more on this theme
|
more from this thinker
Single Idea 10159
[filed under theme 5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
]
Full Idea
Before Tarski's work in the 1930s, the main results in model theory were the Löwenheim-Skolem Theorem, and Gödel's establishment in 1929 of the completeness of the axioms and rules for the classical first-order predicate (or quantificational) calculus.
Gist of Idea
Löwenheim-Skolem Theorem, and Gödel's completeness of first-order logic, the earliest model theory
Source
Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V)
Book Ref
Feferman,S/Feferman,A.B.: 'Alfred Tarski: life and logic' [CUP 2008], p.281
The
12 ideas
from 'Alfred Tarski: life and logic'
10156
|
'Recursion theory' concerns what can be solved by computing machines
[Feferman/Feferman]
|
10146
|
Cantor's theories needed the Axiom of Choice, but it has led to great controversy
[Feferman/Feferman]
|
10147
|
The Axiom of Choice is consistent with the other axioms of set theory
[Feferman/Feferman]
|
10150
|
The Trichotomy Principle is equivalent to the Axiom of Choice
[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]
|
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]
|