On the question of absolute undecidability. (English) Zbl 1113.03011
Summary: The paper begins with an examination of Gödel’s views on absolute undecidability and related topics in set theory. These views are sharpened and assessed in light of recent developments. It is argued that a convincing case can be made for axioms that settle many of the questions undecided by the standard axioms and that in a precise sense the program for large cardinals is a complete success “below” CH. It is also argued that there are reasonable scenarios for settling CH and that there is not currently a convincing case to the effect that a given statement is absolutely undecidable.
MSC:
03A05 | Philosophical and critical aspects of logic and foundations |
03-03 | History of mathematical logic and foundations |
01A60 | History of mathematics in the 20th century |
03E15 | Descriptive set theory |
03E35 | Consistency and independence results |
03E50 | Continuum hypothesis and Martin’s axiom |
03E55 | Large cardinals |
03E60 | Determinacy principles |