Positive abstraction and extensionality

Abstract

It is proved in this paper that the positive abstraction scheme is consistent with extensionality only if one drops equality out of the language. The theory obtained is then compared with GPK, a well-known set theory based on an extended positive comprehension scheme.

Positive Frege and its Scott‐style semantics

“…So we call the resulting system positive Frege (PF). As we shall see, the logical notion of set underlying that system is closely related to the positive set theory studied in [6,7].…”

“…If T is consistent, so is PF[T] 6). As usual, an n-ary relation symbol r of T is interpreted on M by a subset r M of M n .…”

Prospects for a Naive Theory of Classes

“…Forti and Hinnion 1989, Hinnion 1994, Hinnion and Libert 20032008) that show the consistency of various forms of Extensionality, but with highly restricted Abstraction: it is restricted to formulas in which neither negation, nor a conditional, nor even the abstraction operator itself occur. Because of these restrictions, these theories aren't naïve in our sense.…”

What are we to accept, and what are we to reject, while saving truth from paradox?

“…When is it the case that hx : /(x)i = hy : w(y)i? 4 Field's theory, which ajoins properties to a base logic using (NC) alone, tells us little to address this issue. Clearly, if x is P and :ðx is QÞ; then P and Q differ, but this is nothing over and above Leibniz's law.…”

“…Despite the seeming straightforwardness of coarse properties, they provide the means to derive inconsistency. The simplest proof I know of is a generalisation of a result of Roland Hinnion [4]. We start with a definition of a property, rather like the problematic hetorological property, except we will not use negation in its definition.…”