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Single Idea 10481

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

Full Idea

The models in model-theory are structures, but there is also a common use of 'model' to mean a formal theory which describes and explains a phenomenon, or plans to build it.

Gist of Idea

Models in model theory are structures, not sets of descriptions

Source

Wilfrid Hodges (Model Theory [2005], 5)

Book Ref

'Stanford Online Encyclopaedia of Philosophy', ed/tr. Stanford University [plato.stanford.edu], p.12

A Reaction

Hodges is not at all clear here, but the idea seems to be that model-theory offers a set of objects and rules, where the common usage offers a set of descriptions. Model-theory needs homomorphisms to connect models to things,

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]
10289 Up Löwenheim-Skolem: if infinite models, then arbitrarily large models [Hodges,W]
10288 Down Löwenheim-Skolem: if a countable language has a consistent theory, that has a countable model [Hodges,W]
10287 If a first-order theory entails a sentence, there is a finite subset of the theory which entails it [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]
10474 |= should be read as 'is a model for' or 'satisfies' [Hodges,W]
10475 A 'structure' is an interpretation specifying objects and classes of quantification [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]