Foundations of Mathematics: Essays in homor of W. Hugh Woodin's 60th birthday, A. Caicedo, P. Koellner, P. Larson, eds.
This paper explores the foundational roles played by set theory, evaluates the suggestion that ca... more This paper explores the foundational roles played by set theory, evaluates the suggestion that category theory would be a better foundation than set theory, and considers the prospects for replacing a foundation based on a set-theoretic universe with one based on a set-theoretic multiverse.
Uploads
Papers by Penelope Maddy
This paper reconstructs Steel’s multiverse project in his ‘Gödel’s program’ (Steel [2014]), first by comparing it with those of Hamkins ([2012]) and Woodin ([2011]), then by detailed analysis of what’s present in Steel’s brief text. In particular, we reconstruct his notion of a ‘natural’ theory, formulate his multiverse axioms and his translation function, and assess the resulting status of the Continuum Hypothesis. In the end, we reconceptualize the defect that Steel thinks CH might suffer from and isolate what it would take to remove it while working within his framework. As our goal is to present as coherent and compelling a philosophical and mathematical story as we can, we allow ourselves to augment Steel’s story in places (e.g., in the treatment of Amalgamation) and to depart from it in others (e.g., the removal of ‘meaning’ from the account). The relevant mathematics is laid out in the appendices.
International Journal for the Study of Skepticism
This paper reconstructs Steel’s multiverse project in his ‘Gödel’s program’ (Steel [2014]), first by comparing it with those of Hamkins ([2012]) and Woodin ([2011]), then by detailed analysis of what’s present in Steel’s brief text. In particular, we reconstruct his notion of a ‘natural’ theory, formulate his multiverse axioms and his translation function, and assess the resulting status of the Continuum Hypothesis. In the end, we reconceptualize the defect that Steel thinks CH might suffer from and isolate what it would take to remove it while working within his framework. As our goal is to present as coherent and compelling a philosophical and mathematical story as we can, we allow ourselves to augment Steel’s story in places (e.g., in the treatment of Amalgamation) and to depart from it in others (e.g., the removal of ‘meaning’ from the account). The relevant mathematics is laid out in the appendices.
International Journal for the Study of Skepticism
philosophy by focusing with some care on the specific conclusions that a
sampling of prominent figures have attempted to draw – the same
theorem might successfully support one such conclusion while failing to
support another. It begins historically, with Dedekind, Zermelo, and Kreisel,
casting doubt on received readings of the latter two and highlighting the
success of all three in achieving what are argued to be their actual goals.
These earlier uses of categoricity arguments are then compared and
contrasted with the more recent work of Parsons and the coauthors Button and Walsh. Highlighting the roles of first- and second-order theorems, of external and (two varieties of) internal theorems, the Element eventually concludes that categoricity arguments have been more effective in historical cases that reflect philosophically on internal mathematical matters than in more recent uses for questions of pre-theoretic metaphysics.