The Prospects of Unlimited Category Theory: Doing What Remains to Be Done

Ernst, Michael

doi:10.1017/s1755020314000495

Cited by 10 publications

(6 citation statements)

References 15 publications

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“…taking unions, products, etc.). He then explains his own result (in [Ernst, 2015]) that (1.)-(3.) are not jointly satisfiable, before discussing the significance of this fact for categorial foundations.…”

Section: Purementioning

confidence: 91%

Elaine Landry, ed. Categories for the Working Philosopher

2018

Philosophia Mathematica

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Section: Purementioning

confidence: 91%

Elaine Landry, ed. Categories for the Working Philosopher

2018

Philosophia Mathematica

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“…Jensen’s model was welcomed by Feferman (see [42] and the papers cited there, but also [43]) for building a model with the category of all categories, so it is more than a curiosity. Jensen [44, p251] gives this axiom:…”

Section: Quine Beckonsmentioning

confidence: 99%

Weak set theories in foundational debates

Mathias

2023

Phil. Trans. R. Soc. A.

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Perception of the relationship of the discipline of logic to other exact sciences changes with the years. No twentieth-century proposal for a single logical system that would support the whole of mathematics satisfied everyone, so weaker formal systems with applications in many different contexts are now sought, in mathematics, philosophy, computer science or elsewhere. As always, logicians help by asking such questions as ‘Is this theory correct for the given context?’ and ‘Can it be used in other contexts?’ The Introduction of this paper surveys the history of logic and its links to other scientific disciplines such as biology, economics, engineering and physics, and some details of current and future research projects are given. Then the very weak set theory P ROVI is introduced and its support for the techniques of constructibility (Gödel 1935) and forcing (Cohen PJ 1963 The independence of the continuum hypothesis, I. Proc. Natl Acad. Sci. USA 50 , 1143–1148. ( doi:10.1073/pnas.50.6.1143 )) is given in outline, full details being already in print. Relations with other known techniques are explored and developed. New results coming from the study of illfounded ω -models of P ROVI and other systems are given; and new formal systems in the style of Quine (1937 Quine WV. 1936 Set-theoretic foundations for logic. J. Symb. Log. 1 , 45–57. ( doi:10.2307/2268548 )) are described. This article is part of the theme issue ‘Modern perspectives in Proof Theory’.

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“…A long history of inconsistent mathematics (infinitesimals from the 17 th to the 19 th century), the formal contradictions that still survive in theoretical physics, and contemporary paraconsistent mathematics (Mortensen 1995) all demonstrate that his position is not as unreasonable as it may sound to a philosopher or mathematician raised on Hilbertian formalism. That category theorists sometimes exceed the awkward bounds of Grothendieck Universes, despite their awareness lurking contradictions (Ernst 2015, Krömer et al 2009) is also witness to this fact.…”

Section: Pluralist Foundationsmentioning

confidence: 99%