Die Wahrheit liegt weder in der unendlichen Annährung an einer objektiv Gegebenes noch in der Mitte, sondern rundherum wie ein Sack, der mit jeder neuen Meinung, die man hineinstopft, seine Form ändert, aber immer fester wird. R. Musil We represent truth sets for a variety of the well known semantic theories of truth as those sets consisting of those sentences for which a player has a winning strategy in an infinite two person game. The classifications of the games are simple, those over the natural model of arithmetic being all within the arithmetical class of Σ 0 3. 1

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This thesis consists of two parts. The first part treats determinacy in (classical) reverse mathematics and the second part treats determinacy in intuitionistic mathematics. The first part investigates the logical strength of the determinacy of infinite games in the Cantor space in terms of second order arithmetic. We define new determinacy schemata inspired by the Wadge classes of Polish spaces and show that the following equivalences hold over the system RCA ∗ 0, which consists of the axioms of discrete ordered semi-ring with exponentiation, ∆0 1 comprehension and Σ0 0 induction, and which is known as a weaker system than the popular base theory RCA0: • ∆ 0 1-Det ∗ ↔ Σ 0 1-Det ∗ ↔ WKL ∗

Abstract. We identify a number of theories of strength that of Π 1 1-CA0. In particular: (a) the theory that the set of points attracted to the origin in a generalised transfinite dynamical system of any n-dimensional integer torus exists; (b) the theory asserting that for any Z ⊆ ω and n, the halting set H Z n of infinite time n-register machine with oracle Z exists. Suppose f: N n − → N n. We are going to consider transfinite iterations of such f: N n − → N n as a generalised dynamical system. If one wishes, one may think of f acting on the points of an n-dimensional lattice torus where we identify ∞ with 0. We set this up as follows. Given a point r = (r1,..., rn) ∈ N n set: r 0 = (r 0 1,..., r 0 n) = (r1,..., rn); r α+1 = (r α+1 1,..., r α+1 n) = f((r α 1,..., r α n)); r λ = (r λ 1,..., r λ n) = (Liminf ∗ α→λ r α 1, Liminf ∗ α→λ r α 2,..., Liminf ∗ α→λ r α n) where we define Liminf ∗ α→λ r α 1 = Liminf α→λ r α 1 if the latter is < ω, and set it to 0 otherwise, thus: r λ i = Liminfα→λ r α i if the latter is < ω = 0 otherwise. We may ask after the behaviour of points under this dynamic. For example which points ultimately end up at the origin O? As a more amusing example let p = (p0, p1, p2) ∈ (N n) 3 be a triple of three points on the n-dimensional lattice. In general they thus form a proper triangle. Then define: Tf = {(p0, p1, p2) ∈ N n3 | ∃α p α 0 = p α 1 = p α 2}. Tf is thus the set of possible starting triangles, which at some point collapse and become coincident after iteration of their vertices (and remain collapsed of course from some point α0 onwards).

We establish the precise bounds for the amount of determinacy provable in second order arithmetic. We show that for every natural number n, second order arithmetic can prove that determinacy holds for Boolean combinations of n many Π 0 3 classes, but it cannot prove that all finite Boolean combinations of Π 0 3 classes are determined. More specifically, we prove that Π 1 n+2-CA 0 ⊢ n-Π 0 3-DET, but that ∆ 1 n+2-CA � n-Π 0 3-DET, where n-Π 0 3 is the nth level in the difference hierarchy of Π 0 3 classes. We also show some conservativity results that imply that reversals for the theorems above are not possible. We prove that for every true Σ 1 4 sentence T (as for instance n-Π 0 3-DET) and