User ant emyy lee - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T09:40:03Z http://mathoverflow.net/feeds/user/3692 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19014/finding-questions-between-functional-analysis-and-set-theory Finding questions between functional analysis and set theory Ant emyy Lee 2010-03-22T14:05:20Z 2013-04-18T23:33:01Z <p>Are there some good questions on functional analysis whose solution depends on tools in set theory? My major is mathematical logic, I think tools in set theory, especially infinity combinatorics and forcing, should be used to solve some questions in functional analysis. For functional analysis, I just have read the main part of Conway's textbook. In this book, I have not found such questions.</p> http://mathoverflow.net/questions/14883/a-question-about-diagonal-prikry-forcing a question about diagonal prikry forcing Ant emyy Lee 2010-02-10T09:39:26Z 2013-03-29T13:13:12Z <p>Suppose &lt;\kappa_n|n&lt;\omega> is a strictly increasing sequence of measurable cardinals, </p> <p>\kappa is the limit of this sequence. For each n&lt;\omega, U_n is a normal measure on </p> <p>\kappa_n. P is the diagonal Prikry forcing corresponding to \kappa_n's and U_n's. Suppose g is P-generic sequence over V. We have known that for each strictly increasing </p> <p>sequence x of length \omega such that each x(i)&lt;\kappa_i and x\in{V}, x is eventually </p> <p>dominated by g. In V[g], suppose A is a subset of \kappa, A is not in V. Is there a strictly </p> <p>increasing sequence y of length \omega such that each y(i)&lt;\kappa_i and y\in{V[A]}, y is not </p> <p>eventually dominated by g?</p> <p>(g can eventually dominate all such sequences in V, V[A] is greater than V, I feel g can not </p> <p>eventually dominate all such sequences in V[A].)</p> http://mathoverflow.net/questions/68533/on-the-independence-of-the-kurepa-hypothesis On the independence of the Kurepa Hypothesis Ant emyy Lee 2011-06-22T16:38:52Z 2011-07-31T18:34:37Z <p>Kurepa Hypothesis says there is a Kurepa tree, which is a $\omega_1$-tree has at least $\omega_2$ many branches. It is known that beginning from a model with an inaccessible cardinal $\kappa$, after collapes $\kappa$ to $\omega_2$ using the Levy collape, then in the generic extension, Kurepa Hypothesis fails. In above generic extension, $\omega_2$ is equal to $\kappa$ and by a counting argument for nice names, $2^{\omega_1}=\omega_2$. My question is that "is it consistent that Kurepa Hypothesis fails and $2^{\omega_1}>\omega_2$?" (The reason I think this question: The biggest possible value of the number of branches is $2^{\omega_1}$, so in the environment of $2^{\omega_1}=\omega_2$, it is most difficult for the living of a Kurepa tree. So I want to know whether this requiement is necessary.)</p> http://mathoverflow.net/questions/13773/a-question-on-ultrapower A question on ultrapower Ant emyy Lee 2010-02-02T02:13:05Z 2010-02-11T00:17:33Z <p>Suppose $\kappa_0$ is a measurable cardinal and $\mu_0$ is a normal measure on $\kappa_0$. $M_1$ is the transitive collapse of $Ult(V,\mu_0)$, $j_{0,1}:V\rightarrow{M_1}$ is the elementary embedding induced by the ultrapower. In $M_1$, $\kappa_1=j_{0,1}(\kappa_0)$ is a measurable cardinal and $\mu_1$ is a normal measure on $\kappa_1$ in $M_1$ such that $\mu_1$ is not in the range of $j_{0,1}$. $M_2$ is the transitive collapse of $Ult(M_1,\mu_1)$, $j_{1,2}:M_1\rightarrow{M_2}$ is the elementary embedding induced by the ultrapower. $j_{0,2}=j_{1,2}\circ{j_{0,1}}$.</p> <p>Is it true that: Suppose $N$ is an inner model, $i:V\rightarrow{N}$ and $k:N\rightarrow{M_2}$ are elementary embeddings such that $k\circ{i}=j_{0,2}$. Then $k''N=j_{0,2}''V$ or $k''N=j_{1,2}''M_1$ or $k''N=M_2$''?</p> http://mathoverflow.net/questions/68533/on-the-independence-of-the-kurepa-hypothesis/68538#68538 Comment by Ant emyy Lee Ant emyy Lee 2011-06-24T14:12:32Z 2011-06-24T14:12:32Z I find a paper: Random trees under CH James Hirschorn Israel Journal of Mathematics, 2007, Volume 157, Number 1, Pages 123-153 I just read its abstract, he says there is a model of $\neg{KH}$, and we can add many random reals and preserve $\neg{KH}$. But I think $Add(\omega,\omega_1)$ does not add Kurepa tree over Silver model is also a question. http://mathoverflow.net/questions/68533/on-the-independence-of-the-kurepa-hypothesis Comment by Ant emyy Lee Ant emyy Lee 2011-06-24T05:00:14Z 2011-06-24T05:00:14Z I see it.Thanks. http://mathoverflow.net/questions/68533/on-the-independence-of-the-kurepa-hypothesis/68538#68538 Comment by Ant emyy Lee Ant emyy Lee 2011-06-23T07:32:02Z 