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The Axiom of Choice, or: One Principle, Many Things

Leeds Postgraduales Seminar zur reinen Mathematik, 19. November 2015

Abstract:
I will discuss the infamous Axiom of Choice, the joker in the deck of the axioms of set theory. I will give examples of the following:

• Some principles that are equivalent to it.
• Situations where you don't need it (but think you do).
• Situations where you do need it (but think you don't).
• Problematic and/or interesting consequences of it.
• Problematic and/or interesting consequences of its negation.
• Stuff in between (if I have time).

I will attempt to bring out the underlying logical and philosophical issues at the heart of the axiom, using curious and seductive words such as 'ontology', 'reverse-logicism' and 'pluralism'. The talk should (hopefully) be accessible to people with little or no knowledge of axiomatic set theory. I will nonetheless pepper the talk with remarks for the experts, but these remarks should be such that not understanding them will not hinder one's general understanding of the talk. (That last comment is probably more confusing than anything in the talk!)