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 2008,978-0-521-71401-3]].
green numbers give full details |
back to texts
|
expand these ideas
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
|
10150
|
The Trichotomy Principle is equivalent to the Axiom of Choice
|
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
|
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
10160
|
Löwenheim-Skolem says if the sentences are countable, so is the model
|
10159
|
Löwenheim-Skolem Theorem, and Gödel's completeness of first-order logic, the earliest model theory
|
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
|