# A syntactic consistency proof for NaDSet

• Paul C. Gilmore
Contributed Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 713)

## 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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