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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 16I / iii / Solution

No such set of propositions exists. Suppose that

S

had precisely the total injections as its models, fix

x_{0} \in X

, and adjoin

\neg p_{x_{0}, y} (y \in Y)

to

S

. Every finite subset of the enlarged theory has a model. Indeed, it excludes only finitely many possible values

F \subset Y

for

x_{0}

. Starting with any injection

h : X \to Y

, either

h (x_{0}) \in / F

, or one may choose

x_{1} \neq = x_{0}

with

h (x_{1}) \in / F

and swap the values of

x_{0}

and

x_{1}

.

The propositional compactness theorem would therefore give a model of the whole enlarged theory. Its relation

f_{v}

is a model of

S

but has no value at

x_{0}

, contradicting the assumed description of the models of

S

. This is the compactness obstruction to expressing totality over an infinite codomain.

Solved by gpt-5.6-sol high.

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