On the Paradox of Grounded ClassesAuthor(s): Richard MontagueSource: The Journal of Symbolic Logic, Vol. 20, No. 2 (Jun., 1955), p. 140Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2266899 .

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THE JOURNAL OF SYMBOLIC LOGIC Volume 20, Number 2, June 1955

ON THE PARADOX OF GROUNDED CLASSES

RICHARD MONTAGUE

Mr. Shen Yuting, in this JOURNAL, vol. 18, no. 2 (June, 1953), stated a new paradox of intuitive set-theory. This paradox involves what Mr. Yuting calls the class of all grounded classes, that is, the family of all classes a f or which there is no infinite sequence b such that . .. e be e ... e b2 e bE e a.

Now it is possible to state this paradox without employing any complex set-theoretical notions (like those of a natural number or an infinite sequence). For let a class x be called regular if and only if (k) (x e k 0 (3y)(y e k . '.(3z)(z e k . z e y))). Let Reg be the class of all regular classes. I shall show that Reg is neither regular nor non-regular.

Suppose, on the one hand, that Reg is regular. Then Reg e Reg. Now Reg e (z = Reg). Therefore, since Reg is regular, there is a y such that y E i(z = Reg) * (3z) (z e = Reg) . z e y). Hence ':(3z) (Z e z(z -- Reg) z e Reg). But there is a z (namely Reg) such that Z e z(z = Reg) . z e Reg. On the other hand, suppose that Reg is not regular. Then, for some k,

Reg Ek . [1] (y)(y Elk D (3z)(z e k . E ey)). It follows that, for some z, z e k . z e Reg. But this implies that (3y)(y e k . '.'(3w)(w, e k . w e y)), which contradicts [1].

It can easily be shown, with the aid of the axiom of choice, that the regular classes are just Mr. Yuting's grounded classes.

UNIVERSITY OF CALIFORNIA, BERKELEY

Received July 22, 1954.

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