Ideas from 'Alfred Tarski: life and logic' by Feferman / Feferman [2004], by Theme Structure
[found in 'Alfred Tarski: life and logic' by Feferman,S/Feferman,A.B. [CUP 2008,9780521714013]].
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
10147

The Axiom of Choice is consistent with the other axioms of set theory

10146

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

10148

Axiom of Choice: a set exists which chooses just one element each of any set of sets

10149

Platonist will accept the Axiom of Choice, but others want criteria of selection or definition

10150

The Trichotomy Principle is equivalent to the Axiom of Choice

5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
10158

A structure is a 'model' when the axioms are true. So which of the structures are models?

10162

Tarski and Vaught established the equivalence relations between firstorder structures

5. Theory of Logic / J. Model Theory in Logic / 3. LöwenheimSkolem Theorems
10159

LöwenheimSkolem Theorem, and Gödel's completeness of firstorder logic, the earliest model theory

10160

LöwenheimSkolem says if the sentences are countable, so is the model

5. Theory of Logic / K. Features of Logics / 4. Completeness
10161

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
10155

Both Principia Mathematica and Peano Arithmetic are undecidable

10156

'Recursion theory' concerns what can be solved by computing machines

6. Mathematics / C. Sources of Mathematics / 7. Formalism
10154

Tarski's theory of truth shifted the approach away from syntax, to set theory and semantics
