Cantor's transfinite numbers
V O R T E X
Hi,
I would be interested to learn the current view of modern mathematics
about the validity of Cantor's work. I read recently strong criticism
claiming that Cantor violated fundamental logic assioms (Russell,
Whitehead). Is this correct?
Thanks,
--
V O R T E X (Eugene M. Berti ->for...@worldnet.att.net)
"Science - the great antidote to the poison of superstition"
David K. Davis
V O R T E X (for...@worldnet.att.net) wrote:
: Hi,
: I would be interested to learn the current view of modern mathematics
: about the validity of Cantor's work. I read recently strong criticism
: claiming that Cantor violated fundamental logic assioms (Russell,
: Whitehead). Is this correct?
: Thanks,
If Cantor's work is invalid, modern mathematics goes up in smoke. The
investment is too great - if something's wrong we'll just change logic.
-Dave D.
Bob Massey
Please see my interactive(javascripted) AI demonstration of super-logic.
I point out falacies in Godel's incompleteness argument: requires
infinite recursive substitution.
And G.Cantor's 'diagonal' argument for proving higher order infinity can
not be symulated by finite memory computers as R Penrose claims.
Real computers only generate a finite number of rational numbers or
repeating integers if let run long enough because their determinism and
only a finite number of different possible states makes them start
repeating results when their count mechanisms overflow(possible after
the universe collapses).
--
RLMassey denver CO, e-mail rma...@orci.com
http://www.csn.net/~pidmass
Negative feedback neural net increases entropy sooner.
Fetus souls romping in paradise thank god for abortion!
Ariel Scolnicov
Bob Massey <rma...@orci.com> writes:
> David K. Davis wrote:
> >
> > V O R T E X (for...@worldnet.att.net) wrote:
> > : Hi,
> > : I would be interested to learn the current view of modern mathematics
> > : about the validity of Cantor's work. I read recently strong criticism
> > : claiming that Cantor violated fundamental logic assioms (Russell,
> > : Whitehead). Is this correct?
> > : Thanks,
> >
> > If Cantor's work is invalid, modern mathematics goes up in smoke. The
> > investment is too great - if something's wrong we'll just change logic.
> >
> > -Dave D.
>
> Please see my interactive(javascripted) AI demonstration of super-logic.
> I point out falacies in Godel's incompleteness argument: requires
> infinite recursive substitution.
That's not what *I* remember from my logic II class; there we had a
pretty straightforward argument, and didn't even get to learn what an
"infinite recursive substitution" is. Doesn't sound like something
you'd want around the house, though. But all our formulae were
strictly finite, so we probably never got round to the infinite
recursive substitutions...
> And G.Cantor's 'diagonal' argument for proving higher order infinity can
> not be symulated by finite memory computers as R Penrose claims.
> Real computers only generate a finite number of rational numbers or
> repeating integers if let run long enough because their determinism and
> only a finite number of different possible states makes them start
> repeating results when their count mechanisms overflow(possible after
> the universe collapses).
So? You're saying that because you can't build a decent computer
something doesn't exist? Since when have the vagaries of 2's complement
arithmetic got *anything* to do with the real world?
Now, if you showed that writing Godel's proof in Sanskrit requires
symbols which can't be etched onto a grain of rice (the standard for
validity over here in Asia), you'd really have something...
Gerhard Niklasch
In article <327D24...@orci.com>, Bob Massey <rma...@orci.com> writes:
[....]
|> And G.Cantor's 'diagonal' argument for proving higher order infinity can
|> not be symulated by finite memory computers as R Penrose claims.
|> Real computers only generate a finite number of rational numbers or
|> repeating integers if let run long enough because their determinism and
|> only a finite number of different possible states makes them start
|> repeating results when their count mechanisms overflow(possible after
|> the universe collapses).
Harrumph.
Real computers -- at any rate all I've seen, which is not too few --
tend to be replaced by new ones every few years or so, doubling or quadrup-
ling the previous processing speed and memory capacity, and inheriting any
data accumulated by their predecessors. (Well, most of the data. The one
item you really didn't want to lose inevitably ends up in bit nirvana.)
==> (a) Real computers aren't finite state machines,
==> (b) unless computers' physical dimensions scale down at the same
rate, the whole earth is going to be covered by those critters after a
finite span of time...
