October 17
Calliope Ryan-Smith,
University of Leeds
The Axiom of Extendable Choice
The Partition Principle (PP) states that if there is a surjection A to B then there is an injection B to A. While this is an immediate consequence of the Axiom of Choice (AC), the question of if PP implies AC is one of the longest-standing open questions in set theory. Partial results regarding this come to us from many sources, including a theorem of Pincus that tells us that if 'for all ordinals A and all sets B, if there is a surjection B to A then there is an injection A to B' implies AC for well-orderable families of sets. We shall dissect this and related results, looking into the history of the structure of the cardinals in choiceless models and following the throughline to modern research on eccentric sets and the structure of cardinals as a partial order.