### Ideas from 'Alfred Tarski: life and logic' by Feferman / Feferman , by Theme Structure

#### [found in 'Alfred Tarski: life and logic' by Feferman,S/Feferman,A.B. [CUP 2008,978-0-521-71401-3]].

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###### 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
 10146 Cantor's theories needed the Axiom of Choice, but it has led to great controversy
 10147 The Axiom of Choice is consistent with the other axioms of set theory
 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 first-order structures
###### 5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
 10159 Löwenheim-Skolem Theorem, and Gödel's completeness of first-order logic, the earliest model theory
 10160 Löwenheim-Skolem 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
 10156 'Recursion theory' concerns what can be solved by computing machines
 10155 Both Principia Mathematica and Peano Arithmetic are undecidable