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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]
10147 The Axiom of Choice is consistent with the other axioms of set theory [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]
10150 The Trichotomy Principle is equivalent to the Axiom of Choice [Feferman/Feferman]
10146 Cantor's theories needed the Axiom of Choice, but it has led to great controversy [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]