Description

The PI is proposing to work on various aspects of the inner model program, which is the program for constructing inner models for large cardinals. He will concentrate on problems coming from descriptive inner model theory and will work on advancing the theory of hod mice. The long term goal of the project is to prove the Mouse Set Conjecture, which is a central conjecture in descriptive inner model theory. Immediate goals include advancing the core model induction technique beyond its current levels, and computing better consistency lower bounds for the failure of square at a measurable cardinal. Mathematics is the language of science; it is what is used to express scientific predictions and theories. Mathematics is simply the most fundamental and indispensable part of science. However, just as scientific achievements must be tested for their validity, mathematical discoveries too must be tested for their consistency. Mathematical theories are based on axioms known as the axioms of Zermelo-Frankel set theory with the Axiom of Choice (ZFC). Provided that the proofs from ZFC and its various extensions are correct, it is believed that the resulting theories are correct. Set theory is the part of mathematics that deals with the consistency of ZFC and its extensions via Large Cardinal Axioms (LCA). Because of the celebrated Godel's incompleteness theorem one cannot hope to prove that ZFC or any of its extensions are consistent. Nevertheless, one can hope to provide naturally occurring models for ZFC + LCA in the same spirit that the set of natural numbers is the natural model of the axioms of Peano Arithmetic (PA). PA is the axiomatic system that guides and controls the usage of arithmetic in everyday life. The inner model program is a set theoretic program whose primary goal is to construct such canonical models for ZFC + LCA. The proposed project is a contribution to the inner model program. One specific objective of the proposed project is to advance the most successful recent method for constructing natural models for various extensions of ZFC, the core model induction, to new levels, and thus to construct natural models for extensions of ZFC that were unreachable before.
StatusFinished
Effective start/end date9/1/148/31/19

Funding

  • National Science Foundation (NSF)

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Inner Models
Axioms
Large Cardinals
Covering
Peano Arithmetic
Model Theory
Set Theory
Mouse
Proof by induction
Model
Measurable Cardinal
Axiom of choice
Canonical Model
Incompleteness
Natural number
Express
Lower bound
Computing
Prediction
Term