## Journal of Symbolic Logic

### The $\sum^1_2$ Theory of Axioms of Symmetry

Galen Weitkamp

#### Abstract

The axiom of symmetry $(A_{\aleph_0})$ asserts that for every function $F: ^\omega 2 \rightarrow ^\omega 2$ there is a pair of reals $x$ and $y$ in $^\omega 2$ so that $y$ is not in the countable set $\{(F(x))_n:n < \omega\}$ coded by $F(x)$ and $x$ is not in the set coded by $F(y). A(\Gamma)$ denotes axiom $A_{\aleph_0}$ with the restriction that $\text{graph}(F)$ belongs to the pointclass $\Gamma$. In $\S 2$ we prove $A(\Sigma^1_1)$. In $\S 3$ we show $A(\Pi^1_1), A(\Sigma^1_2)$ and $^\omega 2 \nsubseteq L$ are equivalent. In $\S 4$ several effective versions of $A(\mathrm{REC})$ are examined.

#### Article information

Source
J. Symbolic Logic, Volume 54, Issue 3 (1989), 727-734.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183743012

Mathematical Reviews number (MathSciNet)
MR1011164

Zentralblatt MATH identifier
0692.03032

JSTOR
Weitkamp, Galen. The $\sum^1_2$ Theory of Axioms of Symmetry. J. Symbolic Logic 54 (1989), no. 3, 727--734. https://projecteuclid.org/euclid.jsl/1183743012