From my undestanding a standard method of deducing relative consistency results is the following:

By a combination of the Levy reflection principle, Skolem-Hulls and Mostowski Collapse we show:

If $ZFC$ is consistent, then "$ZFC$, $M$ is transitive and countable, $ZFC^M$" is consistent, where $M$ is a new variable.

So we can assume there is a countable transitive Model $M$ of $ZFC$ form the forcing extension $M[G]$ and show $M[G]\models ZFC+\phi$

But why do we have to invoke the Reflection principle, what is for example wrong with the following reasoning:

Assume $ZFC$ is consistent, then by the Löwenheim-Skolem theorem there is a countable model $N$ of $ZFC$, taking its Mostowski transitivisation we get a transitive, countable model $N'$ of $ZFC$ and thus can construct $N'[G]$. So the reflection principle becomes superfluos? Or is there something fundamentaly wrong with my understanding of the former method in the first place?

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    $\begingroup$ Although the consistency of ${\sf T}$ implies it has a model, it doesn't in general imply that it has a well-founded model (which is required to apply Mostowski's lemma). $\endgroup$ – GME Oct 28 '15 at 17:40
  • $\begingroup$ The "statement" in line 3 is not a sentence. Please rewrite. $\endgroup$ – Andrés E. Caicedo Oct 28 '15 at 22:11
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    $\begingroup$ Line 5 also makes no sense as currently written. What is $\phi$? $\endgroup$ – Andrés E. Caicedo Oct 28 '15 at 22:19
  • $\begingroup$ $\phi$ is the statement whose relativ consistency i want to show. $\endgroup$ – Achilles Oct 29 '15 at 5:17

No, you're quite missing a point there.

From "$\sf ZFC$ is consistent" to "$\sf ZFC$ has a countable transitive model" there is a leap in consistency strength. In fact it might be the case that $\sf ZFC$ is consistent, but there is no model of $\sf ZFC$ which agrees with the universe on the natural numbers.

Using the reflection principle we now pick a large enough $\theta$ such that $V_\theta$ satisfies some large enough (but finite) fragment of $\sf ZFC$, now we use Lowenheim-Skolem to produce a countable submodel, and the Mostowski collapse to make it transitive. Then we use forcing or whatever to produce a model of that fragment of $\sf ZFC$ and $\lnot\phi$.

Since we can do this for arbitrarily large finite fragments of $\sf ZFC$, it cannot be the case that $\sf ZFC$ proves $\phi$, otherwise there would be a $\theta$ where the above has to fail. This last part is a meta-theoretic statement, though. And that distinction is important.

I do think, however, that for pedagogical reasons one should think about countable transitive models of $\sf ZFC$, at least until one is comfortable with the distinction between theory, meta-theory and the theory of the finite fragment's model.


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