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ALGEBRAIC AND TOPOLOGICAL METHODS IN NON-CLASSICAL LOGICS III (TANCL'07)
August 5-9, 2007
St Anne's College, University of Oxford
Oxford, England

Organizers
Mai Gehrke and Hilary Priestley

On an arithmetic in a set theory within ukasiewicz logic
by
Shunsuke Yatabe
Kobe University

On an arithmetic in a set theory within \L ukasiewicz logic

On an arithmetic in a set theory within ukasiewicz logic

Shunsuke Yatabe

A significance of the set theory with the comprehension principle is to allow a general form of the recursive definition []: For any formula j(x, ..., y), the comprehension principle implies (∃z)(∀x)[x ∈ z ≡ j(x, ..., z)] within CFLew, i.e. we can define a set z by using a parameter z itself. This allows us to represent any partial recursive function on w.

Let H is a set theory with the comprehension principle within ukasiewicz infinite-valued predicate logic with its standard semantics. It has been conjectured that H is enough strong to develop an arithmetic because the recursive definition on w can be used in place of mathematical induction: We review about this. The arithmetic in H is somehow similar to one in "non-standard models" of PA. For example, we can prove an overspill-like phenomenon. However H is w-inconsistent and the mathematical induction on w implies a contradiction in H [], and their proofs can be modified to be very simple and effective even in any linearly ordered MV-algebra [].

References

[]: Andrea Cantini.: The undecidability of Grisn's set theory. Studia logica 74 345-368 (2003)
[]: Shunsuke Yatabe.: Distinguishing non-standard natural numbers in a set theory within ukasiewicz logic. Accepted to Archive for Mathematical Logic.
[]: Shunsuke Yatabe.: Recursion contradicts to induction within Lukasiewicz logic. Accepted to Many Valued Logic and Cognition - Trends in Logic V Conference in July 2007.

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Date received: May 13, 2007