59 research outputs found
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 ); Grothendieck-Zermelo potentialism (true in all larger
for inaccessible cardinals ); 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
Hierarchies Ontological and Ideological
Godel claimed that Zermelo-Fraenkel set theory is `what becomes of the theory of
types if certain superfluous restrictions are removed'. The aim of this paper is to develop
a clearer understanding of Godel's remark, and of the surrounding philosophical terrain. In
connection with this, we discuss some technical issues concerning in finitary type theories and the programme of developing the semantics for higher-order languages in other higher-order languages
Abstraction and Grounding
The idea that some objects are metaphysically “cheap” has wide appeal. An influential version of the idea builds on abstractionist views in the philosophy of mathematics, on which numbers and other mathematical objects are abstracted from other phenomena. For example, Hume’s Principle states that two collections have the same number just in case they are equinumerous, in the sense that they can be correlated one-to-one:
(HP) #xx=#yy iff xx≈yy.
The principal aim of this article is to use the notion of grounding to develop this sort of abstractionism. The appeal to grounding enables a unified response to the two main challenges that confront abstractionism. First, we must explicate the metaphor of meta- physical “cheapness.” Second, we must rebut the “bad company” objection, which rejects abstraction principles like (HP) as tarnished by their similarity to inconsistent principles like Frege’s Basic Law V. By enforcing a simple requirement that all abstraction be properly grounded, we propose a unified solution to these two hard, and prima facie unrelated, problems. On our view, grounded abstraction simultaneously ensures “cheap” abstracta and permissible abstraction
Only up to isomorphism? Category Theory and the Foundations of Mathematics
Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can be made for its logical and conceptual autonomy. Its justificatory autonomy turns on whether the objects of a foundation for mathematics should be specified only up to isomorphism, as is customary in other branches of contemporary mathematics. If such a specification suffices, then a category-theoretical approach will be highly appropriate. But if sets have a richer 'nature' than is preserved under isomorphism, then such an approach will be inadequate
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