We recently found a kind of orbit equivalence relations, denoted by E ( G ) $E(G)$ upper E left parenthesis upper G right parenthesis , which can well describe structures and properties of Polish groups from the perspective of Borel reducibility. Given a Polish group G , let E ( G ) $E(G)$ upper E left parenthesis upper G right parenthesis be the right coset equivalence relation G ω / c ( G ) $G^\omega /c(G)$ upper G Superscript omega Baseline divided by c left parenthesis upper G right parenthesis , where c ( G ) $c(G)$ c left parenthesis upper G right parenthesis is the group of all convergent sequences in G . We list some main results. Let G and H be two Polish groups. (1) G is a discrete countable group containing at least two elements iff E ( G ) ∼ B E 0 $E(G)\sim _BE_0$ upper E left parenthesis upper G right parenthesis tilde Subscript upper B Baseline upper E 0 ; (2) G is non-archimedean iff E ( G ) ≤ B = + $E(G)\le _B=^+$ upper E left parenthesis upper G right parenthesis less than or equals Subscript upper B Baseline equals Superscript plus Baseline ; (3) if H is CLI but G is not, then
Abstract We investigate the relationship between various versions of Ramsey’s theorem and bounding schemes in a model N $\mathcal {N}$ script upper N of a fragment of arithmetic F $F$ upper F . Our primary objective is to recast and extend the seminal results of Hirst [15] (see Theorem 1) through the modern lens of Weihrauch reducibility. By extracting explicit Weihrauch reductions from classical proofs, we expose the uniform computational content underlying these combinatorial principles, yielding a deeper and more refined understanding of their reverse mathematical strength. Our results, informally stated in our terminology and established inside N $\mathcal {N}$ script upper N , are as follows: The following are equivalent: B Σ 2 0 $\mathrm {B}\Sigma ^0_2$ normal upper B normal upper Sigma 2 Superscript 0 , the statement that the finite union of finite c.e. sets is finite, and the Infinite Pigeonhole Principle (see Theorem 3). We also discuss the Weihrauch relations between these logically equivalent principles (see Section 4). The Infinite Pigeonhole Principle is Weihrauch reducible to RT 2 2 $\mathrm {RT}^2_2$ upper R upper T 2 squared (see Theorem 4). There is also another principle logically equivalent to B Σ 2 0 $\mathrm {B}\Sigma ^0_2$ normal upper B normal upper Sigma 2 Superscript 0 which is Weihrauch reducible to SRT 2 2 $\mathrm {SRT}^2_2$ upper S upper R upper T 2 squared (see Theorem 5). We show that there is a principle which is equivalent to B Σ 3 0 $\mathrm {B}\Sigma ^0_3$ normal upper B normal upper Sigma 3 Superscript 0 (see Theorem 6) and Weihrauch reducible to SRT < ∞ 2 $\mathrm {SRT}^2_{<\infty }$ upper S upper R upper T Subscript less than infinity Superscript 2 (Theorem 7). We discuss some equivalences with B Σ n 0 $\mathrm {B}\Sigma ^0_{n}$ normal upper B normal upper Sigma Subscript n Superscript 0 (see Section 6.1) and end with a problem Weihrauch reducible to RT 2 n + 1 $\mathrm {RT}^{n+1}_{2}$ upper R upper T 2 Superscript n plus 1 (Section 6.2). Since we work within the model N $\mathcal {N}$ script upper N , many standard definitions must be adjusted. Due to the expository nature of this article, these definitions are introduced throughout the text as needed. Reading the article from start to finish will provide a better understanding of the ideas involved than focusing only on individual theorems.
