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 Incoming links: First-order axiomatization of an automorphism

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 16H / b / i / Solution Created 2026-09-24 Updated 2026-09-25

Expand

L

by a unary function symbol

f

. Let

T^{*}

contain:

every sentence of $T$ ;
the first-order axiomatization of an automorphism saying that $f$ is bijective and preserves every function and relation of $L$ ;
the sentence $\exists x [f (x) \neq = x]$ ;
for each $n \geq 1$ , a sentence $θ_{n}$ asserting the existence of at least $n$ distinct elements.

Every finite subset

S \subseteq T^{*}

contains only finitely many size requirements, say no requirement beyond

θ_{N}

. By hypothesis there is a finite nonrigid model

A ⊨ T

with at least

N

elements. Interpret

f

as a nonidentity automorphism of

A

. This expanded structure satisfies

S

; indeed it satisfies every automorphism axiom, including any finitely many that occur in

S

.

Thus every finite subset of

T^{*}

is satisfiable. By the compactness theorem,

T^{*}

M^{*}

. The sentences

θ_{n}

make its domain infinite, while

\exists x [f (x) \neq = x]

and the automorphism axioms make

f^{M^{*}}

a nonidentity automorphism. The

L

M

is therefore an infinite nonrigid model of

T

. This is the compactness transfer of finite nonrigidity.

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