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Single Idea 10214

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

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]

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]