Skip to main content
MathOverflow

Return to Question

added 178 characters in body
Source Link
user213800
user213800

Gödel's Completeness Theorem shows that first-order logic is (semantically) complete, namely, provability and validity coincide.

Gödel's Incompleteness Theorem shows that there are theories where certain true (in the intended interpretation) formulas are undecidable, and thus such theory is (syntactically) incomplete. So there must exist models of such a theory, where the aforementioned formula is false.

Are there systematic ways to construct such a model, given the theory and formula?

EDIT: Sorry for the confusion. Perhaps a better way to frame the question is: what are the/some known systematic ways to construct such a model, given the theory and formula?

Gödel's Completeness Theorem shows that first-order logic is (semantically) complete, namely, provability and validity coincide.

Gödel's Incompleteness Theorem shows that there are theories where certain true (in the intended interpretation) formulas are undecidable, and thus such theory is (syntactically) incomplete. So there must exist models of such a theory, where the aforementioned formula is false.

Are there systematic ways to construct such a model, given the theory and formula?

Gödel's Completeness Theorem shows that first-order logic is (semantically) complete, namely, provability and validity coincide.

Gödel's Incompleteness Theorem shows that there are theories where certain true (in the intended interpretation) formulas are undecidable, and thus such theory is (syntactically) incomplete. So there must exist models of such a theory, where the aforementioned formula is false.

Are there systematic ways to construct such a model, given the theory and formula?

EDIT: Sorry for the confusion. Perhaps a better way to frame the question is: what are the/some known systematic ways to construct such a model, given the theory and formula?

Source Link
user213800
user213800

Construction of models for true but unprovable formulas

Gödel's Completeness Theorem shows that first-order logic is (semantically) complete, namely, provability and validity coincide.

Gödel's Incompleteness Theorem shows that there are theories where certain true (in the intended interpretation) formulas are undecidable, and thus such theory is (syntactically) incomplete. So there must exist models of such a theory, where the aforementioned formula is false.

Are there systematic ways to construct such a model, given the theory and formula?