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Single Idea 10146

[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX ]

Full Idea

The Axiom of Choice is a pure existence statement, without defining conditions. It was necessary to provide a foundation for Cantor's theory of transfinite cardinals and ordinal numbers, but its nonconstructive character engendered heated controversy.

Gist of Idea

Cantor's theories needed the Axiom of Choice, but it has led to great controversy

Source

Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I)

Book Ref

Feferman,S/Feferman,A.B.: 'Alfred Tarski: life and logic' [CUP 2008], p.43

The 12 ideas from 'Alfred Tarski: life and logic'

10156 'Recursion theory' concerns what can be solved by computing machines [Feferman/Feferman]
10147 The Axiom of Choice is consistent with the other axioms of set theory [Feferman/Feferman]
10148 Axiom of Choice: a set exists which chooses just one element each of any set of sets [Feferman/Feferman]
10149 Platonist will accept the Axiom of Choice, but others want criteria of selection or definition [Feferman/Feferman]
10150 The Trichotomy Principle is equivalent to the Axiom of Choice [Feferman/Feferman]
10146 Cantor's theories needed the Axiom of Choice, but it has led to great controversy [Feferman/Feferman]
10155 Both Principia Mathematica and Peano Arithmetic are undecidable [Feferman/Feferman]
10158 A structure is a 'model' when the axioms are true. So which of the structures are models? [Feferman/Feferman]
10162 Tarski and Vaught established the equivalence relations between first-order structures [Feferman/Feferman]
10160 Löwenheim-Skolem says if the sentences are countable, so is the model [Feferman/Feferman]
10159 Löwenheim-Skolem Theorem, and Gödel's completeness of first-order logic, the earliest model theory [Feferman/Feferman]
10161 If a sentence holds in every model of a theory, then it is logically derivable from the theory [Feferman/Feferman]