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.
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Abbreviations
- 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*
References
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Gilmore, Paul C.: Natural Deduction Based Set Theories: A New Resolution of the Old Paradoxes. J. Symbolic Logic 51 (1986) 393–411
Gilmore, Paul C.: How Many Real Numbers are There?. TR89-7, Dept of Computer Science, UBC, revised November 1992. To appear in The Annals of Pure and Applied Logic.
Gilmore, Paul C.: The Consistency and Completeness of an Extended NaDSet. TR91-17, Dept of Computer Science, UBC, revised November 1992.
Gilmore, Paul C.: Logic, Sets and Mathematics. The Mathematical Intelligencer, 15 (1993) 10–19.
Gilmore, Paul C.: Lecture Notes on NaDSet. In preparation
Gilmore, Paul C., Tsiknis, George K.: Logical Foundations for Programming Semantics. To appear in Theoretical Computer Science
Gilmore, Paul C., Tsiknis, George K.: A Logic for Category Theory. To appear in Theoretical Computer Science
Gilmore, Paul C., Tsiknis, George K.: Formalizations of an Extended NaDSet. TR91-15, Dept of Computer Science, UBC.
Szabo, M.E.: The Collected Papers of Gerhard Gentzen. North-Holland, Amsterdam, London (1969)
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Gilmore, P.C. (1993). A syntactic consistency proof for NaDSet. In: Gottlob, G., Leitsch, A., Mundici, D. (eds) Computational Logic and Proof Theory. KGC 1993. Lecture Notes in Computer Science, vol 713. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0022568
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DOI: https://doi.org/10.1007/BFb0022568
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