Compactness Theorem for First-Order Logic
... G |-F j, ÿj • There is a proof in F from G for both j and ÿj ...
... G |-F j, ÿj • There is a proof in F from G for both j and ÿj ...
course notes - Theory and Logic Group
... Proof. Suppose that such a Γ exists and let I Γ. We have M ( I iff M is finite. Consider ∆ t I u Y tLn | n ¥ 1u. Let ∆0 be a finite subset of ∆, then ∆0 t I u Y tLn | 1 ¤ n ¤ mu for some m and every structure of size m 1 is a model of ∆0 . So by the compactness theorem ∆ would have a model whi ...
... Proof. Suppose that such a Γ exists and let I Γ. We have M ( I iff M is finite. Consider ∆ t I u Y tLn | n ¥ 1u. Let ∆0 be a finite subset of ∆, then ∆0 t I u Y tLn | 1 ¤ n ¤ mu for some m and every structure of size m 1 is a model of ∆0 . So by the compactness theorem ∆ would have a model whi ...
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... 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 ...