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Single Idea 13636
[filed under theme 5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
]
Full Idea
An axiomatization is 'categorical' if all its models are isomorphic to one another; ..hence it has 'essentially only one' interpretation [Veblen 1904].
Gist of Idea
An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation
Source
Stewart Shapiro (Foundations without Foundationalism [1991], 1.2.1)
Book Ref
Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.12
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]
|
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]
|