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
Category Archives: Math
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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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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Set Theory Jottings 17. Ordinals Revisited
In post 13 I sketched proof by transfinite induction, and definition by transfinite recursion. Let’s take a closer look at that. By definition, < means ∈ for ordinals—treat that also as vernacular. Sometimes one symbol or the other makes the meaning clearer. We must keep in mind though that we have not shown that < is a well-ordering on Ω, or even a simple ordering. Continue reading
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Set Theory Jottings 16. Axioms of ZFC
The Axioms of ZFC
The language of ZF (ℒ(ZF)) consists of basic first-order syntax, with a single binary predicate symbol ∈. Here is a list of the axioms, with a tag-line (an imprecise description) for each. Continue reading
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Set Theory Jottings 15. From Zermelo to ZFC: Formal Logic
The Role of Formal Logic Continue reading
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Set Theory Jottings 14. From Zermelo to ZFC: Replacement and Foundation
Replacement and Foundation
Thirteen years after Zermelo published his axioms, Fraenkel pointed out that they weren’t quite strong enough. For example, you couldn’t prove the existence of the set ⋃n∈ω𝒫n(0). Continue reading
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Set Theory Jottings 13. From Zermelo to ZFC: Everything’s a Set!
Zermelo’s 1908 system differs from ZFC in three respects. (1) ZFC is a “pure” set theory, without atoms. (2) ZFC includes two additional axioms, Replacement and Foundation. (3) ZFC is a first-order theory. Zermelo’s axioms are not formalized in this way. In this and the next two posts I will go into details. Continue reading
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Set Theory Jottings 12. Zermelo on “definiteness”
In the last post, I mentioned Zermelo’s 1929 paper “On the concept of definiteness in axiomatics”. By this time, people had suggested replacing “definite” with “definable in first-order logic”. Zermelo did not agree with this. Continue reading
Filed under History, Logic, Set Theory
