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GROUP ACTIONS ON V (B) AND THE INDEPENDENCE OF THE AXIOM OF CHOICE

John Bell

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Abstract

The idea of a group acting on a Boolean-valued model is introduced in this chapter and the properties of invariant elements of such models are used to establish the independence of the existence of definable well-orderings of the real numbers. It is shown that the axiom of choice can be falsified in certain submodels of Boolean-valued models on which certain groups act, thereby establishing the independence of the axiom of choice.

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What this paper is about

The idea of a group acting on a Boolean-valued model is introduced in this chapter and the properties of invariant elements of such models are used to establish the independence of the existence of definable well-orderings of the real numbers. It is shown that the axiom of choice can be falsified in certain submodels of Boolean-valued models on which certain groups act, thereby establishing the independence of the axiom of choice.

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Available abstract

The idea of a group acting on a Boolean-valued model is introduced in this chapter and the properties of invariant elements of such models are used to establish the independence of the existence of definable well-orderings of the real numbers. It is shown that the axiom of choice can be falsified in certain submodels of Boolean-valued models on which certain groups act, thereby establishing the independence of the axiom of choice.

Key concepts: Axiom independence, Axiom of choice, Zermelo–Fraenkel set theory, Axiom, Independence (probability theory), Urelement, Constructive set theory, Mathematics

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