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June 6

Banach-Tarski paradox quantifier depth

I'm trying to figure out the formal statement of the Banach-Tarski (BT) paradox. Something like: there exist sets A,B,C,D,E in R³ whose union A+B+C+D+E is the unit ball B1, and there are rigid motions S,T,U,V so that A+S(B)+T(C)=B1 and U(D)+V(E)=another copy of B1. A rigid motion is just a rotation and translation so it can be written as a 5-tuple of reals. Meanwhile, to say two sets of real triples are equal (like G=H) uses a universal quantifier over real triples, $\mathrm{\forall }x.(x\in G\wedge x\in H)\vee (x\notin G\wedge x\notin H).$

Since the outer (existential) quantifier is over sets of reals, would we say it is a third-order arithmetic quantifier? The inner one is (nested) universal quantifiers over reals so that would be second-order arithmetic. Does the whole thing become ${\mathrm{\Sigma }}_2^2$?

At first I didn't notice that the outer quantifiers were over sets of reals, so the BT paradox seemed to conflict with Schoenfeld's absoluteness theorem. But I think the set quantifiers mean BT is not analytic, so no issue there. Thanks.

173.228.123.207 (talk) 20:48, 6 June 2019 (UTC)