# A weak form of Global Choice under the GCH, part I

10.01.2023 15:00 - 16:30

P. Holy (TU Wien)

In 2012, Joel Hamkins asked (on MathOverflow) whether it is possible for the universe of sets to have a linear ordering, but no wellordering (that is, global choice fails). This question, which I consider very interesting, appears to still be open. In my talk, I want to present a somewhat related result. After providing a gentle introduction to second order set theory and the principle of global choice (no knowledge on these matters is assumed), we consider a different weakening of global choice under the GCH: The minimal ordinal-connection axiom MOC due to Rodrigo Freire. It is equivalent to the statement that the universe of sets can be stratified by a subset-increasing hierarchy $$\langle K_\alpha | \alpha\in Ord\rangle$$ with each $$K_\alpha$$ of the same size as $$\alpha$$, and such that $$K_\kappa=H(\kappa)$$, the collection of sets of hereditary size less than $$\kappa$$, for every regular infinite cardinal $$\kappa$$. In this form, it clearly implies the GCH, and is easily seen to be a weak form of global choice under the GCH. We will show, using class forcing products of adding Cohen subsets of regular cardinals (without assuming any particular knowledge regarding the technique of class forcing), that MOC can consistently fail in models of the GCH, and that MOC can consistently hold while global choice fails.

This is joint work with Rodrigo Freire (University of Brasilia).

Students at Uni Wien are required to attend in person.

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#### SR 10, 1. Stock, Koling. 14-16, 1090 Wien

SR 10, 1. Stock, Koling. 14-16, 1090 Wien