In a set of notes for the Cabal Seminar a theorem is attributed to Moschovakis which includes the statement that for certain pointclasses $\Gamma$ the singletons of $\Gamma$ are a basis for $\Gamma$. I've been having difficulty locating a definition of a basis for a pointclass and would appreciate any help.
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$\begingroup$ could you link those notes if they are available online? $\endgroup$– Alessandro CodenottiCommented Mar 21 at 11:55
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1$\begingroup$ I think this is supposed to mean that every nonempty set in class $\Gamma$ contains ($\supseteq$) a singleton in class $\Gamma$. But it could also conceivably mean that every nonempty set in class $\Gamma$ contains ($\ni$) a real in class $\Gamma$. This “basis” terminology is abominable, and totally at odds with the meaning of “basis” in other parts of mathematics; but the general idea is that $B$ being a “basis” for $P$ means $(\exists x.(P(x))) \Rightarrow (\exists x\in B.(P(x)))$. $\endgroup$– Gro-TsenCommented Mar 21 at 12:43
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$\begingroup$ I'm afraid I only have a hardback copy of the notes. I appreciate Gro-Tsen's discussion. $\endgroup$– RupertCommented Mar 21 at 15:51
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