;^)
--
* Gerhard Niklasch <ni...@mathematik.tu-muenchen.de> *** Some or all of the con-
* http://hasse.mathematik.tu-muenchen.de/~nikl/ ******* tents of the above mes-
* sage may, in certain countries, be legally considered unsuitable for consump-
* tion by children under the age of 18. Me transmitte sursum, Caledoni... :^/
Bill Dubuque
V O R T E X <for...@worldnet.att.net> writes:
>
> I would be interested to learn the current view of modern mathematics
> about the validity of Cantor's work. I read recently strong criticism
> claiming that Cantor violated fundamental logic assioms (Russell,
> Whitehead). Is this correct?
There is much misinformation in the "usual story" of Cantor's development
of set theory. This is argued very forcefully in an excellent recent book
by Lavine:
Lavine, Shaughan. Understanding the Infinite.
Harvard University Press, Cambridge, Massachusetts, 1994, 372 pp. $40
In the introduction to this book (p. 3), Lavine says
Cantor's original set theory was neither naive nor subject to paradoxes.
It grew seamlessly out of a single coherent idea: sets are collections
_that can be counted_. He treated infinite collections as if they were
finite to such an extent that the most sensitive historian of Cantor's
work, Michael Hallett, wrote of Cantor's "finitism." Cantor's theory is
a part of the one we use today.
Russell was the inventor of the naive set theory so often attributed to
Cantor. Russell was building on work of Giuseppe Peano. Russell was also
the one to discover paradoxes in the naive set theory he had invented.
Cantor, when he learned of the paradoxes, simply observed that they did
not apply to his own theory. He never worried about them, since they had
nothing to do with him. Burali-Forti didn't discover any paradoxes either,
though his work suggested a paradox to Russell.
Further info on Lavine's book can be found in a recent post of mine, which
I've appended below.
There are also other opinions on these matters, e.g.
------------------------------------------------------------------------------
Frapolli, Maria J. (E-GRAN-Q)
The status of Cantorian numbers.
Modern Logic 2 (1992), no. 4, 365--382. MR 94g:03008
------------------------------------------------------------------------------
Summary: "A critical evaluation of Cantor's number conception is undertaken
against which the interpretations of Cantorian set theory by H. Wang [From
mathematics to philosophy, see pp. 181--223, Humanities Press, New York, 1974]
and M. Hallett [Cantorian set theory and limitation of size, Oxford
Univ. Press, New York, 1984; MR 86e:03003] are measured. Wang took
Cantor's theory to tend to be a theory of numbers rather than a theory of
sets, while Hallett took Cantor as proposing an ordinal theory of cardinal
numbers which permitted Cantor to accept ordinal numbers as given without
defining them. The evidence presented, however, shows that Cantor conceived
numbers, both cardinals and ordinals, as extensional objects, and while either
Wang's or Hallett's interpretation eliminates certain difficulties of
Cantorian set theory, neither one of them is an accurate depiction of Cantor's
theory."
------------------------------------------------------------------------------
Frapolli, Maria J. (E-GRAN)
Is Cantorian set theory an iterative conception of set?
Modern Logic 1 (1991), no. 4, 302--318. MR 92g:04001
------------------------------------------------------------------------------
Summary: "The aim of this paper is to argue against the view that Cantor's is
an iterative conception of set. We distinguish between the theory found in
Grundlagen and Beitrage, which we call the `first theory', and that expounded
in Cantor's correspondence with Dedekind and with Jourdain. We consider that
Cantor's first theory embraces a naive and unrestricted concept of set, and
that the set-theoretical paradoxes do therefore follow from it. In this sense,
we do not support the idea that Cantorian set theory is an iterative
conception of set, as maintained by Boolos, Parsons and Wang, among others, or
Hallett's interpretation, which considers it from the outset as a theory of
limitation of size."
------------------------------------------------------------------------------
The following paper of Maddy contains a nice discussion on believing the
axioms of ZFC based upon the three rules of thumb: limitation of size,
Cantorian finitism, and the iterative conception of set.
Maddy, Penelope. Believing the axioms. I.
J. Symbolic Logic 53 (1988), no. 2, 481--511. MR 89i:03007
See also my post to sci.math,sci.logic of Oct. 23 titled
"Infinity, Believing the Axioms".
The standard historical references on Cantor's work are Hallett's
book "Cantorian set theory and limitation of size", and Dauben's
book "Georg Cantor. His mathematics and philosophy of the infinite.".
See below for the Math Reviews of each.
-Bill
Newsgroups: sci.math,sci.logic
Cc: EDU...@leeds.ac.uk, math...@cs.arizona.edu
Subject: Understanding Infinity, Technical Literature [was: Infinity]
From: Bill Dubuque <w...@martigny.ai.mit.edu>
Date: 01 Nov 1996 13:16:20 -0500
EDU...@leeds.ac.uk (P.C. Wood) writes:
>
> I'm just starting out as a trainee maths teacher and I don't believe
> in infinity. Am I going to run into problems?
