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Single Idea 14212

[filed under theme 5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms ]

Full Idea

A consistent theory is, by definition, one satisfied by some model; an isomorphic image of a model satisfies the same theories as the original model; to provide the making of an isomorphic image of any given model, a domain need only be large enough.

Gist of Idea

A consistent theory just needs one model; isomorphic versions will do too, and large domains provide those

Source

David Lewis (Putnam's Paradox [1984], 'Why Model')

Book Ref

Lewis,David: 'Papers in Metaphysics and Epistemology' [CUP 1999], p.68

A Reaction

This is laying out the ground for Putnam's model theory argument in favour of anti-realism. If you are chasing the one true model of reality, then formal model theory doesn't seem to offer much encouragement.

Related Idea

Idea 14207 If cats equal cherries, model theory allows reinterpretation of the whole language preserving truth [Putnam]

The 13 ideas with the same theme [ways in which two models or theories map together]:

14207 If cats equal cherries, model theory allows reinterpretation of the whole language preserving truth [Putnam]
14212 A consistent theory just needs one model; isomorphic versions will do too, and large domains provide those [Lewis]
10078 An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P]
10077 A 'surjective' ('onto') function creates every element of the output set [Smith,P]
10079 A 'bijective' function has one-to-one correspondence in both directions [Smith,P]
13636 An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro]
13670 Categoricity can't be reached in a first-order language [Shapiro]
10214 Theory ontology is never complete, but is only determined 'up to isomorphism' [Shapiro]
10238 The set-theoretical hierarchy contains as many isomorphism types as possible [Shapiro]
10105 Differences between isomorphic structures seem unimportant [George/Velleman]
13537 An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS]
10884 A theory is 'categorical' if it has just one model up to isomorphism [Horsten]
10758 If models of a mathematical theory are all isomorphic, it is 'categorical', with essentially one model [Rossberg]