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Projective determinacy.

D A Martin1, J R Steel

  • 1Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90024.

Proceedings of the National Academy of Sciences of the United States of America
|September 1, 1988
PubMed
Summary
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Projective determinacy is proven to follow from weaker large cardinal axioms. This finding is significant for set theory, showing implications from axioms less powerful than supercompact cardinals.

Area of Science:

  • Set Theory
  • Mathematical Logic

Background:

  • Large cardinal axioms are fundamental in modern set theory.
  • Projective determinacy is a major statement about the definable sets of real numbers.

Purpose of the Study:

  • To investigate the relationship between large cardinal axioms and projective determinacy.
  • To determine if weaker axioms than the existence of supercompact cardinals can imply projective determinacy.

Main Methods:

  • Axiomatic set theory
  • Model theory
  • Forcing techniques

Main Results:

  • Projective determinacy is shown to be a consequence of certain large cardinal axioms.
  • These axioms are demonstrably weaker than the existence of supercompact cardinals.

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Conclusions:

  • The consistency strength of projective determinacy is reduced.
  • This result refines our understanding of the hierarchy of large cardinal axioms and their implications.