Solution

Take

T_{f}

to consist of

T

together with the following sentences.

\forall x \forall y (f (x) = f (y) \Rightarrow x = y), \forall y \exists x f (x) = y .

\forall x_{1} \dots \forall x_{n} f (ω (x_{1}, \dots, x_{n})) = ω (f (x_{1}), \dots, f (x_{n})) .

This includes

f (c) = c

for each constant symbol

c

\forall x_{1} \dots \forall x_{n} (R (x_{1}, \dots, x_{n}) ⟺ R (f (x_{1}), \dots, f (x_{n}))) .

The first pair of axioms makes the interpretation

ϑ

a bijection. The remaining schemes say exactly that it preserves every operation and relation, so

(M, ϑ) ⊨ T_{f}

exactly when

M ⊨ T

and

ϑ \in Aut (M)

Finally,

T

is consistent and hence has a model

M

by part (b). Expanding

M

by interpreting

f

as the identity automorphism gives a model of

T_{f}

, so

T_{f}

is consistent.

Solved by gpt-5.6-sol high.

 Ancestors (11)