If in a consistent and complete system we can deduce $A$, that is $⊢_J A$, what can be said about the valuation $V(A)$ or the truth value of $A$?
2026-03-25 01:20:39.1774401639
Is a theorem of a consistent and complete extension of L a tautology?
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If a formula $A$ is derivable from a set $J$ of axioms (i.e. it is a theorem of the theory having the said axioms), it is a logical consequence of the axioms, i.e. it is true in every interpretation satisfying the axioms.
It is not necessarily valid (i.e. true in every interpretation): the formula $1+1=2$ is provable from the axioms of arithmetic, but it is not valid.
And it is not necessarily a tautology: the formula $\forall x (x=x)$ is valid but it is not a tautology.