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Abstract
Cohen's first model is a model of ZermeloFraenkel set theory in which there is a Dedekindfinite set of real numbers, and it is perhaps the most famous model where the Axiom of Choice fails. We force over this model to add a function from this Dedekindfinite set to some infinite ordinal κ. In the case that we force the function to be injective, it turns out that the resulting model is the same as adding κ Cohen reals to the ground model, and that we have just added an enumeration of the canonical Dedekindfinite set. In the case where the function is merely surjective it turns out that we do not add any reals, sets of ordinals, or collapse any Dedekindfinite sets. This motivates the question if there is any combinatorial condition on a Dedekindfinite set A which characterises when a forcing will preserve its Dedekindfiniteness or not add new sets of ordinals. We answer this question in the case of 'Adding a Cohen subset' by presenting a varied list of conditions each equivalent to the preservation of Dedekindfiniteness. For example, 2 A is extremally disconnected, or [A] <ω is Dedekindfinite.
Original language  English 

Article number  20190782 
Journal  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 
Volume  476 
Issue number  2239 
DOIs  
Publication status  Published  29 Jul 2020 
Keywords
 Axiom of Choice
 Cohen forcing
 Cohen's first model
 Dedekindfinite sets
 symmetric extensions
Profiles

Asaf Karagila
 School of Mathematics  UKRI Future Leaders Fellow
 Logic  Member
Person: Research Group Member, Academic, Teaching & Research
Projects
 1 Finished

High Forcing Axioms: Forcing Axioms for the Uncountable. Newton International Fellowship
Aspero, D. & Karagila, A.
1/03/18 → 31/03/20
Project: Fellowship