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one function from Px into x only if there is a one-one function from a
subclass of x onto Px. Suppose, if possible, that f is a one-one function
from a subclass Y of x onto Px. Put
( )
D = {zhY: not z R f z }.
(I follow Bernays in using R for the relation of set to set and h for the
relation of set to class.) Evidently D is represented by a set d, for D is
44 Richard L. Cartwright
a subclass of Y and hence of the set x. So d is in Px. But, for an arbi-
trary u in Y:
( )
uhD iff not u R f u
and hence
( )
u Rd iff not u R f u .
So d is not identical with f(u), and thus not in the range of f, contrary
to the supposition that f is onto Px. It follows that there is no one-one
function from Px into x.
Employed in the proof are Class Existence (roughly, every formula that
contains no bound class variables determines a class of sets),29 Exten-
sionality, the so-called Axioms of the Small Sets, and the Subclass
Axiom. Notice in particular that the Power Axiom is not used.
And notice that (unless the Bernays theory is inconsistent) it is not a
theorem that every class is cardinally smaller than the class of all its
subsets. For V is identical with the class of all its subsets.
3. Nothing in the foregoing requires that classes be understood to
be other than collections, or classes as many. But the following defini-
tions may help to allay certain doubts.
A function is a collection of ordered pairs (i.e., sets of the form
{{x},{x,y}}) no two of which have the same first term.
The domain of a function is the collection of things x such that for
some y, ·x,yÒ is one of the pairs of which the function consists.
The range of a function is the collection of things y such that for
some x, ·x,yÒ is one of the pairs of which the function consists.
A function is from a collection X into a collection Y iff X is the
domain of the function and Y is some of its range; and the func-
tion is onto Y just in case Y is its range.
Notice that the domain and range of a function, though themselves col-
lections, consist of sets (and perhaps urelemente). Collections are neither
arguments to nor values of functions.30
Notes
1. Speaking of Everything, p. 8.
2. Skeptical Essays, p. 43. George Boolos called the quotation to my attention. To
separate issues, it would be preferable to have a in place of exactly one ; exten-
sionality is not in question. And I take Mates s class to be synonymous with my
set .
3. Material Beings, p. 74.
4. Mathematics in Philosophy, p. 294.
5. Logic, Logic, and Logic, p. 72.
A Question about Sets 45
6. See, e.g., Abraham A. Fraenkel, Yehoshua Bar-Hillel, and Azriel Levy, Foundations of
Set Theory, p. 27; Thomas Jech, Set Theory, p. 4; and Azriel Levy, Basic Set Theory,
p. 5.
7. Contrary, I take it, to Frege: see Die Grundlagen der Arithmetik, section 46. See also
Michael Dummett, Frege s Philosophy of Mathematics, p. 93.
8. See, e.g., Kurt Gödel, The Consistency of the Axiom of Choice and of the Generalized
Continuum Hypothesis with the Axioms of Set Theory, p. 3.
9. Genesis 1:26 7 and 31, King James version.
10. See Penelope Maddy, Realism in Mathematics, chapter 2.
11. Georg Cantor, Beiträge zur Begründung der transfiniten Mengenlehre, 1.
12. Über unendliche, lineare Punktmannigfaltigkeiten, 3. Reprinted in Gesammelte
Abhandlungen. The quotation is on p. 150 of that volume.
13. See Gesammelte Abhandlungen, p. 150.
14. Gesammelte Abhandlungen, p. 150.
15. Gesammelte Abhandlungen, p. 204.
16. Gesammelte Abhandlungen, p. 443; English translation in J. van Heijenoort (ed.), From
Frege to Gödel, p. 114.
17. Logic, Logic, and Logic, pp. 70 71.
18. On Plural Reference and Elementary Set Theory, p. 213.
19. Russell s collections are his classes as many, and are accordingly distinguished
from what he calls classes as one. See The Principles of Mathematics, pp. 68 69 and
especially p. 76, where he speaks of an ultimate distinction between a class as many
and a class as one.
20. Principles, p. 78.
21. The empty set, if there is one, and singletons present problems that cannot be
addressed here.
22. Here, and at several other places, I lean heavily on Paul Bernays, A System of
Axiomatic Set Theory.
23. It is perhaps of some interest to notice that if the Power Axiom is strengthened to:
For every set x, there is a set y such that for every class Z, Z
resented by a member of y
(where Z
the Subclass Axiom follows.
24. Fraenkel, Bar-Hillel, and Levy, Foundations of Set Theory, p. 63.
25. J. M. E. McTaggart, The Nature of Existence, vol. 1, chapter 17.
26. In the discussion of (20) (23) I have drawn on Bernays, A System of Axiomatic Set
Theory, 4.
27. Principles, p. 368.
28. That is, the theory expounded in A System of Axiomatic Set Theory. The theory is
sometimes referred to as VNB, on the ground that its leading idea derives from John
von Neumann, Eine Axiomatisierung der Mengenlehre.
29. I follow Levy (in Fraenkel, Bar-Hillel, and Levy, Foundations of Set Theory) in taking
as an axiom-schema what Bernays proves from finitely many class-existence axioms.
30. An earlier version of this paper was presented as part of a symposium, held at Tufts
University in September 1998, honoring the work of Helen Cartwright. Other par-
ticipants were Sydney Shoemaker and Judith Thomson.
In thinking about the topics discussed in the paper, I have benefited from conver-
sations with Gabriel Uzquiano; and to those who know their work, it will be obvious
that I have benefited enormously from reading papers by the late George Boolos and
by Helen Cartwright. I am especially indebted to Helen, with whom I have discussed
46 Richard L. Cartwright
these matters at great length. I am very grateful to Ralph Wedgwood for pointing
out a mistake in the early version.
References
Bernays, Paul. A System of Axiomatic Set Theory, Journal of Symbolic Logic 2 (1937):
65 77; 6 (1941): 1 17; 7 (1942): 65 89, 133 145; 8 (1943): 89 106; 13 (1948): 65 79;
19 (1954): 81 96. Reprinted with minor alterations in Gert H. Muller, ed., Sets and
Classes (Amsterdam: North-Holland, 1976), pp. 1 119.
Boolos, George. Logic, Logic, and Logic (Cambridge, MA: Harvard University Press, 1998).
Cantor, Georg. Beiträge zur Begründung der transfiniten Mengenlehre, 1, Mathematis-
che Annalen 46 (1895): 481 512. Translated into English by P. E. B. Jourdain as
Contributions to the Founding of the Theory of Transfinite Numbers (New York: Dover,
1952).
. Über unendliche, lineare Punktmannigfaltigkeiten, 3, Mathematische Annalen
21 (1882): 113 121. Reprinted in E. Zermelo (ed.), Gesammelte Abhandlungen.
. Gesammelte Abhandlungen (Berlin: Springer, 1932). Excerpts in English translation
in J. van Heijenoort (ed.), From Frege to Gödel (Cambridge, MA: Harvard Univer-
sity Press, 1967).
Cartwright, Helen. On Plural Reference and Elementary Set Theory, Synthese 96 (1993):
201 254.
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