[FOM] Iterative Set Theory: Historical References

A.P. Hazen a.hazen at philosophy.unimelb.edu.au
Wed Oct 5 07:07:06 EDT 2005

Roger Bishop Jones asked what literature (beyond George Boolos's 
papers) might be relevant to defending the "standard" philosophy of 
set theory: what is called the "Iterative Conception," that set 
theory describes the things occurring in the (cumulative and 
transfinite) hierarchy of ranks.
This conception has a long, largely hidden, history.  Hao Wang (in 
the "Concept of set" chapter of his "From mathematics to Philosophy", 
repr. in the 1983 ed'n of Benacerraf & Putnam's "Philosophy of 
Mathematics: selected readings") argued that Cantor held it.  It 
seems NOT to have been well-known to the general philosophical public 
before the 1970s, however: Boolos's "The iterative conception of set" 
("Journal of Philosophy" 1971, repr. in the 1983 Benacerraf and 
Putnam and as ch. 1 of Boolos's "Logic, Logic and Logic"), which did 
much to bring it to the awareness of philosophers, says of it:
    "The conception is well known among logicians.... I learned of it
     principally from Putnam,  Kripke, and Donald Martin.  Authors of
     set-theory textbooks either omit it or relegate it to back pages;
     philosophers, in the main, seem to be unaware of it, or of the
     preeminence of ZF, which may be said to embody it.  It is due
     primarily to Zermelo and Russell."
If I may ***speculate*** a bit about the history....  I have a 
feeling that Gödel did a lot to make logicians aware of it.  His 
consistency proof of the continuum hypothesis and axiom of choice was 
one of the most charismatic results  in set theory in the middle 
third of the 20th C, and though the universe of "Constructible" sets 
(L) appealed to in the proof form a "ramified" cumulative hierarchy 
rather than a "simple"  one, I can't help thinking that it did much 
to accustom logicians to thinking of "the" sets as living in a 
well-founded cumulative hierarchy!
		...  Gödel made a couple of efforts to propagandize 
for the conception more explicitly.  His 1933 Mathematical 
Association of America lecture is quite explicit about set theory 
being best thought of as a generalization-- cumulative types, and 
transfinite ones-- of Simple Type Theory.  This paper, however, was 
not published at the time (it wasn't published until it came out as 
1933o in volume 3 of Gödel's "Collected Works"), and doesn't seem to 
have been widely known.  (Quine, much later and independently, 
noticed that Zermelo-style  set theories could be viewed in this 
light: cf. his "Unification of universes in set theory" (JSL, 1956) 
or chapters 11 and 12 of his "Set Theory and its Logic."  At the San 
Marino conference in his honor-- remarks somewhere in the volume "On 
Quine" edited by Leonardi and Santambrogio-- Quine said he had not 
been aware of Gödel's paper until it came out in the "Collected 
	... And, if you are already familiar with this conception of 
set, it is clear that Gödel is talking  about it in "What is Cantor's 
continuum  hypothesis?" ("American  Mathematical Monthly," 1947; 
expanded version in both 1964 and 1983 ed'ns of Benacerraf & Putnam; 
now in vol. 2 of Gödel's "Collected  Works.")  The description was, 
however, not explicit enough to introduce the conception to 
     One other logician tried to alert the general philosophical 
community about this conception in the 1950s.  Robert McNaughton 
wrote two papers distinguishing between axiomatic theories and what 
he called the "conceptual schemes" they embody, in "Philosophy of 
Science" v. 21 (1954), pp. 44-53 and in "Philosophical Review" v. 66 
(1957), pp. 66-80.  They don't seem to have had much impact.
     Boolos's 1971 paper effectively popularized the "iterative" 
conception of set  among philosophers.  (There had been a rather 
similar discussion of a mixed, 
sets-AND-the-stages-at-which-they-are-formed, account a few years 
earlier: the first section of the set theory chapter of Shoenfield's 
"Mathematical Logic" (1967, I think) uses it to motivate the ZF 
     The conception lent itself to metaphysical interpretation: sets, 
it tempts one to think, are  somehow "generated" by their members, 
exist because and only because the members do.  Formally, without 
much EXPLICIT metaphysics, Dana Scott drew on the iterative 
conception to find and motivate a very elegant reaxiomatization of ZF 
in his "Axiomatizing set theory" in T. Jech, ed.,"Axiomatic Set 
Theory," (= Part 2 of Volume 13 in the AMS "Proceedings of Symposia 
in Pure Mathematics" series: Part 2 didn't appear until 1974, but all 
or most of the papers in it had been given at a 1967 (I think) summer 
   	...	This was developed further by James Van Aken in his 
"Axioms for the set-theoretic hierarchy," JSL v. 51 (1986), pp. 
992-1004.  Van Aken uses some suspiciously metaphysical-sounding 
language: he speaks of sets as "presupposing" their members.
			...	Perhaps the high point in the 
metaphysical interpretation of the Iterative Conception was in David 
Lewis's  (1991) monograph "Parts of Classes."  Lewis presents (and 
defends as philosophically particularly perspicuous & doubt-free) a 
certain higher-order "framework," amounting in effect to Monadic 3rd 
Order Logic.  In it he he formulates an axiomatic theory of sets as 
generated by "lower" entities, showing how to interpret in it the 
Zermelo-Fraenkel axioms.  (Given the background logic, the result is 
not standard ZF: just as MK set theory can be thought of as  2nd 
Order  ZF, Lewis's amounts to 3rd Order ZF.)
    The reaction soon set in.  Even in his 1971 article, Boolos 
suggested that the axiom of Replacement  -- the principle that 
distinguishes ZF from the much weaker Zermelo  set theory -- was not 
justified by the Iterative Conception.  In his "Iteration again" 
("Philosophical Topics," v. 17 (1989); repr. in "Logic, Logic and 
Logic") he argued that this axiom was justified by (what Russell had 
dubbed) the  "Limitiation of Size" principle, and that that principle 
gave an alternative way of motivating something like ZF that was 
independent of the Iterative Conception. Even in David Lewis's 
monograph there is a shift between the main text and the appendix. 
(I think it reflected a shift in Lewis's own thinking,not just the 
influence  of the  co-authors of the appendix.  Lewis did the actual 
writing of the appendix, though scrupulously seeking the approval of 
the co-authors on every point.)  The metaphysical absoluteness of the 
"generation"  of sets recedes into the background, and a much more 
"structuralist" attitude is taken toward the iterative hierarchy.  In 
the technical development of the appendix,  limitation of size rather 
than the iterative  hierarchy does the work.  It is even admitted 
that, from a "structuralist" perspective, a set-conception like Peter 
Aczel's, postulating many non-well-founded sets, is just as 
legitimate as (and, even equivalent to, under "ramsification") the 
well-founded iterative conception.
Allen Hazen
Philosophy Department
University of Melbourne

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