Fri Oct 1, 2010
60 Evans Hall, 4–6 PM
Solomon Feferman (Stanford University)
What’s Definite? What’s Not?
Most axiomatizations of set theory that have been treated metamathematically have been based either entirely on classical logic or entirely on intuitionistic logic. But a natural conception of the set-theoretic universe is as an indefinite (or “potential”) totality, to which intuitionistic logic is more appropriately applied, while each set is taken to be a definite (“or completed”) totality, for which classical logic is appropriate; so on that view, set theory should be axiomatized on some correspondingly mixed basis. Similarly, in the case of predicative analysis, the natural numbers are conceived to form a definite totality, while the universe of sets (or functions) of natural numbers are viewed as an indefinite totality, so that, again, a mixed semi-constructive logic should be the appropriate one to treat the two together. In the first part of the talk I will present ways of formulating such semi-constructive systems of analysis and set theory and survey some results characterizing their proof-theoretic strength. Interestingly, though the logic is weakened, one can usefully strengthen certain principles. In the last part of the talk I will relate this work to the controversial discussion as to whether certain statements in the language of set theory such as the Continuum Hypothesis express definite or indefinite problems.
The biweekly LOGIC TEA will be held in the Alfred Tarski Room (727 Evans Hall) immediately following the Colloquium (with support from the Graduate Assembly).