I recommend that you familiarize yourself with the literature
before committing to a (possibly naive) viewpoint of the infinite.
An excellent entry point is the following lucid exposition:
Lavine, Shaughan. Understanding the Infinite.
Harvard University Press, Cambridge, Massachusetts, 1994, 372 pp.
Following are a synopsis and reviews by Feferman, Field, and McGee
(all from the dust jacket), followed by MR 95k:00009 by Mendelson.
Synopsis from the dust jacket:
How can the infinite, a subject so remote from our finite experience,
be an everyday tool for the working mathematician? Blending history,
philosophy, mathematics and logic, Shaughan Lavine answers this
question with exceptional clarity. An engaging account of the origins
of the modern mathematical theory of the infinite, his book is also a
spirited defense against the attacks and misconceptions that have
dogged the theory since its introduction in the late nineteenth
century.
With his development of set theory in the 1880's, Georg Cantor
introduced the infinite into mathematics. But his theory, both critics
and supporters have charged, was subject to paradoxes proceeding from
Cantor's "naive intuitions"--and this verdict has had an enormous
impact on the philosophy of mathematics. Lavine effectively reverses
this charge by showing that set theory is in fact an excellent example
of the positive and necessary role of intuition in mathematics. His
history, moving from Greek geometry to the development of calculus to
the evolution of set theory, ultimately leads to the crux of the issue:
the source of our intuitions concerning the infinite. Along the way,
he offers a careful and sympathetic critical discussion of differing
views across the philosophical spectrum.
Making use of the mathematical work of Jan Mycielski, formerly
accessible only to logicians, Lavine demonstrates that knowledge of the
infinite is possible, even according to strict standards that require
some intuitive basis for knowledge. In rigorous detail, he shows that
the source of our intuitions concerning Cantor's infinite, as a matter
of historical and psychological fact, is extrapolation from ordinary
experience of the indefinitely large.
Reviews on the dust jacket:
"This book is a defense of set theory that is more persuasive and more
compelling than many of the main-line defenses. In form it is an
unusual combination of history, philosophy, mathematics, logic and
exposition, which is woven together in a very skillful and seamless
way ... If Lavine is right, received views of what various important
figures in this history actually thought will have to be modified."
---Solomon Feferman, Stanford University
"Highly original ... [This book] deserves to become a focus of
philosophical discussion."
---Hartry Field, Graduate Center, City University of New York
"What one mostly finds in the recent philosophy of mathematics are
variations on a few basic themes. Often the variations are pursued
with originality, ingenuity, and verve, but, still, the number of
fundamental ideas and positions is rather small. In Lavine's book,
we find a new idea. It is an important idea which is likely to
significantly alter the way we think about the infinite, and it is
developed with enormous clarity, both in thinking and writing. The
result is a book which is sure to leave a lasting impression on the
philosophy of mathematics."
---Vann McGee, Rutgers University
------------------------------------------------------------------------------
95k:00009 00A30 03A05 03E30
Lavine, Shaughan (1-CLMB-Q)
Understanding the infinite. (English. English summary)
Harvard University Press, Cambridge, MA, 1994. xii+372 pp. $39.95.
ISBN 0-674-92096-1
------------------------------------------------------------------------------
The first three-fourths of this interesting book are devoted to a
sophisticated exposition of the development of set theory and some of the
traditional philosophical problems about the source of our set-theoretic
concepts in general and the basis for our knowledge of the infinite in
particular. The author accepts Bernays' distinction between the Peano-Russell
"logical" notion of set determined by a formula and Cantor's "combinatorial"
notion of set. The latter allegedly pictures a set as a collection gathered by
enumerating its elements, "in an arbitrary way, not necessarily in virtue of a
rule"; one might describe a set somewhat more precisely as the range of a
function defined on an initial segment of ordinal numbers (assuming that we
have a better understanding of these other notions). This picture would then
have had to have been abandoned by Cantor in order to allow for the power set
operation, which was needed in the construction of the real
numbers. Nevertheless, the author continues to refer to a presumably more
general "combinatorial" notion of set as if it were a clear-cut idea, on the
basis of which we can determine the acceptability of various axioms. Aside
from this issue, the author does an excellent job of covering many of the most
important questions on the history and foundations of set theory.
