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I have a (seeming) contradiction and I can't seem to figure out the (obvious) mistake in my reasoning. The following (related to my previous question):

(1) $\omega$, defined to be the least infinite ordinal, by definition has the same cardinality in every model of set theory (model of ZF or ZFC I'm not sure it's true in every theory), namely it is countably infinite. I think one can prove the countability by showing that its cardinality is less equals the cardinality of every infinite set (give an injection and apply Cantor-Schroeder-Bernstein). Then use the existence of a countably infinite set which follows from the axiom of infinity.

(2) We can force $\omega$ to be finite! To this end use the Levy collapse to collapse $\aleph_0$ to $42$. Let's call this new model $M[G]$.

Now, presumably, $M[G]$ is no longer a model of ZF(C).

Questions:

1.Is this correct? Is $M[G]$ no longer a model of ZFC?

2.And if yes: why not? Because it doesn't have an element representing the natural numbers?

2.a)If yes: how do I know that it doesn't?

3.And more generally: how do I know, after applying forcing to a model $M$ of ZFC, whether the resulting model $M[G]$ is still a model of ZFC or not?

Thanks for your help!

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    In your previous question, you showed that $\omega$ has the same cardinality in every **transitive** model of ZF(C). It's worth noting that in ZF, $\omega$ isn't necessarily injectable into every infinite set (look up Dedekind-finite, or D-finite, sets for more on this). To show that $\omega$ is countable depends on how you define "countable". If you define it, for example, as "injectable into $\omega$", this is trivial to show.2012-11-04
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    Dear @CameronBuie, excellent point. I have been thinking about this, actually. And I was going to post a question about how to define countable.2012-11-04
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    @Matt: Please don't. There are like 10 questions about the definition of countable on this site as it is.2012-11-04
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    @AsafKaragila Can you point me to these 10 questions, I can't seem to find what I'm looking for. That would be very kind of you!2012-11-04
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    [One](http://math.stackexchange.com/questions/3048/), [two](http://math.stackexchange.com/questions/205563/), [three](http://math.stackexchange.com/questions/185234/), and [four](http://math.stackexchange.com/questions/192496/). [And one you might want to read anyway](http://math.stackexchange.com/questions/150575/). All this and I'm not counting the endless repetitions and discussions in other questions about countability of $\mathbb N\times\mathbb N$ and so on.2012-11-04
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    @AsafKaragila Thank you but none of these answer my question.2012-11-05
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    Matt, all of these deal with the definition of countable. If you wish to discuss internal vs. external definitions, that you can find in the last linked post as well in pichael (the OP of that last link) other questions.2012-11-05

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