Characterization of consistency. Prove that for a σ-theory T the following are equivalent: (1) T is consistent. (2) T ⊬ F , where F := ¬T and T := ∀x(x = x). (3) T ⊬ ϕ for some σ-formula ϕ.

Characterization of consistency. Prove that for a σ-theory T the following are equivalent: (1) T is consistent. (2) T ⊬ F , where F := ¬T and T := ∀x(x = x). (3) T ⊬ ϕ for some σ-formula ϕ.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Characterization of consistency. Prove that for a σ-theory T the following are equivalent:
(1) T is consistent.
(2) T ⊬ F , where F := ¬T and T := ∀x(x = x).
(3) T ⊬ ϕ for some σ-formula ϕ.

Characterization of consistency. Prove that for a o-theory T the following are equiva-
lent:
(1) T is consistent.
(2) TF, where F
(3) Tro for some o-formula q.
=
¬T and T :=\x(x = x).

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