structure for 'Theory of Logic'    |     alphabetical list of themes    |     unexpand these ideas

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]
     Full Idea: If the number of cats happens to equal the cherries, then it follows from the theory of models that there is a reinterpretation of the entire language that leaves all sentences unchanged in truth value while permuting the extensions of 'cat' and 'cherry'.
     From: Hilary Putnam (Reason, Truth and History [1981], Ch.2)
     A reaction: This horrifying result seems to come simply from the fact that there is an isomorphism between two models, which in turn seems to rest largely on the cardinality of the models. There seems to be something wrong with model theory here (?).
A consistent theory just needs one model; isomorphic versions will do too, and large domains provide those [Lewis]
     Full Idea: A consistent theory is, by definition, one satisfied by some model; an isomorphic image of a model satisfies the same theories as the original model; to provide the making of an isomorphic image of any given model, a domain need only be large enough.
     From: David Lewis (Putnam's Paradox [1984], 'Why Model')
     A reaction: This is laying out the ground for Putnam's model theory argument in favour of anti-realism. If you are chasing the one true model of reality, then formal model theory doesn't seem to offer much encouragement.
An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P]
     Full Idea: An 'injective' function is 'one-to-one' - each element of the output set results from a different element of the original set.
     From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1)
     A reaction: That is, two different original elements cannot lead to the same output element.
A 'surjective' ('onto') function creates every element of the output set [Smith,P]
     Full Idea: A 'surjective' function is 'onto' - the whole of the output set results from the function being applied to elements of the original set.
     From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1)
A 'bijective' function has one-to-one correspondence in both directions [Smith,P]
     Full Idea: A 'bijective' function has 'one-to-one correspondence' - it is both surjective and injective, so that every element in each of the original and the output sets has a matching element in the other.
     From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1)
     A reaction: Note that 'injective' is also one-to-one, but only in the one direction.
An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro]
     Full Idea: An axiomatization is 'categorical' if all its models are isomorphic to one another; ..hence it has 'essentially only one' interpretation [Veblen 1904].
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 1.2.1)
Categoricity can't be reached in a first-order language [Shapiro]
     Full Idea: Categoricity cannot be attained in a first-order language.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 7.3)
Theory ontology is never complete, but is only determined 'up to isomorphism' [Shapiro]
     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.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 2.5)
     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 set-theoretical hierarchy contains as many isomorphism types as possible [Shapiro]
     Full Idea: Set theorists often point out that the set-theoretical hierarchy contains as many isomorphism types as possible; that is the point of the theory.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.8)
     A reaction: Hence there are a huge number of models for any theory, which are then reduced to the one we want at the level of isomorphism.
Differences between isomorphic structures seem unimportant [George/Velleman]
     Full Idea: Mathematicians tend to regard the differences between isomorphic mathematical structures as unimportant.
     From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3)
     A reaction: This seems to be a pointer towards Structuralism as the underlying story in mathematics. The intrinsic character of so-called 'objects' seems unimportant. How theories map onto one another (and onto the world?) is all that matters?
An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS]
     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.
     From: Robert S. Wolf (A Tour through Mathematical Logic [2005], 5.4)
A theory is 'categorical' if it has just one model up to isomorphism [Horsten]
     Full Idea: If a theory has, up to isomorphism, exactly one model, then it is said to be 'categorical'.
     From: Leon Horsten (Philosophy of Mathematics [2007], §5.2)
If models of a mathematical theory are all isomorphic, it is 'categorical', with essentially one model [Rossberg]
     Full Idea: A mathematical theory is 'categorical' if, and only if, all of its models are isomorphic. Such a theory then essentially has just one model, the standard one.
     From: Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3)
     A reaction: So the term 'categorical' is gradually replacing the much-used phrase 'up to isomorphism'.