Axiomatic set theory

Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties.

Independence in ZFC

Many important statements are independent of ZFC, see the list of statements undecidable in ZFC. The independence is usually proved by forcing, that is it is shown that every countable transitive model of ZFC (plus, occasionally, large cardinal axioms) can be expanded to satisfy the statement in question, and (through a different expansion) its negation. An independence proof by forcing automatically proves independence from arithmetical statements, other concrete statements, and large cardinal axioms. Some statements independent of ZFC can be proven to hold in particular inner models, such as in the constructible universe. However, some statements that are true about constructible sets are not consistent with hypothesised large cardinal axioms.

Related Topics:
Independent - List of statements undecidable in ZFC - Forcing - Inner model - Constructible universe

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Here are some statements whose independence is provable by forcing:

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• Continuum hypothesis
• Diamond principle
• Suslin hypothesis
• Kurepa hypothesis
• Martin's axiom (Note despite the name this is NOT an axiom of ZFC)
• Axiom of Constructibility (V=L) (also not an axiom of ZFC)

Notes:

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• Consistency of V=L is not provable by forcing, but is provable through inner models: every model of ZF can be trimmed to be a model of ZFC+V=L.
• The Diamond Principle implies the Continuum Hypothesis and the negation of the Suslin Hypothesis.
• Martin's axiom plus the negation of the Continuum Hypothesis implies the Suslin Hypothesis.
• The constructible universe satisfies the Generalized Continuum Hypothesis, the Diamond Principle, Martin's Axiom and the Kurepa Hypothesis.

A variation on the method of forcing can also be used to demonstrate the consistency and unprovability of the axiom of choice, i.e., that the axiom of choice is independent of ZF. The consistency of choice can be (relatively) easily verified by proving the inner model L satisfies choice (thus every model of ZF contains a submodel of ZFC hence Con(ZF) implies Con(ZFC)). Since forcing preserves choice we cannot directly produce a model contradicting choice from a model satisfying choice. However, we can use forcing to create a model which contains a suitable submodel, namely one which satisfies ZF but not C. In particular the model constructed by adding a cohen generic and then considering only the hereditarily ordinal definable sets in that model satisfies ZF but not choice.

Related Topics:
Forcing - Axiom of choice - Cohen generic

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Forcing is perhaps the most useful method of proving independence results but not the only method. In particular Godel's 2nd incompleteness theorem which asserts that no sufficiently complex recursively axiomatizable system can prove its own consistency can be used to prove independence results. In this approach it is demonstrated that a particular statement in set theory can be used to prove the existence of a set model of ZFC and thereby demonstrate the consistency of ZFC. Since we know that Con(ZFC) (the sentence asserting the consistency of ZFC in the language of set theory) is unprovable in ZFC no statement allowing such a proof can itself be provable in ZFC. For instance this method can be used to demonstrate the existence of large cardinals is not provable in ZFC (but it is essentially impossible to shown they are consistent).

Related Topics:
Godel's 2^nd incompleteness theorem - Large cardinals

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