I was trying to solve the following exercise from Kunen's new book but I'm not sure if my answer its right or less formal than would it be. The exercise says that assuming the consistency of $ZFC+CON(ZFC)$ prove that $ZFC+CON(ZFC)\nvdash TM$ where $TM$ is the statement "The exists a transitive model of $ZFC$".
My attempt to prove it is the following:
Let's assume that $ZFC+CON(ZFC)$ is consistent and $ZFC+CON(ZFC)\vdash TM$, then there exists models $M$ of $ZFC+CON(ZFC)$ and for every of these models $M\vDash TM$. Now working in $ZFC+TM$ define $o(M)$ as the least ordinal not in $M$ (this kind of ordinals exist because we can obtain, for example, countable models via the Downward Lowënheim-Skolem theorem) and pick $o(M_0)$ the lest ordinal among all of that transitive models of $ZFC$. Now it's enough to see that $M_0\vDash ZFC+CON(ZFC)+\neg TM$:
- $M_0\vDash CON(ZFC)$ because $CON(ZFC)$ is absolute.
- If $M_0\vDash TM$ there would exists a $M'\in M$ such that $M'$ is a transitive model of ZFC with $o(M')\in o(M_0)$ (by transitivity of $M_0$) which is a contradiction.
I'm not pretty sure about the point 1. (in fact I don't know how to prove it) so if something is willing to help me I would be more than gratefull.
Thanks in advance.