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@article{moroni2020zhou, title = {The {Z}hou Ordinal of Labelled {M}arkov Processes over Separable Spaces}, author = {Moroni, Martín Santiago and S\'anchez Terraf, Pedro}, journal = {The Review of Symbolic Logic}, month = dec, year = 2023, volume = 16, number = 4, pages = {1011--1032}, eprint = {2005.03630}, archive = {arXiv}, primaryclass = {cs.LO}, doi = {10.1017/S1755020322000375}, url = {https://doi.org/10.1017/S1755020322000375}, abstract = {There exist two notions of equivalence of behavior between states of a Labelled Markov Process (LMP): state bisimilarity and event bisimilarity. The first one can be considered as an appropriate generalization to continuous spaces of Larsen and Skou's probabilistic bisimilarity, while the second one is characterized by a natural logic. C. Zhou expressed state bisimilarity as the greatest fixed point of an operator $\mathcal{O}$, and thus introduced an ordinal measure of the discrepancy between it and event bisimilarity. We call this ordinal the "Zhou ordinal" of $\mathbb{S}$, $\mathfrak{Z}(\mathbb{S})$. When $\mathfrak{Z}(\mathbb{S})=0$, $\mathbb{S}$ satisfies the Hennessy-Milner property. The second author proved the existence of an LMP $\mathbb{S}$ with $\mathfrak{Z}(\mathbb{S}) \geq 1$ and Zhou showed that there are LMPs having an infinite Zhou ordinal. In this paper we show that there are LMPs $\mathbb{S}$ over separable metrizable spaces having arbitrary large countable $\mathfrak{Z}(\mathbb{S})$ and that it is consistent with the axioms of $\mathit{ZFC}$ that there is such a process with an uncountable Zhou ordinal.} }
@article{2022arXiv221015609G, author = {Gunther, Emmanuel and Pagano, Miguel and S{\'a}nchez Terraf, Pedro and Steinberg, Mat{\'i}as}, title = {The formal verification of the ctm approach to forcing}, journal = {Annals of Pure and Applied Logic}, issn = {0168-0072}, url = {https://www.sciencedirect.com/science/article/pii/S0168007224000101}, keywords = {forcing, Isabelle/ZF, countable transitive models, continuum hypothesis, proof assistants, interactive theorem provers, generic extension}, year = 2024, volume = 175, month = may, number = 5, archiveprefix = {arXiv}, eprint = {2210.15609}, primaryclass = {math.LO}, adsurl = {https://ui.adsabs.harvard.edu/abs/2022arXiv221015609G}, adsnote = {Provided by the SAO/NASA Astrophysics Data System}, doi = {10.1016/j.apal.2024.103413}, abstract = {We discuss some highlights of our computer-verified proof of the construction, given a countable transitive set-model $M$ of $\mathit{ZFC}$, of generic extensions satisfying $\mathit{ZFC} + \neg\mathit{CH}$ and $\mathit{ZFC} + \mathit{CH}$. Moreover, let $\mathcal{R}$ be the set of instances of the Axiom of Replacement. We isolated a 21-element subset $\Omega\subseteq\mathcal{R}$ and defined $\mathcal{F}:\mathcal {R}\to\mathcal{R}$ such that for every $\Phi\subseteq\mathcal{R}$ and $M $-generic $G$, $M\models \mathit{ZC} \cup \mathcal{F}\text{``}\Phi \cup \Omega$ implies $M[G]\models \mathit{ZC} \cup \Phi \cup \{\neg\mathit{CH}\}$, where $\mathit{ZC}$ is Zermelo set theory with Choice. To achieve this, we worked in the proof assistant \emph{Isabelle}, basing our development on the Isabelle/ZF library by L.~Paulson and others.} }
@article{ciem40, title = {Set Theory at {C}\'ordoba}, author = {S{\'a}nchez Terraf, Pedro}, pages = {61--65}, volume = 16, month = jul, year = 2024, journal = {Actas de la Academia Nacional de Ciencias}, url = {https://www.anc-argentina.org.ar/wp-content/uploads/sites/36/2024/07/ANC_XVI.pdf}, note = {Extended abstract for invited talk at the 40th anniversary of the Center for Research and Studies in Mathematics (C\'ordoba)}, publisher = {Academia Nacional de Ciencias}, address = {Córdoba, Argentina}, abstract = {Set Theory is a new research area in Argentina, still with very few practitioners. We present some of the first steps towards its development at the National University of C\'ordoba. Cantor's \emph{continuum problem}, that of the determining which place does the cardinality of the reals occupy in the cardinal line, provides an appropriate frame for this exposition (and for the whole of Set Theory indeed).} }
