<logic> A system is omega-consistent iff there is no wff W with one free variable such that (1) Wn is a theorem for every natural number n, and (2) ~(x)Wx is also a theorem.
[Glossary of First-Order Logic]
omega-inconsistency
There is a wff W with one free variable such that (1) Wn is a theorem for every natural number n, and (2) ~(x)Wx is also a theorem.
[Glossary of First-Order Logic]
omega-incompleteness
There is a wff W with one free variable such that (1) Wn is a theorem for every natural number n, and (2) (x)Wx is not a theorem. Wx is true for every n by instantiation but not by generalization.
See k-validity
[Glossary of First-Order Logic]
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