[FOM] Falsify Platonism

Irving ianellis at iupui.edu
Tue Apr 27 22:22:32 EDT 2010

Bill Taylor wrote:

> Popularizers DO like to speak of "the paradoxes", and their allegedly
> devastating effect.  But the only really famous paradox (Russell vs Frege)
> came in logic, not set theory; the set theoretical paradoxes are virtually
> all identical (whether framed as Cantor's, Burali-Forti's, Russell's or
> whatever), and were seen immediately and IN ADVANCE by Cantor himself,
> during the pre-formal stage of set theory, one might almost call it
> the pre-mathematical stage.  Cantor airily dismissed them with a wave
> of the hand about "inconsistent totalities".


> So, the whole "disastrous paradoxes" thing is really just an academic
> folklore.

It is fair to say that the Burali-Forti, Cantor, and Russell paradoxes 
share the same structure; but at the same time, one might well argue, 
as, e.g., Jean van Heijenoort ("Logical Paradoxes", Paul Edwards (ed.), 
Encyclopedia of Philosophy, vol. 5 (New York: Macmillan, 1963), 45-51) 
did, that the Russell paradox is far more basic than either the 
Burali-Forti or Cantor paradoxes, insofar as the Russell paradox 
explicitly involves the very concepts of "set" and "elementhood" in a 
way that neither the Burali-Forti nor Cantor paradox do.

It should also be noted that Russell presented two versions of his 
paradox, namely the set-theoretic, which involves the set of all sets, 
which Russell discovered possibly as early as 1900, and which John 
Newsome Crossley ("A Note on Cantor's Theorem and Russell's Paradox", 
Australian Journal of Philosophy 51 (1973), 70-71) derived from 
Cantor's Theorem; and the function-theoretic, which he traced to 
Frege's allowing, in the Grundgesetze (and commonly associated with 
Basic Rule V) a function to serve as an indeterminate argument for a 
higher-order function. (For some of the historical background and 
notations for additional reading, see, e.g., my "The First Russell 
Paradox", in Thomas Drucker (ed.), Perspectives on the History of 
Mathematics of Mathematical Logic (Boston/Basel/Berlin: Birkhäuser, 
1991), 33-46.)

So far as the allegedly "devastating effect" of the Russell paradox, it 
is enough to recall that, from the standpoint at least of philosophical 
foundations, it sufficed to induce Russell to finally abandon logicism, 
and its effect on Frege, which has been widely reported, was to induce 
him to confess, in his reply to Russell and in the appendix that he 
added at the last minute to the second volume of the Grundgesetze, to 
determine that his entire life's work was undone. The relevant 
correspondence, in English translation, between Russell and Frege on 
the Russell paradox, can be found, e.g., at pp. 124-128 in Jean van 
Heijenoort (ed.), From Frege to Gödel: A Source Book in Mathematical 
Logic, 1879-1931 (Cambridge, Mass.: Harvard University Press, 1967), 
where Frege tells Russell, in his reply of 22 June 1902, that it caused 
him great consternation "since it shaken the basis on which I intended 
to build arithmetic." Historian of mathematics Ivor Grattan-Guinness 
(p. 327, The Search  for Mathematical Roots, 1870-1940: Logics, Set 
Theories and the Foundations  of Mathematics from Cantor through 
Russell to Gödel (Princeton/London: Princeton University Press, 2000), 
in describing Frege's reactions, suggested that Russell's "Appendix A" 
on Frege for his 1903 Principles of Mathematics "greatly helped to give 
Frege a less tiny audience," mainly by adding a few more British 
readers to that "tiny audience".

Cantor's letter to Dedekind of 28 July 1899 does not seem to be in 
accord with the notion that "Cantor airily dismissed them with a wave 
of the hand about "inconsistent totalities"." Rather, Cantor there, 
although retaining his belief in the well-ordering of multiplicities  
in the face of the Burali-Forti paradox of the ordinal of the 
multiplicity of all ordinals, rejects their being sets, and in 
distinguishing, in that letter, between consistent and inconsistent 
multiplicities, allowing that only consistent multiplicities are sets, 
anticpates by a quarter of a century John Von Neumann's distinction 
between sets and classes (and Zermelo's distinction between proper 
subsets and subsets), which suggests to me that he is not "airily 
dismissing, with a wave of the hand", but taking the problem seriously.

 From the historical standpoint, Gregory H. Moore & Alejandro R. 
Garciadiego ("Burali-Forti's Paradox: A Reappraisal of Its Origins", 
Historia Mathematica 8 (1981), 319-350) contend that while the argument 
leading to the Burali-Forti paradox appeared first in Burali-Forti's 
paper of 1897, he did not recognize the paradox because of a mistake he 
made. And Cantor did not see a paradox because he adopted an attitude 
similar to that of ZF set theory, for which this paradox is just a 
proof that the class of all ordinals is not a set. The interpretation 
of Burali-Forti's argument as a paradox was first given by Russell in 
his Principles of Mathematics.

Prof. Taylor also wrote:

> True set theory began with Zermelo's 1908 axiomatization, and was designed
> NOT to counter paradoxes, but to clarify what people were talking about
> and in particular find a decent proof of the well-ordering theorem.
> This was all achieved, and any lingering whiff of paradox banished
> simultaneously, and has never been seriously challenged since.  Indeed, it
> has been a real struggle to INTRODUCE paradoxes (!) by means of super-large
> cardinals, alternative axiomatizations and whatnot.

First, I am guessing that by "true set theory" is here meant ZF (or 
perhaps ZFC) in distinction from Cantor's naïve set theory.

Second, assuming that the reference is intended to be Zermelo's "Über 
die Grundlagen der Mengenlehre" of 1908 rather than Zermelo's earlier 
paper (1908) with a new (second) proof of well-ordering, i.e. is "Neuer 
Beweis für die Möglichkeit einer Wohlordnung", it should be noted that 
Zermelo himself opens the "Untersuchung..." paper in the first 
paragraph by specifically referencing at the very outset the "Russell 
antinomy" as the reason for undertaking in that paper to challenge 
Cantor's original definition of a set as, in Zermelo's words "no longer 
admissible", which suggests that, in this paper at least, Zermelo saw 
the motives of defining more precisely the concept of a set and doing 
so for the purpose of restricting sets so as to avoid the pathological 
sets that result in the Russell, Cantor, and Burali-Forti paradoxes. 
This is done by refusing to take sets as collections that are "too 
big", such as the set of all sets, the set of all cardinals, the set of 
all ordinals, so that it resembles in this respect Russell's limitation 
of size theory. The axiomatization that Zermelo developed in the 
"Untersuchung..." was supposed to have a sequel, in which well-ordering 
was to have again been proven, in this case using the new 
axiomatization; and in fact the introduction to the "Untersuchung..." 
ends with the promise to do just that in the sequel, but that sequel, 
in which the axiomatization developed in the "Untersuchung..." was to 
be used to prove well-ordering, was never published. (For an historical 
account of the origins of Zermelo's axiomatization, see, e.g. Gregory 
H. Moore, "The Origins of Zermelo's Axiomatization of Set Theory", 
Journal of Philosophical Logic 7 (1978), 307-329.)

Irving H. Anellis
Visiting Research Associate
Peirce Edition, Institute for American Thought
902 W. New York St.
Indiana University-Purdue University at Indianapolis
Indianapolis, IN 46202-5159
URL: http://www.irvinganellis.info

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