The last quarter of the book presents an original theory that is claimed to
show, in a psychologically and historically accurate way, how the notion of
infinity derives from the concept of "indefinitely large" sets by means of an
extrapolation. "Ordinary infinitary set theory arose essentially as the result
of extrapolation from natural principles of finite set theory the principles
of finite set theory can be motivated as self-evident principles concerning
finite sets, including indefinitely large ones." Using ideas of J. Mycielski
[e.g., J. Symbolic Logic 51 (1986), no. 1, 59--62; MR 87f:03085], the
author shows how, from a first-order theory $T$, one can construct a
corresponding theory ${\rm Fin}(T)$ that can be thought of as referring to
indefinitely large finite sets. By a result of Mycielski and Pawlokowski,
$T$ is consistent if and only if every finite subset of ${\rm Fin}(T)$ has a
finite model. The language of ${\rm Fin}(T)$ is obtained from that of $T$ by
adding monadic predicates $\Omega\sb p(x)$, where $p$ is a rational
number. (Often, $\Omega\sb p(x)$ is written as $x\in\Omega\sb p$.) The axioms
of ${\rm Fin}(T)$ are: (i) $\Omega\sb p(c)$ for any constant symbol $c$ of
$T$; (ii) $(\forall x\sb 1,\cdots,x\sb n\in\Omega\sb p)\Omega\sb q(f(x\sb
1,\cdots,x\sb n))$ for $p<q$ and any function letter $f$ in $T$; (iii)
$(\forall x\in\Omega\sb p)\Omega\sb q(x)$ for $p<q$; (iv) $(\forall x\sb
1,\cdots,x\sb n\in\Omega\sb p)((\forall x\in\Omega\sb q)\phi\ \Leftrightarrow
(\forall x\in\Omega\sb r)\phi)$, where $p<q$, $p<r$, $q$ and $r$ are less than
$s$ for all $s$ such that $\Omega\sb s$ appears in $\phi$, $\phi$ is regular,
and the free variables of $\phi$ are among $x,x\sb 1,\cdots,x\sb n$ (to say
that $\phi$ is regular means that each quantifier in it is relativized with
respect to some $\Omega$ and whenever $\Omega\sb q$ occurs within the scope of
a quantifier relativized with respect to $\Omega\sb p$ in $\phi$, then $p<q$);
(v) $\psi\sp \Omega$ where $\psi\sp \Omega$ is obtained from any axiom $\psi$
of $T$ by relativizing all quantifiers with respect to suitable $\Omega\sb
p$'s in a special way (described on p. 271). The author maintains that the
metalanguage for ${\rm Fin}(T)$ can be formulated in such a way that it
involves no commitment to infinite sets. If $T$ is ZFC (Zermelo-Fraenkel set
theory with Choice), then ${\rm Fin}({\rm ZFC})$ can be interpreted as a
theory of hereditarily finite pure sets with the $\Omega\sb p$'s playing the
role of indefinitely large such sets, and the axioms of Fin(ZFC) can be seen
to be plausible facts about such sets. On the other hand, by starting from
Fin(ZFC) and omitting all the $\Omega\sb p$'s (which the author calls
"extrapolation"), we arrive at ZFC. Although this may not be a convincing
reconstruction of the historical and/or psychological process by which we were
led to our ideas about infinite sets, it is certainly of interest in its own
right.
Reviewed by E. Mendelson
------------------------------------------------------------------------------
86e:03003 03A05 00A25 01A60 03-03 04-03
Hallett, Michael (4-OXW)
Cantorian set theory and limitation of size. (English)
Oxford Logic Guides, 10.
The Clarendon Press, Oxford University Press, New York, 1984. xxii+343pp.
$32.50. ISBN 0-19-853179-6
------------------------------------------------------------------------------
The primary purpose of this book is to trace the foundations of set theory to
their historical and conceptual sources (p. ix). The author is interested,
above all, in Cantor, the creator of set theory. The book is divided into
two parts. Part I aims at presenting an "integrated account of both Cantor's
metaphysical and mathematical theories of infinity" (p. x). The author claims
that "there is a direct route from Cantor's metaphysics to the substance and
nature of modern set theory". Part II deals with the influence and status of
the "limitation of size" idea. The volume contains some interesting and novel
materials and remarks; it represents a healthy, historic-philosophical
approach, really "foundational", to a topic which supposedly lies in the
"foundations" of mathematics but is not always treated accordingly. Once this
is said, however, the reviewer has two central objections to the author's
project, concerning the treatment of theology and abstraction.
