Realizing Mahlo set theory in type theory

Abstract After introducing the large set notion of Mahloness, this paper shows that constructive set theory with an axiom asserting the existence of a Mahlo set has a realizability interpretation in an extension of Martin-Löf type theory developed by A. Setzer

Realization of analysis into Explicit Mathematics

We define a novel interpretation of second order arithmetic into Explicit Mathematics. As a difference from standard -interpretation, which was used before and was shown to interpret only subsystems proof-theoretically weaker than T 0. our interpretation can reach the full strength of T 0. The -interpretation is an adaptation of Kleene's recursive readability, and is applicable only to intuitionistic theorie

Proof theory and Martin-Löf Type Theory

In this article an overview over the work of the author on developing proof theoretic strong extensions of Martin-Loef Type Theory including precise proof theoretic bounds is given. It presents the first publication of the proof theoretically strongest known extensions of Martin-Loef Type Theory, namely the hyper-Mahlo Universe, the hyper-alpha-Mahlo universe, the autononomous Mahlo universe and the Pi_3-reflecting universe. This is part of a proof theoretic program in developing proof theoretic as strong as possible constructive theories in order to obtain a constructive underpinning of strong classical theories with a full proof theoretic analysis

Incompatible bounded category forcing axioms

We introduce bounded category forcing axioms for well-behaved classes $\Gamma$. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe $H_{\lambda_\Gamma^+}$ modulo forcing in $\Gamma$, for some cardinal $\lambda_\Gamma$ naturally associated to $\Gamma$. These axioms naturally extend projective absoluteness for arbitrary set-forcing--in this situation $\lambda_\Gamma=\omega$--to classes $\Gamma$ with $\lambda_\Gamma>\omega$. Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms, but can be forced under mild large cardinal assumptions on $V$. We also show the existence of many classes $\Gamma$ with $\lambda_\Gamma=\omega_1$, and giving rise to pairwise incompatible theories for $H_{\omega_2}$.Comment: arXiv admin note: substantial text overlap with arXiv:1805.0873

The modal logic of set-theoretic potentialism and the potentialist maximality principles

We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on those of Hamkins, Leibman and L\"owe, including the use of buttons, switches, dials and ratchets. Among the potentialist conceptions we consider are: rank potentialism (true in all larger $V_\beta$); Grothendieck-Zermelo potentialism (true in all larger $V_\kappa$ for inaccessible cardinals $\kappa$); transitive-set potentialism (true in all larger transitive sets); forcing potentialism (true in all forcing extensions); countable-transitive-model potentialism (true in all larger countable transitive models of ZFC); countable-model potentialism (true in all larger countable models of ZFC); and others. In each case, we identify lower bounds for the modal validities, which are generally either S4.2 or S4.3, and an upper bound of S5, proving in each case that these bounds are optimal. The validity of S5 in a world is a potentialist maximality principle, an interesting set-theoretic principle of its own. The results can be viewed as providing an analysis of the modal commitments of the various set-theoretic multiverse conceptions corresponding to each potentialist account.Comment: 36 pages. Commentary can be made about this article at http://jdh.hamkins.org/set-theoretic-potentialism. Minor revisions in v2; further minor revisions in v

Survey on the Tukey theory of ultrafilters

This article surveys results regarding the Tukey theory of ultrafilters on countable base sets. The driving forces for this investigation are Isbell's Problem and the question of how closely related the Rudin-Keisler and Tukey reducibilities are. We review work on the possible structures of cofinal types and conditions which guarantee that an ultrafilter is below the Tukey maximum. The known canonical forms for cofinal maps on ultrafilters are reviewed, as well as their applications to finding which structures embed into the Tukey types of ultrafilters. With the addition of some Ramsey theory, fine analyses of the structures at the bottom of the Tukey hierarchy are made.Comment: 25 page

Set Theory

This workshop included selected talks on pure set theory and its applications, simultaneously showing diversity and coherence of the subject