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Single Idea 10287

[filed under theme 5. Theory of Logic / K. Features of Logics / 6. Compactness ]

Full Idea

Compactness Theorem: suppose T is a first-order theory, ψ is a first-order sentence, and T entails ψ. Then there is a finite subset U of T such that U entails ψ.

Gist of Idea

If a first-order theory entails a sentence, there is a finite subset of the theory which entails it

Source

Wilfrid Hodges (First-Order Logic [2001], 1.10)

Book Ref

'Blackwell Guide to Philosophical Logic', ed/tr. Goble,Lou [Blackwell 2001], p.29

A Reaction

If entailment is possible, it can be done finitely.

The 16 ideas from Wilfrid Hodges

10282 Logic is the study of sound argument, or of certain artificial languages (or applying the latter to the former) [Hodges,W]
10287 If a first-order theory entails a sentence, there is a finite subset of the theory which entails it [Hodges,W]
10288 Down Löwenheim-Skolem: if a countable language has a consistent theory, that has a countable model [Hodges,W]
10289 Up Löwenheim-Skolem: if infinite models, then arbitrarily large models [Hodges,W]
10283 A formula needs an 'interpretation' of its constants, and a 'valuation' of its variables [Hodges,W]
10284 There are three different standard presentations of semantics [Hodges,W]
10285 I |= φ means that the formula φ is true in the interpretation I [Hodges,W]
10286 A 'set' is a mathematically well-behaved class [Hodges,W]
10473 Model theory studies formal or natural language-interpretation using set-theory [Hodges,W]
10475 A 'structure' is an interpretation specifying objects and classes of quantification [Hodges,W]
10474 |= should be read as 'is a model for' or 'satisfies' [Hodges,W]
10476 The idea that groups of concepts could be 'implicitly defined' was abandoned [Hodges,W]
10478 Since first-order languages are complete, |= and |- have the same meaning [Hodges,W]
10477 |= in model-theory means 'logical consequence' - it holds in all models [Hodges,W]
10480 First-order logic can't discriminate between one infinite cardinal and another [Hodges,W]
10481 Models in model theory are structures, not sets of descriptions [Hodges,W]