Let’s put it all together. Recall that Gödel proved three main results about L: Continue reading
Set Theory Jottings 23. Absoluteness of Constructibility
Now we turn to the absolutness of the notion of constructibility. There is a formula Λ(x) which says that x is constructible, and which holds in L iff it holds in V. Λ(x) is not Δ0, nor is it absolute over all transitive classes, so some subtleties come into play. (It is absolute between models of ZF.) Continue reading
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Set Theory Jottings 22. Absoluteness
Let’s look again at the notion of definability, rewritten slightly: for any set A, x⊆A is definable over A if there is a first-order formula φ(y,ū) and elements ā∈A such that Continue reading
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Set Theory Jottings 21. The Constructible Universe
The constructible universe is traditionally denoted L. L is a subclass of V and is a proper class. Gödel proved three things about L: Continue reading
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From Kepler to Ptolemy 23
The Astronomia nova: “One Sustained Argument”
In his classic The Sleepwalkers, Arthur Koestler said this about the Astronomia nova: Continue reading
Set Theory Jottings 20. Consistency of GCH and AC: Overview
In 1938 Gödel published “The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis”. This paper introduces the constructible universe, a so-called inner model of ZFC. This is a class L that satisfies the ZFC axioms, plus GCH, provided that V satisfies the ZF axioms. So if ZF is consistent, then so is ZF+AC+GCH. Continue reading
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Set Theory Jottings 19. GCH implies AC.
Sierpiński’s Theorem: GCH implies AC Continue reading
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Set Theory Jottings 18. The Axiom of Determinacy
Just denying the axiom of choice doesn’t buy you much. If you’re going to throw away AC, you should add some powerful incompatible axiom in its place. The Axiom of Determinacy (AD) has been studied in this light. Continue reading
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Diagonal Argument 