Online research in philosophy

We show that the fact that the first player (“white”) wins every instance of Galvin’s “racing pawns” game (for countable trees) is equivalent to arithmetic transfinite recursion. Along the way we analyze the satisfaction relation for infinitary formulas, of “internal” hyperarithmetic comprehension, and of the law of excluded middle for such formulas.

We explore the interaction between Lebesgue measure and dominating functions. We show, via both a priority construction and a forcing construction, that there is a function of incomplete degree that dominates almost all degrees. This answers a question of Dobrinen and Simpson, who showed that such functions are related to the proof-theoretic strength of the regularity of Lebesgue measure for Gδ sets. Our constructions essentially settle the reverse mathematical classification of this principle.

A real is called properly n-generic if it is n-generic but not n+1-generic. We show that every 1-generic real computes a properly 1-generic real. On the other hand, if m > n ≥ 2 then an m-generic real cannot compute a properly n-generic real.

We show, however, that this is not always the case.

We show that if d $\in GH_1$ then D( $\leq$ d) has the complementation property, i.e.. for all a < d there is some b < d such that a $\wedge$ b = 0 and a $\vee$ b = d.