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Full Idea
An 'isomorphism' is a bijection between two sets that preserves all structural components. The interpretations of each constant symbol are mapped across, and functions map the relation and function symbols.
Gist of Idea
An 'isomorphism' is a bijection that preserves all structural components
Source
Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.4)
Book Ref
Wolf,Robert S.: 'A Tour Through Mathematical Logic' [Carus Maths Monographs 2005], p.181
Related Idea
Idea 10079 A 'bijective' function has one-to-one correspondence in both directions [Smith,P]
| 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] |