On Tao's "finitary" infinite pigeonhole principle

In 2007, Terence Tao wrote on his blog an essay about soft analysis, hard analysis and the finitization of soft analysis statements into hard analysis statements. One of his main examples was a quasi-finitization of the infinite pigeonhole principle IPP, arriving at the "finitary" infinite pigeonhole principle FIPP1. That turned out to not be the proper formulation and so we proposed an alternative version FIPP2. Tao himself formulated yet another version FIPP3 in a revised version of his essay. We give a counterexample to FIPP1 and discuss for both of the versions FIPP2 and FIPP3 the faithfulness of their respective finitization of IPP by studying the equivalences IPP FIPP2 and IPP FIPP3 in the context of reverse mathematics. In the process of doing this we also introduce a continuous uniform boundedness principle CUB as a formalization of Tao's notion of a correspondence principle and study the strength of this principle and various restrictions thereof in terms of reverse mathematics, i.e., in terms of the "big five" subsystems of second order arithmetic

The cohesive principle and the Bolzano-Weierstra{\ss} principle

The aim of this paper is to determine the logical and computational strength of instances of the Bolzano-Weierstra{\ss} principle (BW) and a weak variant of it. We show that BW is instance-wise equivalent to the weak K\"onig's lemma for $\Sigma^0_1$-trees ($\Sigma^0_1$-WKL). This means that from every bounded sequence of reals one can compute an infinite $\Sigma^0_1$-0/1-tree, such that each infinite branch of it yields an accumulation point and vice versa. Especially, this shows that the degrees d >> 0' are exactly those containing an accumulation point for all bounded computable sequences. Let BW_weak be the principle stating that every bounded sequence of real numbers contains a Cauchy subsequence (a sequence converging but not necessarily fast). We show that BW_weak is instance-wise equivalent to the (strong) cohesive principle (StCOH) and - using this - obtain a classification of the computational and logical strength of BW_weak. Especially we show that BW_weak does not solve the halting problem and does not lead to more than primitive recursive growth. Therefore it is strictly weaker than BW. We also discuss possible uses of BW_weak.Comment: corrected typos, slightly improved presentatio

On proximal mappings with Young functions in uniformly convex Banach spaces

It is well known in convex analysis that proximal mappings on Hilbert spaces are $1$-Lipschitz. In the present paper we show that proximal mappings on uniformly convex Banach spaces are uniformly continuous on bounded sets. Moreover, we introduce a new general proximal mapping whose regularization term is given as a composition of a Young function and the norm, and formulate our results at this level of generality. It is our aim to obtain the corresponding modulus of uniform continuity explicitly in terms of a modulus of uniform convexity of the norm and of moduli witnessing properties of the Young function. We also derive several quantitative results on uniform convexity, which may be of interest on their own.Comment: Accepted in J. Convex Ana

From Bolzano-Weierstra{\ss} to Arzel\`a-Ascoli

We show how one can obtain solutions to the Arzel\`a-Ascoli theorem using suitable applications of the Bolzano-Weierstra{\ss} principle. With this, we can apply the results from \cite{aK} and obtain a classification of the strength of instances of the Arzel\`a-Ascoli theorem and a variant of it. Let AA be the statement that each equicontinuous sequence of functions f_n: [0,1] --> [0,1] contains a subsequence that converges uniformly with the rate 2^-k and let AA_weak be the statement that each such sequence contains a subsequence which converges uniformly but possibly without any rate. We show that AA is instance-wise equivalent over RCA_0 to the Bolzano-Weierstra{\ss} principle BW and that AA_weak is instance-wise equivalent over WKL_0 to BW_weak, and thus to the strong cohesive principle StCOH. Moreover, we show that over RCA_0 the principles AA_weak, BW_weak + WKL and StCOH + WKL are equivalent

Arithmetical conservation results

In this paper we present a proof of Goodman's Theorem, a classical result in the metamathematics of constructivism, which states that the addition of the axiom of choice to Heyting arithmetic in finite types does not increase the collection of provable arithmetical sentences. Our proof relies on several ideas from earlier proofs by other authors, but adds some new ones as well. In particular, we show how a recent paper by Jaap van Oosten can be used to simplify a key step in the proof. We have also included an interesting corollary for classical systems pointed out to us by Ulrich Kohlenbach