2011-06-23T07:32:02Z Thank you for your answer, Joel. Another thing is unknown for me: In Silver's model, no new real has been added, and inaccessible $\kappa$ is collaped to $\omega_2$, so Continumm hypothesis holds in this model. But if adding some Cohen reals to this model, maybe some new Kurepa tree are created. This should be my second question: is it consistent that &quot;Kurepa hypothesis fails and $2^{\omega_1}&gt;\omega_2$ and $2^{\omega}&gt;\omega_1$&quot;? http://mathoverflow.net/questions/68533/on-the-independence-of-the-kurepa-hypothesis Comment by Ant emyy Lee Ant emyy Lee 2011-06-23T05:10:45Z 2011-06-23T05:10:45Z sorry,I do not know what is community wiki. So I just choose it. http://mathoverflow.net/questions/19014/finding-questions-between-functional-analysis-and-set-theory/19015#19015 Comment by Ant emyy Lee Ant emyy Lee 2010-03-22T14:48:01Z 2010-03-22T14:48:01Z Thank you for your solution. Axiom of choice or transfinite induction, I think, they are general tools in math, but not just in logic. http://mathoverflow.net/questions/19014/finding-questions-between-functional-analysis-and-set-theory/19017#19017 Comment by Ant emyy Lee Ant emyy Lee 2010-03-22T14:44:49Z 2010-03-22T14:44:49Z This book should be what I was searching for. Thank you. http://mathoverflow.net/questions/19014/finding-questions-between-functional-analysis-and-set-theory/19016#19016 Comment by Ant emyy Lee Ant emyy Lee 2010-03-22T14:32:24Z 2010-03-22T14:32:24Z en. This is a good example. Also, do you know any questions as this style which is still open now? http://mathoverflow.net/questions/13773/a-question-on-ultrapower/13795#13795 Comment by Ant emyy Lee Ant emyy Lee 2010-02-11T06:53:36Z 2010-02-11T06:53:36Z Thank you for your astonishing solution. http://mathoverflow.net/questions/14883/a-question-about-diagonal-prikry-forcing Comment by Ant emyy Lee Ant emyy Lee 2010-02-11T02:52:12Z 2010-02-11T02:52:12Z This forcing has Prikry property. Similarly to Prikry forcing, if \kappa is the limit of this measurable cardinal, then this forcing does not add any new bounded subset of \kappa. This forcing appears in the chapter &quot;Prikry-type forcing&quot; of Handbook of set theory written by Moti Gitik. It is in the section 1.3 of this chapter. http://mathoverflow.net/questions/14883/a-question-about-diagonal-prikry-forcing Comment by Ant emyy Lee Ant emyy Lee 2010-02-11T02:47:49Z 2010-02-11T02:47:49Z yes. I mean for every A, such that A\subseteq{\kappa} and A\notin{V}, there is such a sequence in V[A] not dominated by g. http://mathoverflow.net/questions/14883/a-question-about-diagonal-prikry-forcing/14932#14932 Comment by Ant emyy Lee Ant emyy Lee 2010-02-11T02:43:17Z 2010-02-11T02:43:17Z Maybe I did not say clearly. You find an A, there is a sequence in V[A], g can not eventually dominate it. But I want to know whether for every A such that A\subseteq{\kappa} and A\notin{V}, there is such a sequence in V[A] not dominated by g. http://mathoverflow.net/questions/14883/a-question-about-diagonal-prikry-forcing Comment by Ant emyy Lee Ant emyy Lee 2010-02-10T16:50:17Z 2010-02-10T16:50:17Z Hi. Every condition of P is a ordered pair (s,F). s is a strictly increasing finite sequence such that each s(i)&lt;\kappa_i. F is a function, dom(F)=\omega, for each i&lt;\omega, F(i)\in{U_i}. (s,F) and (t,H) are two conditions. (s,F) is stronger than (t,H) means: (i) s end extends t; (ii) for each i&lt;\omega, F(i) is a subset of H(i); (iii) for each i, if |t|\leqslant{i}&lt;|s|, s(i)\in{H(i)}. Also, I do not know whether this forcing should be called diagonal prikry forcing''. http://mathoverflow.net/questions/13773/a-question-on-ultrapower/13795#13795 Comment by Ant emyy Lee Ant emyy Lee 2010-02-03T03:20:28Z 2010-02-03T03:20:28Z So the model-theoretic fact you said in your answer may not make sense in this case, since $\mu_1$ is not a normal measure in $V$. How to deal with this case? http://mathoverflow.net/questions/13773/a-question-on-ultrapower/13795#13795 Comment by Ant emyy Lee Ant emyy Lee 2010-02-03T03:20:13Z 2010-02-03T03:20:13Z I am the student exchange you with email, you introduce me to go here, so I register this new name. In my question, I require $\mu_1$ is not in the range of $j_{0,1}$, i.e. the second step of the iterated ultrapower is done by a normal measure not in $j_{0,1}''V$. For example, suppose $\kappa$ has $\kappa$ many normal measures &lt;U_\alpha:\alpha&lt;\kappa&gt;, the first step of the iterated ultrapower I use $U_0$, and the second step of the iteration, I use the \kappa-th measure in the sequence j_{0,1}(&lt;U_\alpha:\alpha&lt;\kappa&gt;) .