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Single Idea 13533

[filed under theme 5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models ]

Full Idea

The three foundations of first-order model theory are the Completeness theorem, the Compactness theorem, and the Löwenheim-Skolem-Tarski theorem.

Gist of Idea

First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem

Source

Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.3)

Book Ref

Wolf,Robert S.: 'A Tour Through Mathematical Logic' [Carus Maths Monographs 2005], p.172

A Reaction

On p.180 he notes that Compactness and LST make no mention of |- and are purely semantic, where Completeness shows the equivalence of |- and |=. All three fail for second-order logic (p.223).

The 19 ideas from 'A Tour through Mathematical Logic'

13519 Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS]
13518 Modern mathematics has unified all of its objects within set theory [Wolf,RS]
13520 A 'tautology' must include connectives [Wolf,RS]
13524 Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS]
13522 Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS]
13521 Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS]
13523 Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS]
13525 Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS]
13526 Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS]
13529 Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS]
13530 An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS]
13531 Model theory reveals the structures of mathematics [Wolf,RS]
13532 Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS]
13534 In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS]
13535 First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS]
13533 First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS]
13537 An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS]
13538 If a theory is complete, only a more powerful language can strengthen it [Wolf,RS]
13539 The LST Theorem is a serious limitation of first-order logic [Wolf,RS]