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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 $M_0\vDash ZFC+CON(ZFC)+\neg TM$:

  1. $M_0\vDash CON(ZFC)$ because $CON(ZFC)$ is absolute.
  2. 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.

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    $\begingroup$ Related: math.stackexchange.com/questions/265379/… $\endgroup$ – GME Feb 19 '16 at 0:05
  • $\begingroup$ We can talk about $o(M)$ only if $M$ is transitive. Our $M$ need not be transitive and might contain no ordinals (though $M$ thinks it has ordinals, such 'ordinals' are not ordinals externally.) $\endgroup$ – Hanul Jeon Feb 19 '16 at 13:13
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It is enough to show that under the hypothesis, there is a model of $ZFC + \mathop{Con}(ZFC) +\neg TM$. Now take $V$ be a model of $ZFC + \mathop{Con}(ZFC)$. If it is already a model of $\neg TM$, you are done. So we may assume that $V$ is a model of $ZFC + \mathop{Con}(ZFC) + TM$.

Your argument using Lövenheim-Skolem looks a bit confusing to me. Actually, once you have a transitive (set) model, that least ordinal immediately exists. (Actually, I think there is no reason for $V$ above to be transitive.) To make things neat you should now work inside $V$. There, take a transitive model $M$. As confirmed by GME's comment, $ \mathop{Con}(ZFC) $ is arithmetical and thus it descends from $V$ to $M$. Now you can perform the reasoning on minimality of $o(M_0)$.

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  • $\begingroup$ Probably I would have say that $TM$ is "There exists a standard and transitive model of ZFC" $\endgroup$ – Ergonvi Feb 20 '16 at 19:06
  • $\begingroup$ @Ergonvi That's the usual interpretation. But there The first model $M$ that you invoke might not be standard nor transitive, despite it satisfies $TM$. (Or perhaps you are using the same $M$ for two different things, I don't know.) $\endgroup$ – Pedro Sánchez Terraf Feb 20 '16 at 20:34

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