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I have read this sentence "If ZFC is consistent, then there is a countable model of it by Lowenheim Skolem theorem" in many text books of mathematical logic and set theory. The theorem tells us a word "countable", or " infinity cardinal". And this theorem is proved by axioms of ZFC. Here is my questions) Athough the Lowenheim Skolem theorem is proved in ZFC, how can we apply this to the set theory(ZFC)? Does this theorem should be applied to subtheory of ZFC? Is the theorem proved only in ZFC? Thank you so much for your answer.

dre rt
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    I think https://math.stackexchange.com/questions/531516/meta-theory-when-studying-set-theory might be helpful here. – Asaf Karagila Mar 12 '20 at 07:30
  • Thank you so much. – dre rt Mar 12 '20 at 22:46
  • @AsafKaragila If you dont mind, could you give me an answer to my new question? The question) ZFC can prove the consistency of Peano Arithmetic, Con(PA). Thus, Lowenheim-Skolem thm can be applied to PA in ZFC. Similarly, Can we state that the theorem can be applied to ZFC in ZFC+Con(ZFC) as ZFC+Con(ZFC) can prove the consistency of ZFC? – dre rt Mar 13 '20 at 06:28
  • Yes, that is correct. – Asaf Karagila Mar 13 '20 at 06:31

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