Reduction and Elimination in Philosophy and the Sciences

Let us summarize: H is not a conservative extension
of PA. The induction scheme implies a contradiction in H
nevertheless any partial recursive function is numerically
representable in H. This is a non-standard arithmetic,
different to PA, but it itself is a generalization based on the
concept of recursion, and that is not ad hoc.
4 Conclusion
We investigated a theory of property which satisfies what
Myhill called Frege’s principle, and we examine how much
arithmetic we can develop by it. It is known that, in many
logics, the comprehension principle (which represents
Frege’s principle) does not imply a contradiction. We concentrated
the case of the set theory H, in Lukasiewicz infinite-valued
predicate logic ∀ L , which is known as the
strongest theory among set theories with the comprehension
principle.
We pointed out two features of sets in H: nonextensionality
and full circularity. First, the basic law V
does not hold in H and any versions of RV has not been
known to be consistent to H yet. Second, H forgives
circular definitions. It is because H proves a general form
of recursive definition, and a certain amount of arithmetic
can be developed: we can define a graph of any recursive
function in H. However, the mathematical induction
scheme leads to a contradiction in H, so the arithmetic
developed in H is not a conservative extension of PA.
These results showed that we do not know about
arithmetic developed by the comprehension principle
enough. The problem how we can develop mathematics in
H seems to be interesting enough from the perspective of
the analysis of the broad sense of Fregean intent.
404
The Comprehension Principle and Arithmetic in Fuzzy Logic — Shunsuke Yatabe
Literature
[Cantini 2003] Cantini, A. 2003 “The undecidability of Grisin’s set
theory”, Studia logica 74, 345-368.
[Girard 1998] Girard, J.-Y. 1998 “Light Linear Logic”, Information
and Computation 143.
[Girard et al. 1989] Girard, J.-Y, Taylor, P, Lafont Y. 1989 Proofs
and Types, Cambridge: Cambridge University Press
[Grisin 1982] Grisin, V. N. 1982 “Predicate and set-theoretic caliculi
based on logic without contractions”, Math. USSR Izvestija 18,
41-59.
[Hajek 2001] Hajek, Petr 2001 Metamathematics of Fuzzy Logic,
Dordrecht: Kluwer academic publishers
[Hajek 2005] Hajek, Petr 2005 “On arithmetic in the Cantor-
Lukasiewicz fuzzy set theory”, Archive for Mathematical Logic.
44(6) 763 - 82.
[Hale 2008] Hale, Bob 2008 “The problem of Mathematical Ojects”
Talks in Kyoto University, March 24, 2008.
[Myhill 1984] Myhill, John 1984 “Paradoxes”, Synthese 60, 129-43
[Shapiro 2003] Shapiro, Stewart 2003 “Prolegomenon to Any Future
Neo-logicist Set Theory: Abstraction and Indefinite Extensibility”
British Journal of Philosophy of Science 54, 59-91.
[Shapiro, Weir 2000] Shapiro, Stewart. Weir, Alan. 2000 ““Neo-
Logicist” logic is not epistemically innocent” Philosophia Mathematica
8, 160-89.
[Terui 2004] Terui, Kazushige 2004 “Light Affine Set Theory: A
Naive Set Theory of Polynomial Time”, Studia Logica, 77, 9-40.
[White 1979] White, Richard B. 1979 “The consistency of the axiom
of comprehension in the infinitevalued predicate logic of Lukasiewicz”
Journal of Philosophical Logic 8, 509-534.
[Yatabe 2007] Yatabe, Shunsuke 2007 “Distinguishing nonstandard
natural numbers in a set theory within Lukasiewicz logic”
Archive for Mathematical Logic 46, 281-287.