6 papers · 1 filter
Class choice and the surprising weakness of Kelley-Morse set theory
Victoria Gitman, Joel David Hamkins, Thomas A. Johnstone
Kelley-Morse set theory KM is weaker than generally supposed and fails to prove several principles that may be desirable in a foundational second-order set theory. Even though KM i…
The modal logic of arithmetic potentialism and the universal algorithm
Joel David Hamkins
I investigate the modal commitments of various conceptions of the philosophy of arithmetic potentialism. Specifically, I shall consider the potentialist conceptions arising from a…
A potentialist conception of ultrafinitism
Joel David Hamkins
I shall explore various senses in which ultrafinitism can be fruitfully understood as engaging with a potentialist perspective in mathematics. First, I explain that every model …
Did Turing prove the undecidability of the halting problem?
Joel David Hamkins, Theodor Nenu
We discuss the accuracy of the attribution commonly given to Turing's 1936 paper "On computable numbers..." for the computable undecidability of the halting problem, coming eventua…
Satisfaction is not absolute
Joel David Hamkins, Ruizhi Yang
We prove that the satisfaction relation of first-order logic is not absolute between models of set theory having the structure and the…
How the continuum hypothesis could have been a fundamental axiom
Joel David Hamkins
I describe a simple historical thought experiment showing how we might have come to view the continuum hypothesis as a fundamental axiom, one necessary for mathematics, indispensab…