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Kreisel's squeezing argument (1967) shows that there is an informal notion of validity which is irreducible with respect to both model-theoretic and proof-theoretic validity of First-Order Logic (FOL), but it is coextensive with both formal notions. Besides Kreisel's informal notion of validitu, it is possible to provide other informal notions which also are coextensive with those formal notions. We will argue that the existence of more than one informal notion squeezed by the formal definitions of validity of FOL shows that the definitions of first-order validity are underdeterminated with respect to informal notions. We will argue that these issues are not only found on FOL. As an example, we will the case of the intuitionist logic.

We will continue to read and discuss the fourth chapter of Luca Incurvati's book Conceptions of Sets and the Foundations of Mathematics, Cambridge University Press, 2020.

The main goal of the proposed research is to account for the nature of the illocutionary force corresponding to hypotheses in logic and mathematics, as well as explaining its role in the logical structure of mathematical language. Thus, it gives way to a convergence of central ideas of two research projects: Hypotheses, leaded by Prof. Peter Schroeder-Heister (Uni-Tübingen), and Genericity and arbitrariness. Or how to speak of the unspeakable, coordinated by the supervisor – respectively: (i) to carry out a proof-theoretic (and) semantic analysis of hypotheses in formal logic; and (ii) to identify and explain the illocutionary elements inherent to the structure of the languages of logic and mathematics. The following guiding assumptions establish the motivation and the departure point of the research: (a) hypotheses are an essential part of the formal logical structure of mathematical language; (b) contemporary speech-act theory cannot account satisfactorily for the illocutionary nature of hypotheses in argumentative contexts; and (c) the formal apparatus of different hypotheses-based deductive calculi (e.g. natural deduction, sequent calculus, etc.) gives way to a proof-theoretic characterisation of the illocutionary role of hypotheses in logic and mathematics. The main goal is thus to be pursued by means of a proof-theoretic semantical study of noteworthy formal representations of this kind of illocution against the background of contemporary speech-act theory. Indirectly, this shall also provide for the attainment of a second goal: a critical analysis of current limitations of speech-act theory, which shall improve its explanatory power over the illocutionary phenomena to be studied – both in their occurrences within formal and mathematical contexts and elsewhere.

Set theory has performed since its formulation – and moreover, its axiomatization by Zermelo – the role of being a language common to all mathematics. The idea of unifying all mathematics, as well as being a maximal model under which all sentences of mathematics can be dealt with seemed a very fitting concept of what the purposes of set theory should be. However, the independence results stroke set-theorists with the possibility of new approaches to the unifying role assumed by set theory. By means of forcing new set theories could be conceived and thus the maximal universe that hitherto served as the foundation for all mathematics could be extended. Should set theorists then resort to assume there are many set theories instead of one? According to Maddy [Maddy, 2017], the universalist view satisfies many of the conditions needed for a theory to be considered a foundational theory for mathematics, to wit, ‘Shared Standard’ – provides a standard way of carrying proofs in mathematics –, ‘Generous Arena’ – is the theory in which every interaction between mathematical objects occur – and ‘Metamathematical Corral’ – is the theory that allows mathematicians to carry out metamathe-matical investigations – whereas the multiversist, despite also satisfying these conditions, do not do so in the best possible way. For Maddy, the main problem with the multiversist view is that it cannot satisfy ’Generous Arena’ in the sense that the universist view does. In turn, Ternullo [Ternullo, 2019] defends that the multiversism not only satisfies every condition posed by Maddy, but does so in a satisfatory way, which better comes in terms with what has been developed in set theory contemporarily. My goal is to compare Maddy’s and Ternullo’s argument and, assuming  that set theory is the best foundational theory for mathematics, give criteria that try to determine the question: which view should we adopt: the universist or the multiversist?

References

[Maddy, 2017] Maddy, P. (2017). Set-theoretic foundations. In Andrés Eduardo Caicedo, James Cummings, P. K. and Larson, P. B., editors, Foundations of Mathematics, volume 690 of Contemporary Mathematics, pages 289–322. American Mathematical Society.

[Ternullo, 2019] Ternullo, C. (2019). Maddy on the multiverse. In Stefania Centrone, Deborah Kant, D. S., editor, Reflections on the Foundations of Mathematics - Univalent Foundations, Set Theory and General Thoughts, volume 407 of Studies in Epistemology, Logic, Methodology,and Philosophy of Science, pages 43–78. Springer, 1st edition.

In this talk we show that the boolean-valued construction of models of set theory can be applied to algebras whose underlying logic is non-classical. We start by presenting the first results in this area by Lowe and Tarafder (2015) and  some recent extensions of their  work. We then show that there are algebra-valued models of ZF, whose underlying logic is neither classical, nor intuitionistic. In the end we will show how to extend independence results for classical ZF to a non-classical setting, showing, as an example, the independence of the Continuum Hypothesis.