arXiv subjects
Publications and source records attributed to Katarzyna W. Kowalik.
We exhibit a well-ordering principle that is equivalent to a theory of partial impredicativity. The latter goes back to Towsner and relates to recent work of Suzuki and Yokoyama. Together with part I of the present paper, this yields a combinatorial upper bound on the axiomatic strength of Fraïssé's conjecture.
We show that the theory $\mathsf{WKL}^*_0+\mathsf{CAC}$ is polynomially simulated by $\mathsf{RCA}_0^*$ with respect to $\forallΠ^0_3$ formulas. For the proof, we use the method of forcing interpretations and syntactically simulate a two-step model-theoretic argument, which involves construction of a restricted definable ultrapower, followed by a generic cut satisfying $\mathsf{CAC}$. Our result sharply contrasts with the previously known fact that $\mathsf{RCA}_0^*+\mathsf{RT}^2_2$ has non-elementary speedup over $\mathsf{RCA}_0^*$.
The paper is devoted to a reverse-mathematical study of some well-known consequences of Ramsey's theorem for pairs, focused on the chain-antichain principle $\mathsf{CAC}$, the ascending-descending sequence principle $\mathsf{ADS}$, and the Cohesive Ramsey Theorem for pairs $\mathsf{CRT}^2_2$. We study these principles over the base theory $\mathsf{RCA}^*_0$, which is weaker than the usual base theory $\mathsf{RCA}_0$ considered in reverse mathematics in that it allows only $Δ^0_1$-induction as opposed to $Σ^0_1$-induction. In $\mathsf{RCA}^*_0$, it may happen that an unbounded subset of $\mathbb{N}$ is not in bijective correspondence with $\mathbb{N}$. Accordingly, Ramsey-theoretic principles split into at least two variants, "normal" and "long", depending on the sense in which the set witnessing the principle is required to be infinite. We prove that the normal versions of our principles, like that of Ramsey's theorem for pairs and two colours, are equivalent to their relativizations to proper $Σ^0_1$-definable cuts. Because of this, they are all $Π^0_3$- but not $Π^1_1$-conservative over $\mathsf{RCA}^*_0$, and, in any model of $\mathsf{RCA}^*_0 + \neg \mathsf{RCA}_0$, if they are true then they are computably true relative to some set. The long versions exhibit one of two behaviours: they either imply $\mathsf{RCA}_0$ over $\mathsf{RCA}^*_0$ or are $Π^0_3$-conservative over $\mathsf{RCA}^*_0$. The conservation results are obtained using a variant of the so-called grouping principle. We also show that the cohesion principle $\mathsf{COH}$, a strengthening of $\mathsf{CRT}^2_2$, is never computably true in a model of $\mathsf{RCA}^*_0$ and, as a consequence, does not follow from $\mathsf{RT}^2_2$ over $\mathsf{RCA}^*_0$.
We study the first-order consequences of Ramsey's Theorem for $k$-colourings of $n$-tuples, for fixed $n, k \ge 2$, over the relatively weak second-order arithmetic theory $\mathrm{RCA}^*_0$. Using the Chong-Mourad coding lemma, we show that in a model of $\mathrm{RCA}^*_0 + \neg \mathrm{I}Σ^0_1$, $\mathrm{RT}^n_k$ is equivalent to its own relativization to any proper $Σ^0_1$-definable cut, so its truth value remains unchanged in all extensions of the model with the same first-order universe. We give an axiomatization of the first-order consequences of $\mathrm{RCA}^*_0 + \mathrm{RT}^n_k$ for $n \ge 3$. We show that they form a non-finitely axiomatizable subtheory of PA whose $Π_3$ fragment is $\mathrm{B}Σ_1 + \exp$ and whose $Π_{\ell+3}$ fragment for $\ell \ge 1$ lies between $\mathrm{I}Σ_\ell \Rightarrow \mathrm{B}Σ_{\ell+1}$ and $\mathrm{B}Σ_{\ell+1}$. We also consider the first-order consequences of $\mathrm{RCA}^*_0 + \mathrm{RT}^2_k$. We show that they form a subtheory of $\mathrm{I}Σ_2$ whose $Π_3$ fragment is $\mathrm{B}Σ_1 + \exp$ and whose $Π_4$ fragment is strictly weaker than $\mathrm{B}Σ_2$ but not contained in $\mathrm{I}Σ_1$. Additionally, we consider a principle $Δ^0_2$-$\mathrm{RT}^2_2$, defined like $\mathrm{RT}^2_2$ but with both the $2$-colourings and the solutions allowed to be $Δ^0_2$-sets. We show that the behaviour of $Δ^0_2$-$\mathrm{RT}^2_2$ over $\mathrm{RCA}_0 + \mathrm{B}Σ^0_2$ is similar to that of $\mathrm{RT}^2_2$ over $\mathrm{RCA}^*_0$, and that $\mathrm{RCA}_0 + \mathrm{B}Σ^0_2 + Δ^0_2$-$\mathrm{RT}^2_2$ is $Π_4$- but not $Π_5$-conservative over $\mathrm{B}Σ_2$. However, the statement we use to witness lack of $Π_5$-conservativity is not provable in $\mathrm{RCA}_0 +\mathrm{RT}^2_2$.