75 research outputs found
Laver and set theory
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
Indeterminateness and `The' Universe of Sets: Multiversism, Potentialism, and Pluralism
In this article, I survey some philosophical attitudes to talk concerning `the' universe of sets. I separate out four different strands of the debate, namely: (i) Universism, (ii) Multiversism, (iii) Potentialism, and (iv) Pluralism. I discuss standard arguments and counterarguments concerning the positions and some of the natural mathematical programmes that are suggested by the various views
Intrinsic Justification for Large Cardinals and Structural Reflection
We deal with the complex issue of whether large cardinals are intrinsically
justified principles of set theory (we call this the Intrinsicness Issue). In
order to do this, we review, in a systematic fashion, (1.) the abstract
principles that have been formulated to motivate them, as well as (2.) their
mathematical expressions, and assess the justifiability of both on the grounds
of the (iterative) concept of set. A parallel, but closely linked, issue is
whether there exist mathematical principles able to yield all known large
cardinals (we call this the Universality Issue), and we also test principles
for their responses to this issue. Finally, we discuss the first author's
Structural Reflection Principles (SRPs), and their response to Intrinsicness
and Universality. We conclude the paper with some considerations on the global
justifiability of SRPs, and on alternative construals of the concept of set
also potentially able to intrinsically justify large cardinals
On -Strongly Measurable Cardinals
We prove several consistency results concerning the notion of
-strongly measurable cardinal in HOD. In particular, we show that is it
consistent, relative to a large cardinal hypothesis weaker than , that every successor of a regular cardinal is -strongly
measurable in HOD
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