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5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms

[ways in which two models or theories map together]

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