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A Generalization of the Second Incompleteness Theorem and Some Exceptions to It
by
Dan E. Willard
University at Albany
This Extended
Abstract considers the generality and partial
Boundary-Case exceptions to the Second
Incompleteness Theorem for a class of axiom
systems \alpha that treat Addition and
Multiplication as 3-way relations A(x, y, z) and
M(x, y, z), and which do not formally recognize
even Successor as a total function
(i.e. \alpha cannot prove
for all x exists y x+1=y ). Such
formalisms will recognize the existence of an
infinite collection of integers 0, 1, 2, ... by
using an infinite number of constant symbols
C1, C2, C3, ... . Assuming that
C1 = 2 , we will say a sequence of
constant symbols satisfies the
Additive Naming Convention iff Ci
satisfies Equation (1)'s identity, and it
satisfies the Multiplicative Naming
Convention iff (2) holds.
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Integers having no formal names will obviously be constructed from ``Named Integers'' via the operations of Subtraction and Division. These two operations are recognized by our axiom system as total functions. (For simplicity, we assume throughout this paper that Subtraction and Division, with rounding, are defined so that x-y=0 when x < y, and \lfloor \frac x y \rfloor =x when y=0.)
Our basic result is that the Additive Naming convention does permit some forms of unusual Boundary-Case exceptions for the Hilbert-styled version of the Second Incompleteness Theorem. On the other hand, Theorem 2 states that there exists no mathematically non-trivial, consistent axiom system \alpha employing the Multiplicative Naming Convention that can recognize the non-existence of a Hilbert-style proof of 0=1 from itself.
Date received: March 17, 2003
Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # cajy-15.