The promised "integrated account" of metaphysics and mathematics is really one
of theology and set theory: Cantor's theology (this expression occurs, for
example, on p. xi) should help us to understand Cantor's philosophy of
sets. But consider the following counterexample. The class of all ordinals,
let us call it $O$, is, according to Cantor, an "adequate symbol" of the
Absolute or God (p. 42). Instead of "is a symbol of" the author also says
"reflects" (p. 174) or "represents" (p. 44). Now the question arises of why
God cannot conceive $O$ as "united together" in one set (pp. 37, 43, 44). The
answer suggested by the author is the following: If $O$ were a set ("united
together") then $O$ would be amenable to mathematization too (pp. 44--45),
contrary to Cantor's principle of Absolute infinity: the Absolute infinite
cannot be mathematically determinable (p. 7). The conclusion "hence" is
totally unwarranted; why should "$x$ is a symbol of $y$" and "$x$ has a
property $P$" entail that $y$ has the property $P$ as well? Which properties
of symbols hold of the symbolized entities? The Cantor-Hallett theology
creates the impression that the class $O$ is, not merely symbolizes, God or
the Absolute. Good scholastic theology had a rigorous, careful theory of
analogy for discourse about God [cf., e.g., I. M. Bochenski , The Thomist 11
(1948), no. 4, 424--447]. Cantor may have been a "deeply religious man"
(p. 10) but this is a different matter. Contrary to the announcement in the
foreword (by M. Dummett), theology is not "deeply illuminating" here but very
confusing. A good scholastic theologian would probably have dealt with the
question of why the class $O$ cannot be united together in one set by God in a
more straightforward manner: The class $O$ is an inconsistent manifold (as
Cantor himself says) and God cannot "unite it" just as God cannot unite $p$
and not-$p$ in one asserted conjunction [cf. the reviewer, Historia Math. 6
(1979), no. 3, 305--309; MR 80g:01010].
The second criticism has to do with abstraction. The author tries hard, in
fact harder than most critics, but being unaware of modern abstraction
(Peano-Weyl-Lorenzen), he falls short of a right understanding of this very
entangled issue. Here is a sketch of the view which the reviewer considers
correct. Cantor and most pre-Fregean philosophers of arithmetic insisted on
defining number as the result of an abstraction on the set of objects to be
counted. But traditional abstraction lacked the proper analysis of the
abstractive process as relative to a previously specified equivalence relation
(in this case bijection between sets) and consequently "injected" abstraction
in the wrong place, namely in each individual unit, generating the nonsensical
"pure Ones" so rightly criticized by Frege. On the other hand, Frege and
Russell were no less wrong, in that they dismissed abstraction and ended up
with what the reviewer has called the "looking around" method [cf. the
reviewer, Studia Leibnitiana Sonderheft 8 (1979), 108--123; Philos. Natur. 21
(1984), no. 2, 453--471; Anuar. Filos. 14 (1981), no. 2, 9--21]. The
traditional and Cantorian insistence on number as an abstractum was
basically right; it only needed a better theory of abstraction. This was done,
apparently for the first time, by P. Lorenzen [Introduction to operative logic
and mathematics (German), Section 13, Springer, Berlin, 1955; MR 17, 223].
Reviewed by Ignacio Angelelli
Article Unavailable
Matthew P Wiener
In article <y8zn2ww...@martigny.ai.mit.edu>, Bill Dubuque <wgd@martigny writes:
>There is much misinformation in the "usual story" of Cantor's development
>of set theory. This is argued very forcefully in an excellent recent book
>by Lavine:
> Lavine, Shaughan. Understanding the Infinite.
> Harvard University Press, Cambridge, Massachusetts, 1994, 372 pp. $40
I second your recommendation of this book.
> Russell was the inventor of the naive set theory so often attributed to
> Cantor. Russell was building on work of Giuseppe Peano.
And Frege.
> Russell was also
> the one to discover paradoxes in the naive set theory he had invented.
> Cantor, when he learned of the paradoxes, simply observed that they did
> not apply to his own theory.
It should be noted that Russell was aware that his, and the other paradoxes,
did _not_ apply to Cantor's set theory. Russell even stated as much in his
famous letter to Frege.
Part of our confusion today, as Lavine states, is due to the fact that
Fraenkel later on rewrote history and conflated Russellian with Cantorian
set theory and erroneously attributed Zermelo's axiomization as a rescue
effort on behalf of "naive set theory", and almost every account of set
theory has echoed this nonsense ever since.
--
-Matthew P Wiener (wee...@sagi.wistar.upenn.edu)