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user38200
user38200
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Shoenfield absoluteness is well known for $\Pi_2^1$-statements, but it does not hold between a countable transitive model of ZFC and the universe (as I heard, but I might be wrong). But it is also known that $\Pi_1^1$ statements are absolute between a countable transitive model of set theory and the universe (as I saw in an answer to my previous question on Scott sentence in models of set theory).

My question is:

Where can I find a good reference for the last fact?

Shoenfield absoluteness is well known for $\Pi_2^1$-statements, but it does not hold between a countable transitive model of ZFC and the universe (as I heard, but I might be wrong). But it is also known that $\Pi_1^1$ statements are absolute between a countable transitive model of set theory and the universe (as I saw in an answer to my previous question on Scott sentence in models of set theory).

My question is:

Where can I find a good reference for the last fact?

Shoenfield absoluteness is well known for $\Pi_2^1$-statements, but it does not hold between a countable transitive model of ZFC and the universe. But it is also known that $\Pi_1^1$ statements are absolute between a countable transitive model of set theory and the universe.

My question is:

Where can I find a good reference for the last fact?

Source Link
user38200
user38200
  • 1.4k
  • 8
  • 13

Absolutness of $\Pi_1^1$ statements

Shoenfield absoluteness is well known for $\Pi_2^1$-statements, but it does not hold between a countable transitive model of ZFC and the universe (as I heard, but I might be wrong). But it is also known that $\Pi_1^1$ statements are absolute between a countable transitive model of set theory and the universe (as I saw in an answer to my previous question on Scott sentence in models of set theory).

My question is:

Where can I find a good reference for the last fact?