Ideas of Feferman / Feferman, by Theme

[American, fl. 2004, He a professor at Stanford, she an independent scholar]

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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
Cantor's theories needed the Axiom of Choice, but it has led to great controversy
The Axiom of Choice is consistent with the other axioms of set theory
The Trichotomy Principle is equivalent to the Axiom of Choice
Axiom of Choice: a set exists which chooses just one element each of any set of sets
Platonist will accept the Axiom of Choice, but others want criteria of selection or definition
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
A structure is a 'model' when the axioms are true. So which of the structures are models?
Tarski and Vaught established the equivalence relations between first-order structures
5. Theory of Logic / J. Model Theory in Logic / 3. L÷wenheim-Skolem Theorems
L÷wenheim-Skolem says if the sentences are countable, so is the model
L÷wenheim-Skolem Theorem, and G÷del's completeness of first-order logic, the earliest model theory
5. Theory of Logic / K. Features of Logics / 4. Completeness
If a sentence holds in every model of a theory, then it is logically derivable from the theory
5. Theory of Logic / K. Features of Logics / 7. Decidability
'Recursion theory' concerns what can be solved by computing machines
Both Principia Mathematica and Peano Arithmetic are undecidable
6. Mathematics / C. Sources of Mathematics / 7. Formalism
Tarski's theory of truth shifted the approach away from syntax, to set theory and semantics