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Register a new userThrowing Darts at a Ruler: Unpacking the Intuition Behind Freilings Axiom of Symmetry
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Anthony B. Morton
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It is well-known that the Continuum Hypothesis (CH) is independent of the other axioms of Zermelo-Fraenkel set theory with choice (ZFC). This raises the question of whether an intuitive justification exists for CH as an additional axiom, or conversely whether it is more intuitive to deny CH. Freilings Axiom of Symmetry (AS) is one candidate for an intuitively justifiable axiom that, when appended to ZFC, is equivalent to the denial of CH. The intuition relies on a probabalistic argument, usually cast in terms of throwing random darts at the real line, and has been defended by researchers as well as popular writers. In this note, the intuitive argument is reviewed. Following William Abram, it is suggested that while accepting CH leads directly to a counterexample to AS, this is not necessarily fatal to our intuition. Instead, we suggest, it serves to alert us to the error in a naive intuition that leaps too readily from the near-certainty of individual events to near-certainty of a joint event.
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A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set forcing extension of any inner model. The Ground Axiom is first-order expressible, and any model of ZFC has a class forcing extension which satisfies it. The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of measurable and supercompact cardinals. The related Bedrock Axiom, asserting that the universe is a set forcing extension of a model satisfying the Ground Axiom, is also first-order expressible, and its negation is consistent.
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