Abstract

The axiom of symmetry (A ℵ 0 ) asserts that for every function F: ω 2 → ω 2 there is a pair of reals x and y in ω 2 so that y is not in the countable set $\{(F(x))_n:n coded by F(x) and x is not in the set coded by F(y). A(Γ) denotes axiom A ℵ 0 with the restriction that graph(F) belongs to the pointclass Γ. In § 2 we prove A(Σ 1 1 ). In § 3 we show A(Π 1 1 ), A(Σ 1 2 ) and $^\omega 2 \nsubseteq L$ are equivalent. In § 4 several effective versions of A(REC) are examined