Abstract. The enterprise of comparing mathematical theorems according to their logical strength is an active area in mathematical logic, with one of the most common frameworks for doing so being reverse mathematics. In this setting, one investigates which theorems provably imply which others in a weak formal theory roughly corresponding to computable mathematics. Since the proofs of such implications take place in classical logic, they may in principle involve appeals to multiple applications of a particular theorem, or to nonuniform decisions about how to proceed in a given construction. In practice, however, if a theorem Q implies a theorem P, it is usually because there is a direct uniform translation of the problems represented by P into the problems represented by Q, in a precise sense formalized by Weihrauch reducibility. We study this notion of uniform reducibility in the context of several natural combinatorial problems, and compare and contrast it with the traditional notion of implication in reverse mathematics. We show, for instance, that for all n, j, k ≥ 1, if j < k then Ramsey's theorem for n-tuples and k many colors is not uniformly, or Weihrauch, reducible to Ramsey's theorem for n-tuples and j many colors. The two theorems are classically equivalent, so our analysis gives a genuinely finer metric by which to gauge the relative strength of mathematical propositions. We also study Weak König's Lemma, the Thin Set Theorem, and the Rainbow Ramsey's Theorem, along with a number of their variants investigated in the literature. Weihrauch reducibility turns out to be connected with sequential forms of mathematical principles, where one wishes to solve infinitely many instances of a particular problem simultaneously. We exploit this connection to uncover new points of difference between combinatorial problems previously thought to be more closely related.

The sequential form of a statement ( †) ∀ξ(B(ξ) → ∃ζA(ξ, ζ)) is the statement ∀ξ(∀nB(ξ n ) → ∃ζ∀nA(ξ n , ζ n )).There are many classically true statements of the form ( †) whose proofs lack uniformity and therefore the corresponding sequential form is not provable in weak classical systems. The main culprit for this lack of uniformity is of course the law of excluded middle. Continuing along the lines of Hirst and Mummert [4], we show that if a statement of the form ( †) satisfying certain syntactic requirements is provable in some weak intuitionistic system, then the proof is necessarily sufficiently uniform that the corresponding sequential form is provable in a corresponding weak classical system. Our results depend on Kleene's realizability with functions and the Lifschitz variant thereof.

If NONEMPTY has a winning strategy against EMPTY in the Choquet game on a space, the space is said to be a Choquet space. Such a winning strategy allows NONEMPTY to consider the entire finite history of previous moves before making each new move; a stationary strategy only permits NONEMPTY to consider the previous move by EMPTY. We show that NONEMPTY has a stationary winning strategy for every second countable T1 Choquet space. More generally, NONEMPTY has a stationary winning strategy for any T1 Choquet space with an open-finite basis. We also study convergent strategies for the Choquet game, proving the following results. (1) A T1 space X is the open image of a complete metric space if and only if NONEMPTY has a convergent winning strategy in the Choquet game on X. (2) A T1 space X is the compact open image of a metric space if and only if X is metacompact and NONEMPTY has a stationary convergent strategy in the Choquet game on X. (3) A T1 space X is the compact open image of a complete metric space if and only if X is metacompact and NONEMPTY has a stationary convergent winning strategy in the Choquet game on X.Comment: 24 page

Using the techniques of reverse mathematics, we analyze the logical strength of statements similar to trichotomy and dichotomy for sequences of reals. Capitalizing on the connection between sequential statements and constructivity, we find computable restrictions of the statements for sequences and constructive restrictions of the original principles. Axiom systems and encoding realsWe will examine several statements about real numbers and sequences of real numbers in the framework of reverse mathematics and in some formalizations of weak constructive analysis. From reverse mathematics, we will concentrate on the axiom systems RCA 0 , WKL 0 , and ACA 0 , which are described in detail by Simpson [8]. Very roughly, RCA 0 is a subsystem of second order arithmetic incorporating ordered semi-ring axioms, a restricted form of induction, and comprehension for ∆ 0 1 definable sets. The axiom system WKL 0 appends König's tree lemma restricted to 0-1 trees, and the system ACA 0 appends a comprehension scheme for arithmetically definable sets. ACA 0 is strictly stronger than WKL 0 , and WKL 0 is strictly stronger than RCA 0 .We will also make use of several formalizations of subsystems of constructive analysis, all variations of E-HA ω , which is intuitionistic arithmetic (Heyting arithmetic) in all finite types with an extensionality scheme. Unlike the reverse mathematics systems which use classical logic, these constructive systems omit the law of the excluded middle. For example, we will use extensions of E-HA ω , a form of Heyting arithmetic with

Given sets X,Y$X,Y$ and a regular cardinal μ, let normalΦfalse(X,Y,μfalse)$\Phi (X,Y,\mu )$ be the statement that for any function f:X×Y→μ$f : X \times Y \rightarrow \mu$, there are functions g1:X→μ$g_1 : X \rightarrow \mu$ and g2:Y→μ$g_2 : Y \rightarrow \mu$ such that for all false(x,yfalse)∈X×Y$(x,y) \in X \times Y$, f(x,y)≤maxfalse{g1(x),g2(y)false}$f(x,y) \le \max \lbrace g_1(x), g_2(y) \rbrace$. In ZFC$\mathsf {ZFC}$, the statement normalΦfalse(ω1,ω1,ωfalse)$\Phi (\omega _1, \omega _1, \omega )$ is false. However, we show the theory sans-serifZF+“the4.ptclub4.ptfilter4.pton4.ptω14.ptis4.ptnormal”+normalΦfalse(ω1,ω1,ωfalse)$\mathsf {ZF}+ \text{``the club filter on $\omega _1$ is normal''} + \Phi (\omega _1, \omega _1, \omega )$ (which is implied by sans-serifZF+sans-serifDC$\mathsf {ZF}+ \mathsf {DC}$+ “V=Lfalse(double-struckRfalse)$V = L(\mathbb {R})$” + “ω1 is measurable”) implies that for every α<ω1$\alpha < \omega _1$ there is a κ∈false(α,ω1false)$\kappa \in (\alpha ,\omega _1)$ such that in some inner model, κ is measurable with Mitchell order ≥α$\ge \alpha$.