@article{2024arXiv240407877K, author = {{Kuperman}, Joel and {Petrovich}, Alejandro and S{\'a}nchez Terraf, Pedro}, title = {Definability of band structures on posets}, journal = {Semigroup Forum}, keywords = {Mathematics - Logic, Mathematics - Rings and Algebras}, year = 2024, month = sep, eid = {arXiv:2404.07877}, archiveprefix = {arXiv}, eprint = {2404.07877}, primaryclass = {math.LO}, adsurl = {https://ui.adsabs.harvard.edu/abs/2024arXiv240407877K}, adsnote = {Provided by the SAO/NASA Astrophysics Data System}, note = {In press}, abstract = {The idempotent semigroups (bands) that give rise to partial orders by defining $a \leq b \Leftrightarrow a \cdot b = a$ are the \emph{right-regular} bands (RRB), which are axiomatized by $x\cdot y \cdot x = y \cdot x$. In this work we consider the class of \emph{associative posets}, which comprises all partial orders underlying right-regular bands, and study to what extent the ordering determines the possible “compatible” band structures and their canonicity. We show that the class of associative posets in the signature $\{\leq\}$ is not first-order axiomatizable. We also show that the Axiom of Choice is equivalent over $\mathit{ZF}$ to the fact that every tree with finite branches is associative. We study the smaller class of “normal” posets (corresponding to right-normal bands) and give a structural characterization.} }
@article{moroni2024classification, title = {A classification of bisimilarities for general {M}arkov decision processes}, author = {Moroni, Mart{\'\i}n Santiago and S{\'a}nchez Terraf, Pedro}, year = {2024}, month = jan, eprint = {2401.09273}, archiveprefix = {arXiv}, primaryclass = {cs.LO}, abstract = { We provide a fine classification of bisimilarities between states of possibly different labelled Markov processes (LMP). We show that a bisimilarity relation proposed by Panangaden that uses direct sums coincides with ``event bisimilarity'' from his joint work with Danos, Desharnais, and Laviolette. We also extend Giorgio Bacci's notions of bisimilarity between two different processes to the case of nondeterministic LMP and generalize the game characterization of state bisimilarity by Clerc et al. for the latter.} }
@misc{chain_bounding, title = {{Chain Bounding} and the leanest proof of {Zorn}'s lemma}, author = {Incatasciato, Guillermo L. and S{\'a}nchez Terraf, Pedro}, year = 2024, url = {preprints/chain_bounding.pdf}, note = {Expository article}, keywords = {Mathematics - Logic, Mathematics - History and Overview}, year = 2024, month = apr, eid = {arXiv:2404.11638}, archiveprefix = {arXiv}, eprint = {2404.11638}, primaryclass = {math.LO}, adsurl = {https://ui.adsabs.harvard.edu/abs/2024arXiv240411638I}, adsnote = {Provided by the SAO/NASA Astrophysics Data System}, abstract = {We present an exposition of the \emph{Chain Bounding Lemma}, which is a common generalization of both Zorn's Lemma and the Bourbaki-Witt fixed point theorem. The proofs of these results through the use of Chain Bounding are amongst the simplest ones that we are aware of. As a by-product, we show that for every poset $P$ and a function $f$ from the powerset of $P$ into $P$, there exists a maximal well-ordered chain whose family of initial segments is appropriately closed under $f$. We also provide a “computer formalization” of our main results using the Lean proof assistant.} }
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