Mervyn Tong (University of Leeds)
- Date
- Friday 3 March 2023, 1.00 PM
- Location
- MALL
- Category
- Postgraduate Logic Seminar
The Axiom of Determinacy and Large Cardinals - Part 1: Strange Beginnings
The Axiom of Determinacy (AD) was proposed by Mycielski and Steinhaus in 1962. By then, it was already known that the Axiom of Choice (AC) implies the falsehood of AD, causing AD to be sidelined by many set-theorists. However, in 1967, Solovay showed that AD implies the measurability of ω1, and the new-found realisation that large cardinal properties are exhibited by small cardinals under AD contributed to the induction of AD into mainstream set-theoretic research.
This is a two-talk series that will culminate in Solovay’s results on some large cardinal properties exhibited by ω1 under AD (and its stronger counterpart ADR). In the first talk, we will lay the groundwork by introducing infinite games and defining AD, then discussing some of its pop-culture properties, such as its annihilation of the Banach-Tarski Paradox as well as its surprising implications on the real numbers (including the Continuum Hypothesis). If time permits, we will define a cardinal Θ that equals the successor of the continuum under AC, but can be much larger under AD.