The limits of determinacy in third-order arithmetic
Thibaut Kouptchinsky
Authors: Thibaut Kouptchinsky and Juan Aguilera
This talk is about the foundations of mathematics, studying determinacy axioms derived from game theory, with a reverse mathematics point of view. Reverse Mathematics is an endeavour to compare theorems according to the “strength” of the axioms a mathematician needs to prove them (see [5]).
We study a refined case of the proof from Martin of Borel determinacy [2], which showed how to use the existence of high-order objects when one wants to show that infinite games of increasing difficulty in the Borel hierarchy have winning strategies (and thus are called determined). The use of such principles had already been shown to be necessary by Friedmann [1]. In the terms that will be ours, Martin provided the final proof that \((2+\gamma)\)th-order arithmetic (\(\mathsf{Z}_{2+\gamma}\)) is the first to witness the determinacy of \(\Pi^0_{1+\gamma+2}\) Gale-Stewart games (\(\gamma < \omega_1^{\mathsf{CK}}\)).
For our part, we will look in more detail at the situation in second-order and third-order arithmetic, examining a paper by Montalbán and Shore [3]. We present a generalisation of their results, in some natural interpretation of third-order arithmetic about infinite games which complexity is described by the difference hierarchy of \(\Pi^0_4\) sets. Along the way, we underline a shift compared to the analogous situation in the countable case (about differences of \(\Pi^0_3\) sets), which is the object of the paper of Montalbán and Shore. Namely, we need less of the separation scheme than in the countable case.
Finally, we use these generalisations following the results of Pacheco and Yokoyama to [4] show that, while
\[\forall m \in \omega \mathsf{Z}_{2+\gamma} \vdash (\Pi^0_{1+\gamma+2})_m\text{-}\mathsf{Det} \qquad \text{but} \qquad \mathsf{Z}_{2+\gamma} \not\vdash \forall m (\Pi^0_{1+\gamma+2})_m\text{-}\mathsf{Det},\]the last theorem is equivalent to a form of reflection principle.
Bibliography
- H. M. Friedmann,Higher set theory and mathematical practice,Annals of Mathematicsvol. 2 (1971), no. 3, pp. 352–357.
- D. A. Martin,Borel determinacy,Annals of Mathematics,vol. 102 (1975), no. 2, pp. 363–371.
- A. Montalbán, R. A. Shore,The limits of determinacy in second order arithmetic,Israel Journal of Mathematics,vol. 104 (2011), no. 2, pp. 223–252.
- L. Pacheco, Ke. Yokoyama,Determinacy and reflection principles in second-order arithmetic,Preprint, arXiv (2022).
- S. G. Simpson,Subsystems of second order arithmetic (Second edition),Perspectives in Logic,Association for Symbolic Logic,2009.