Timeline for Preserving $\omega_1$ is Inaccessible to the reals

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Jun 30, 2015 at 20:17	comment	added	William		It appears that the iteration of all the forcing you used will be proper. Schindler showed that if $L[G]$ is the Levy collapse extension of a remarkable cardinal, then every real added by proper forcing extension of $L[G]$ is generic over $L$ for a forcing of size less than the remarkable cardinal. This appears to imply that if $x$ is a real added by a proper forcing extension of $L[G]$, then $\omega_1^{L[x]} < \omega_1$. It seems that $V$, you started with, could not be this $L[G]$. I think there should be some assumption on your ground model.
Jun 30, 2015 at 20:07	comment	added	William		Regarding your comment to my answer: The question was to find a ground model $V$ satisfying $\omega_1$ is inaccessible to the reals and a $\omega_1$ preserving forcing such that $V[G]$ does not satisfy $\omega_1$ is inaccessible to the reals. In your answer, you made some assumptions on the structure of your ground model $V$. I am concerned that without assuming $V$ has very weak large cardinals, you may not be able to get such a ground model $V$ which will also satisfy $\omega_1$ is inaccessible to the reals.
Jun 30, 2015 at 4:07	history	answered	Mohammad Golshani	CC BY-SA 3.0