{\sharp}$ $\subseteq$ $V_{\lambda + 1}$)? | |
| Oct 30, 2018 at 12:48 | comment | added | Gabe Goldberg | andrescaicedo.files.wordpress.com/2008/04/sharps.pdf | |
| Oct 30, 2018 at 12:45 | comment | added | Thomas Benjamin | Thanks. Caicedo's "A review of sharps" can be found online, correct? | |
| Oct 30, 2018 at 12:29 | comment | added | Gabe Goldberg | Yes $V_{\lambda+1}^\#$ can be viewed as a subset of $V_{\lambda+1}$. It is the theory of $\omega$-many Silver indiscernibles of $L(V_{\lambda+1})$ with parameters from $V_{\lambda+1}$. You should look at Caicedo's "A review of sharps." | |
| Oct 27, 2018 at 4:20 | comment | added | Thomas Benjamin | (cont.) Also, for the above $X$=$V_{\lambda + 1}^{\sharp}$ in the first line of the second part of my comment. | |
| Oct 27, 2018 at 4:12 | comment | added | Thomas Benjamin | (cont.) (That is, $V_{\lambda + 1}^{\sharp}$ contains the 'truth' of $L$($V_{\lambda + 1}$), so it cannot be in $L$($V_{\lambda + 1}$)). | |
| Oct 27, 2018 at 4:03 | comment | added | Thomas Benjamin | (cont.) $X$ $\subset$ $V_{\lambda + 1}^{\sharp}$. This means that for $V_{\lambda + 1}^{\sharp}$ to be Icarus, $V_{\lambda + 1}^{\sharp}$ $\subseteq$ $V_{\lambda + 1}$. Considering that $V_{\lambda+ 1}^{\sharp}$ is, (intuitively, according to Dimonte from his slide presentation, "Non proper elementary embeddings beyond $L$($V_{\lambda + 1}$)," found on his website), "$X^{\sharp}$ contains the 'truth' of $L$($X$), so it cannot be in $L$($X$)." If this is the case then how can $V_{\lambda + 1}^{\sharp}$ $\subseteq$ $V_{\lambda + 1}$? | |
| Oct 27, 2018 at 0:55 | comment | added | Thomas Benjamin | Here is the defnition of Icarus set from Cramer's "Implications of very large cardinals", pg. 4: "Suppose $X$ $\subseteq$ $V_{\lambda + 1}$. We say that $X$ is Icarus if or $I_{0}$($X$) holds if there exists a non-trivial elementary embedding $j$: $L$($X$, $V_{\lambda + 1}$) $\rightarrow$ $L$($X$, $V_{\lambda + 1}$) such that $crit$($j$) $\lt$ $\lambda$. Now look above at (4) "$I_{0}^{\sharp}$ holds at $\lambda$". If you substitute $X$ for $V_{\lambda+ 1}^{\sharp}$, you (seemingly) get the definition of Icarus set except that for the Icarus set $X$, | |
| Oct 26, 2018 at 21:59 | comment | added | Gabe Goldberg | Does Icarus mean that there is a $j :L(V_{\lambda+1},V_{\lambda+1}^\#)\to L(V_{\lambda+1},V_{\lambda+1}^\#)$ with $\text{crt}(j) < \lambda$? If so, the answer is that this is probably consistent, but it is stronger than $I_0$. | |
| Oct 26, 2018 at 15:32 | comment | added | Thomas Benjamin | For Theorem 3.9, can $V_{\lambda +1}^{\sharp}$ be an Icarus set? | |
| Oct 24, 2018 at 15:04 | comment | added | Gabe Goldberg | It is in "Inverse limit reflection and the structure of $L(V_{\lambda+1})$." The result is Theorem 3.9. | |
| Oct 24, 2018 at 13:58 | comment | added | Thomas Benjamin | By the way, I have started reading some of Cramer's papers--what paper of his contains the result you mentioned in your comment? | |
| Oct 23, 2018 at 23:08 | comment | added | Gabe Goldberg | Cramer showed that if $I^\#_0$ holds (i.e., there is a $j :L(V_{\lambda+1}^\#)\to L(V_{\lambda+1}^\#)$ with sup of critical sequence $\lambda$) then $I_0$ holds in $V_\lambda$. If you do the AC forcing and carry out Cramer's proof in DC then you get that $\text{ZF + DC + }I_0^\#$ implies $\text{Con(ZFC + }I_0)$. The fact that a Reinhardt implies $I_0^\#$ requires proving $V_{\lambda+1}^\#$ exists from a Reinhardt. In fact a Reinhardt implies every set has a sharp, another result of mine from that paper (which is still unpublished). | |
| Oct 23, 2018 at 17:32 | comment | added | Thomas Benjamin | Also, are there weaker assumptions than $NGB$ + $DC$ + $j$: $V$ $\rightarrow$ $V$ (the weaker assumptions being in the realm of Icarus sets, so to speak) that would imply $Con$($ZFC$ + $I0$)? | |
| Oct 23, 2018 at 17:23 | comment | added | Thomas Benjamin | By the way, has your proof of $NBG$ + $DC$ + $j$: $V$ $\rightarrow$ $V$ implies $Con$($ZFC$ + $I0$) been published somewhere? I would definitely like to read your paper. Thanks in advance. | |
| Oct 23, 2018 at 17:09 | vote | accept | Thomas Benjamin | ||
| Oct 15, 2018 at 14:53 | comment | added | Thomas Benjamin | Thanks for the answer, it is very much appreciated. | |
| Oct 15, 2018 at 14:49 | history | answered | Gabe Goldberg | CC BY-SA 4.0 |