Effect of large cardinals on the value of $\omega_1^L$ in $L$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T05:59:16Z http://mathoverflow.net/feeds/question/110877 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110877/effect-of-large-cardinals-on-the-value-of-omega-1l-in-l Effect of large cardinals on the value of $\omega_1^L$ in $L$ Nate Ackerman 2012-10-28T05:47:52Z 2012-10-28T13:19:43Z <p>I have three three questions, the first two of which probably have the same answer and the third of which is more vague. </p> <p>For a set $A$ let $L_\alpha(A)$ be the constructible universe up to $\alpha$, built from $A$ as a set (and not a predicate). Further let $X = (B, f)$ where $B$ is a transitive set and $f$ is a bijection from $\omega$ to $B$. </p> <p>Also assume that the background universe has whatever large cardinals you would like (or that would be helpful). In particular though there is at least one inaccessible cardinal in $L$. </p> <p>(1) Suppose $L_\alpha\models ZFC$. Is it the case that $\omega_1^L = \omega_1^{L_\alpha}$?</p> <p>(2) Suppose $L_\alpha(X)\models ZFC$. Is it the case that $\omega_1^{L(X)} = \omega_1^{L_\alpha(X)}$?</p> <p>(3) If the answer to (1), (2) is yes, is there any simpler way for $L_\alpha$ to know that $\omega_1^{L_\alpha} = \omega_1^L$ (other than $L_\alpha\models ZFC$)?</p> <p>Finally I will just make one observation to highlight why this question isn't trivial. If you replace $ZFC$ with $KP$ then there are many countable admissible sets $L_\alpha\models KP$ with countable (in $L$) ordinals $\beta\in L_\alpha$ such that $L_\alpha \models \omega_1 = \beta$. </p> <p>Thanks</p> http://mathoverflow.net/questions/110877/effect-of-large-cardinals-on-the-value-of-omega-1l-in-l/110880#110880 Answer by Andres Caicedo for Effect of large cardinals on the value of $\omega_1^L$ in $L$ Andres Caicedo 2012-10-28T06:07:23Z 2012-10-28T06:07:23Z <p>The answer is no. If there is a transitive set model $M$ of set theory (and this is all you need), then if $\alpha$ is its height (that is, if $\alpha=\mathsf{ORD}^M$), then $L_\alpha$ is a model of $\mathsf{ZFC}+V=L$. Note that the assumption is strictly stronger than the existence of an $\omega$-model of $\mathsf{ZFC}$, which in turn is strictly stronger than the mere consistency of $\mathsf{ZFC}$, but it is strictly weaker than the existence of inaccessible cardinals. </p> <p>Now work in $L$, and consider a countable elementary substructure $Y\preceq L_\alpha$. Note that $Y$ is well-founded and satisfies $V=L$. Its collapse is then $L_\beta$ for some $\beta$ that, by necessity, is countable in $L$.</p> <p>(Note that this also addresses (2), by taking $X=\emptyset$.)</p> <hr> <p>By the way, this highlights some of the difficulties a challenger of "$V=L$" must face. Even if there are no inaccessibles, the model $M$ we began with may perfectly well be constructible, and satisfy that there are supercompact cardinals, or whatever. About this, you may also want to look at <a href="http://jdh.hamkins.org/multiverse-perspective-on-constructibility/" rel="nofollow">this post</a> by Joel. </p> http://mathoverflow.net/questions/110877/effect-of-large-cardinals-on-the-value-of-omega-1l-in-l/110902#110902 Answer by Andreas Blass for Effect of large cardinals on the value of $\omega_1^L$ in $L$ Andreas Blass 2012-10-28T13:19:43Z 2012-10-28T13:19:43Z <p>Andres Caicedo has answered Questions 1 and 2 as stated, but, just in case you intended <code>$\kappa&gt;\omega_1^L$</code> in Question 1 (since that's obviously necessary for the proposed conclusion), let me add that this is also sufficient for an affirmative answer to Question 1. The point is that, if an ordinal $\alpha$ is countable in $L$ then a bijection between $\alpha$ and $\omega$ appears in the construction of $L$ well before stage <code>$\omega_1^L$</code>. This is essentially contained in Gödel's proof of GCH in $L$. It uses much less than the full strength of ZFC in <code>$L_\kappa$</code>; you just need enough set theory to talk sensibly about countability of ordinals.</p>