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Post-Doctoral Research Fellow, School of Mathematics, Institute for Research in Fundamental Sciences (IPM).

My Shelah number is close to one.


Apr
6
revised The origins of forcing in mathematical logic and other branches of mathematics
added 307 characters in body
Apr
6
answered Forcing as a replacement of induction and diagonal arguments
Apr
5
awarded  Necromancer
Apr
4
answered Is there a “large powerset axiom” so extreme that it disproves the existence of strongly inaccessible cardinals?
Apr
3
awarded  Popular Question
Apr
2
revised Existence of $\kappa$-Suslin trees above a measurable cardinal
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Apr
2
comment Existence of $\kappa$-Suslin trees above a measurable cardinal
It just forces the tree property at $\kappa,$ and in fact it make the weakly compact cardinal in the extension to become $\kappa.$
Apr
1
answered Existence of $\kappa$-Suslin trees above a measurable cardinal
Apr
1
revised Consistency strength of $\aleph_2$-Souslin hypothesis
added 2 characters in body
Apr
1
asked Consistency strength of $\aleph_2$-Souslin hypothesis
Mar
31
comment Can the first ordinal in which $V\neq HOD$ be $\aleph_\omega$?
I think the above idea does not work, the whole forcing is enough homogeneous so it seems $HOD$ of final extension is just the original core model $K.$
Mar
31
answered Antichain on $\mathcal{P}(\omega)/fin$ of cardinality $2^{\aleph_0}$?
Mar
31
comment Can the first ordinal in which $V\neq HOD$ be $\aleph_\omega$?
@AsafKaragila The forcing is definable at least in intermediate submodel, and that's enough.
Mar
31
comment Can the first ordinal in which $V\neq HOD$ be $\aleph_\omega$?
In fact we don't need to add many Prikry sequences, it suffices to add two different ones, such that they are not constructible from each other. This is possible if we consider the $\alpha-$th Prikry sequence added, where $\alpha>\kappa$ is $\kappa_1$ as defined in Def. 2.1 of Merimovich's paper "Prikry on extenders revisited".
Mar
31
comment Can the first ordinal in which $V\neq HOD$ be $\aleph_\omega$?
Consider the paper "Supercompact extender based Prikry forcing" by Merimovich. His forcing is not difficult to see has enough weak homogenerity. By extender based Prikry forcing, I essentially mean a revised version of it, which works like Gitik-Magidor forcing, but has a much simpler presentation. So unlike in Merimovich paper, in this version we just add subsets of $\kappa,$ and preserve all cardinals. We mix it with collapses as in Gitik-Magidor paper. The tail essentially adds the rest of Prikry sequences.
Mar
31
comment Can the first ordinal in which $V\neq HOD$ be $\aleph_\omega$?
What about the following idea: Start with a core model for a strong cardinal $\kappa$. Do extender based Prikry forcing to turn $\kappa$ into $\aleph_{\omega},$ and make $2^{\aleph_\omega}=\aleph_{\omega+2}.$ Consider an intermediate submodel in which the normal Prikry sequence and the collapses are added, so that $\kappa$ becomes $\aleph_\omega$ in it. The tail forcing is enough homogeneous, and it adds no bounded subsets of $\kappa=\aleph_\omega,$ so the tail forcing extension is as required
Mar
30
comment Notions of infinity in $\mathsf{ZF}$ without choice
Now the question has changed and I removed my comment. With this version, the answer is no, as is shown in Asaf's answer.
Mar
30
comment Proofs of the uncountability of the reals.
Welcome to Mathoverflow.
Mar
29
awarded  Nice Question
Mar
28
revised Reference for proof that consistency of $\omega_1$-Erdos cardinal implies Con(Chang's Conjecture)
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