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zf - Introduction to Axiomatic Set Theory Prepared by Prof...

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Introduction to Axiomatic Set Theory Prepared by: Prof. Kendra Cooper Contents Axiomatic Set Theory Frege Russell Zermelo-Fraenkel ZF Set Theory Axioms - Summary Continuum Hypothesis Problems Naïve set theory is “elegant, simple, intuitive, and generally lovely; the only problem is that it leads straight to disaster” (Curtis Brown 2002) Frege’s Axiomatization of Naïve Set Theory Naïve set theory extends predicate logic: - add a new style of variable, exclusively for sets (e.g. a,b,c) o i.e., these variables only range over sets - add a new relation symbol, , to represent set membership (e.g., c S) - add two axioms concerning the properties of the set membership relation Axiom Axioms are well formed formulae (wff) that are “known to be true” Axioms are used as a basis from which to reason An example of an axiom from propositional logic: p ∨¬ p o truth table shows this is a Tautology o Axiom of Extensionality a b [ x (x a x b) a=b] What does this axiom mean?

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o Axiom of Comprehension a x [x a P(x)] where P(x) can be replaced by any formula in predicate logic that contains x as a free (unbound) variable P(x) may also contain additional free variables z1, z2, …zn; the axiom schema must be prefixed by universal quantifiers z1 z2 … zn) o e.g., z1 z2 a x [x a P(x, z1,z2)] What does this axiom schema mean? For any predicate that can be formulated in predicate logic (plus the set membership relation), there is a set of all items that satisfy that formula Naïve set theory was shown to be inconsistent e.g. Russell’s paradox (the barber shaves everyone who doesn’t shave himself) in an inconsistent theory, you can derive a contradiction o i.e., can derive p and can also derive ¬ p other mathematicians attempted to correct this… Russell pointed out the flaw in Frege’s set theory (1902) advocated correcting the flaw with Type Theory Russell’s sets were classified into multiple levels o Level 0 had only basic elements that are not sets (these are called urelements) o Level 1 had sets of urelements o Level 2 sets could contain urelements or Level 1 sets o Level 3 sets could contain urelements, Level 1 sets, or Level 2 sets o And so on… In this set theory no loops can occur. As a result, there is no set of all sets that do not contain themselves, and the paradox is averted.
Zermelo-Fraenkel Zermelo-Fraenkel (ZF) set theory o Zermelo 1908 – first axiomatization o Fraenkel 1922 – found a weakness in Zermelo’s system, proposed a correction o Skolem – 1922 Fraenkel’s correction reformulated as a new

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zf - Introduction to Axiomatic Set Theory Prepared by Prof...

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