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Ideas for 'Philosophy of Mathematics', 'Letters to William Molyneux' and 'Existentialism and Humanism'

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5 ideas

5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
10239 The central notion of model theory is the relation of 'satisfaction' [Shapiro]
     Full Idea: The central notion of model theory is the relation of 'satisfaction', sometimes called 'truth in a model'.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.9)
10240 Model theory deals with relations, reference and extensions [Shapiro]
     Full Idea: Model theory determines only the relations between truth conditions, the reference of singular terms, the extensions of predicates, and the extensions of the logical terminology.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.9)
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
10214 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]
10238 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.
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
10234 Any theory with an infinite model has a model of every infinite cardinality [Shapiro]
     Full Idea: The Löwenheim-Skolem theorems (which apply to first-order formal theories) show that any theory with an infinite model has a model of every infinite cardinality.
     From: Stewart Shapiro (Philosophy of Mathematics [1997], 4.8)
     A reaction: This aspect of the theorems is the Skolem Paradox. Shapiro argues that in first-order this infinity of models for arithmetic must be accepted, but he defends second-order model theory, where 'standard' models can be selected.