[FOM] question

[Aatu Koskensilta]

  Roman Muwarski asks:

      I was told about  Kreisel's result stating that ZF + AC + GCH is a
      conservative extension of ZF with respect to sentences about natural
      numbers. Is it true? Where one can find it?

  This observation, often which is often attributed to Kreisel but was surely known to Gödel already, follows immediately from the fact that the relativization of an arithmetical statement to the universe L of constructible sets is, provably in ZF, equivalent to the original statement. More generally, by Shoenfield's absoluteness theorem, we know that any sentence of complexity below Delta-1-3 is necessarily decided the same way in any inner model. In the case of AC and GCH in particular we can do even better and show that conservativity holds in even higher reaches a bit farther up in the analytical hierarchy. 

  Best,

Aatu Koskensilta (koskensilta.aatu.j at student.uta.fi)

"Wovon man nicht sprechen kann, darüber muss man schweigen."
  -- Ludwig Wittgenstein, Tractatus Logico-Philosophicus