# A syntactic consistency proof for NaDSet

## Abstract

NaDSet is a natural deduction based logic and set theory with applications in programming semantics, category theory and the theory of non-well-founded sets. The paradoxes are resolved through a nominalist interpretation of atomic formulas requiring a distinction between use and mention. A form of second order arithmetic can be derived within it. Here an outline of a syntactic consistency proof of the theory is provided in contrast to the existing semantic proofs for cut-elimination in second order logic.

## Notation

- dg1
the degree of a degree path of largest degree in Derv that does not pass through the right premiss of cut 1 and that has the

*G1*in Γ′ → Θ′,*G1*as first element- dg2
the degree of a degree path of largest degree in Derv that does not pass through the left premiss of cut 2 and that has the

*G2*in*G2*, Δ′ → Λ′ as first element- dpA1 the degree of a degree path of largest degree in Derv that does not pass through the right premiss of cut 1 or the right cut formula of cut 3 and that has the
*A*in the left premiss Γ → Θ,*A*of cut 3 as first element dpA2 the degree of a degree path of largest degree in Derv that does not pass through the left premiss of cut 2 or the left cut formula of cut 3 and that has the

*A*in the right premiss A, Δ → Λ of cut 3 as first element- d1, d2, d3
the degrees of cuts 1, 2 & 3 in Derv

- d4
the degree of cuts 4 in Derv and 4′ and 4 in Derv

^{*}- h3
the height of the premisses of cuts 3 & 4 in Derv

- h
the height of the conclusion of cut 4 in Derv and cut 5 in Derv

^{*}- d3′, d3, d5
the degrees of cuts 3′, 3 & 5 in Derv

^{*}- h5
the height of the premisses of cut 5 in Derv

^{*}- h4
^{*} the height of the premisses of cuts 4′ and 4 in Derv

^{*}

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## References

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