Continuous Model Theory - Math @ McMaster University
... We fix a language L and a complete theory T in this language. For a tuple of sorts S from L, we define the set SS (T ) to be all complete types defined on FS . The logic topology on SS (T ) is the restriction of the weak-* topology on the dual space of FS . Equivalently, the collection of sets {p ∈ ...
... We fix a language L and a complete theory T in this language. For a tuple of sorts S from L, we define the set SS (T ) to be all complete types defined on FS . The logic topology on SS (T ) is the restriction of the weak-* topology on the dual space of FS . Equivalently, the collection of sets {p ∈ ...
Notes
... does it. This is a function that takes a pair of functions as its argument and returns their composition. The proof tree that establishes the typing of this function is essentially an intuitionistic proof of the transitivity of implication. Here is another example. Consider the formula ∀P, Q, R . (P ...
... does it. This is a function that takes a pair of functions as its argument and returns their composition. The proof tree that establishes the typing of this function is essentially an intuitionistic proof of the transitivity of implication. Here is another example. Consider the formula ∀P, Q, R . (P ...
Propositions as types
... think of ¬φ as corresponding to a function τ → 0. We have seen functions that accept a type and don’t return a value before: continuations have that behavior. If φ corresponds to τ , a reasonable interpretation of ¬φ is as a continuation expecting a τ . Negation corresponds to turning outputs into i ...
... think of ¬φ as corresponding to a function τ → 0. We have seen functions that accept a type and don’t return a value before: continuations have that behavior. If φ corresponds to τ , a reasonable interpretation of ¬φ is as a continuation expecting a τ . Negation corresponds to turning outputs into i ...
PDF
... 3. We will in due course explore the notion of constructive “truth” or evidence. We will see that we can’t decide whether there is evidence for a given proposition, i.e. whether a programming task is solvable. 4. The notion of evidence/truth depends on a theory of types and programs which we are gra ...
... 3. We will in due course explore the notion of constructive “truth” or evidence. We will see that we can’t decide whether there is evidence for a given proposition, i.e. whether a programming task is solvable. 4. The notion of evidence/truth depends on a theory of types and programs which we are gra ...