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Full Idea
No object-language theory determines its ontology by itself. The best possible is that all models are isomorphic, in which case the ontology is determined 'up to isomorphism', but only if the domain is finite, or it is stronger than first-order.
Gist of Idea
Theory ontology is never complete, but is only determined 'up to isomorphism'
Source
Stewart Shapiro (Philosophy of Mathematics [1997], 2.5)
Book Ref
Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.55
A Reaction
This seems highly significant when ontological claims are being made, and is good support for Shapiro's claim that the structures matter, not the objects. There is a parallel in Tarksi's notion of truth-in-all-models. [The Skolem Paradox is the problem]
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] |
10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [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] |