How to have more things by forgetting how to count them^†

Area of Science:

• Set Theory
• Mathematical Logic
• Foundations of Mathematics

Background:

• Introduces Cohen's first model, a notable example in Zermelo-Fraenkel set theory where the Axiom of Choice fails and Dedekind-finite sets of real numbers exist.
• Highlights the significance of Dedekind-finite sets in understanding models of set theory without the Axiom of Choice.

Purpose of the Study:

• Investigates the impact of forcing functions onto Dedekind-finite sets within Cohen's model.
• Aims to identify combinatorial conditions that determine whether forcing preserves Dedekind-finiteness and avoids introducing new sets of ordinals.

Main Methods:

• Employs forcing techniques over Cohen's model to introduce functions from a Dedekind-finite set to an infinite ordinal κ.
• Analyzes two cases: forcing an injective function and forcing a surjective function.
• Examines the resulting models for changes in real numbers, sets of ordinals, and Dedekind-finite sets.

Main Results:

• Forcing an injective function results in a model equivalent to adding κ Cohen reals, enumerating the Dedekind-finite set.
• Forcing a surjective function preserves Dedekind-finiteness and does not add new reals or sets of ordinals.
• Characterizes conditions equivalent to the preservation of Dedekind-finiteness under 'Adding a Cohen subset' forcing.

Conclusions:

• Provides combinatorial criteria for preserving Dedekind-finiteness in set theory models under specific forcing conditions.
• Demonstrates that the nature of the forced function (injective vs. surjective) significantly influences the resulting model's properties.
• Establishes equivalences, such as 2^A being extremally disconnected or [A]^< being Dedekind-finite, for preserving Dedekind-finiteness.