Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 16G / b

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- Solution b

Solution

Suppose the finitary set of formulae

S

is equivalent to the finite set

F = {f_{1}, \dots, f_{m}} .

For each

f_{j}

, the relation

S ⊢ f_{j}

has a finite proof, so it uses a finite set

R_{j} \subseteq S

of assumptions. Put

R = ⋃_{j = 1}^{m} R_{j} .

This is a finite subset of

S

, and

R ⊢ f_{j}

for every

j

, hence

R ⊢ F

. Since equivalence also gives

F ⊢ S

, transitivity of deduction yields

R ⊢ S .

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