[FOM] Iterative Set Theory: Historical References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
Works".)
... 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
non-specialists.
---
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
axioms.)
---
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
school).
... 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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