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STEEL’S PROGRAMME: EVIDENTIAL FRAMEWORK, THE CORE AND ULTIMATE-L
- Part of
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- Journal:
- The Review of Symbolic Logic / Volume 16 / Issue 3 / September 2023
- Published online by Cambridge University Press:
- 31 August 2021, pp. 788-812
- Print publication:
- September 2023
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- Article
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We address Steel’s Programme to identify a ‘preferred’ universe of set theory and the best axioms extending
$\mathsf {ZFC}$ by using his multiverse axioms 
$\mathsf {MV}$ and the ‘core hypothesis’. In the first part, we examine the evidential framework for 
$\mathsf {MV}$, in particular the use of large cardinals and of ‘worlds’ obtained through forcing to ‘represent’ alternative extensions of 
$\mathsf {ZFC}$. In the second part, we address the existence and the possible features of the core of 
$\mathsf {MV}_T$ (where T is 
$\mathsf {ZFC}$+Large Cardinals). In the last part, we discuss the hypothesis that the core is Ultimate-L, and examine whether and how, based on this fact, the Core Universist can justify V=Ultimate-L as the best (and ultimate) extension of 
$\mathsf {ZFC}$. To this end, we take into account several strategies, and assess their prospects in the light of 
$\mathsf {MV}$’s evidential framework.
