Talk:Axiomatic set theory

message: (Beginning of moved text)
In fact, this idea can
be made very precise. If we follow the logic of set theory, we conclude that
the universe of objects which have something to do with computation is a
countable set, but since in set theory, there "exist" uncountable sets, there
must exist objects that have nothing to do with computation.
However, we could define the mathematical universe to be the collection of
objects which have something to do with computation. Set theory definitely has
not shown that there is some inconsistency if we do so. And nothing of the
mathematics which has the potential to be applied would be lost if we did
so. The conclusion being that set theory, as it is currently conceived, 
introduces a world beyond what we can see and experience - a fantasy world.
The constructivists view the debate between themselves and the classical
mathematicians as a science versus religion debate, very much analogous to
the debate between the evolutionist and the creationists. The constructivist's
idea is that the study of phenomena that can be observed within the universe
of computation is a true science - it studies a universe which has an
objective existence, something we all agree exists. The basic claim
of the constructivists is that we do not need to believe in a world beyond
what we can see and experience, to understand the world that we do see and
experience.
To further the analogy, it should be noted that both creationism and set
theory have similar origins. Creationism comes from the ancient jewish
religious writings, and set theory has it origins in medieval jewish
mystical thought known as the Kabbalah, where the search for 
knowledge of the infinite leads to knowledge of god. Certainly 
this is the view that was held by
Cantor and many of the originators of set theory.
What the set theorist have accepted is essential a religious
world view, and of course, they have a right to their views, but nevertheless,
there are strong arguments as to why the religious view does not belong in
the public domain, but the scientific view does. See 
science vs religion debate.
Set theorists of course do not accept this view (indeed, not all constructivists believe that there is a link between set theory and mysticism or religion). Set theory can be framed in computational terms by pointing out that sets are described with finite numbers of symbols, which are manipulated according to rules which can be followed by a computer. There is then no need to believe that sets "exist" (or for that matter that "functions", "numbers" or "Turing machines" exist) other than as abstract concepts to aid mathematicians in manipulating these symbols. Application to the world which we experience then becomes a simple question of whether or not set theory provides results which are useful in science, not whether or not we can reconcile sets with physical objects. The characterisation of this debate as science vs religion is, according to this defence of set theory, a straw man argument, which attempts to defeat the set theorists' position without correctly representing it.''
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I don't think this is too much to do with axiomatic set theory. What the constructivists argue, in detail, can be put on separate page under a suitable title.
Charles Matthews 13:04, 17 Jan 2004 (UTC)
thread_title:  Moved out of the page 

message: This article is about axiomatic set theory but seems to me to be too much oriented towards ZF(C). That has its own article. I propose moving this stuff to there and if possible and desirable to replace it with something NPOV. -MarSch 16:13, 19 Apr 2005 (UTC)
replies: text: I suppose the section axioms for set theory could be reduced to something like the first and last paragraphs. When it comes to independence results, I imagine independence from ZFC is most interesting case, and/or the only case anyone is going to bother to write up properly. Saying that isn't POV; it just reflects the fact that most mathematics implicitly is set in ZFC. Charles Matthews 16:27, 19 Apr 2005 (UTC)
thread_title:  ZF(C) POV 

message: Seeing as there is a separate article on set theory, shouldn't this article define Axiomatic set theory, and mention early on the reason for the presence of the adjective axiomatic? Grayum 10:41, 12 September 2005 (UTC)
thread_title:  Axiomatic set theory vs. set theory 

message: I inserted some brief references to the fact that there are other systems of axiomatic
set theory (New Foundations and positive set theory).  But I agree that ZFC is the dominant system of axiomatic set theory in practice
and is also the first system of this kind (in the earlier form of Zermelo set theory).
Randall Holmes 04:02, 15 December 2005 (UTC)
thread_title: Other systems of axiomatic set theory

message: The use of the term 'manifold' (in "Origins of rigorous set theory") is needlessly confusing in this context.  If 'manifold' is meant to refer to a specific type of mathematical usage, this should be pointed out.  If it just means 'many', then just say 'many'.
66.91.249.23 05:57, 16 January 2006 (UTC)
replies: text: I've changed "manifold" to "many". Paul August ☎ 23:19, 22 January 2006 (UTC)
thread_title:  Style comments 

message: Should Principia Mathematica be metioned here? --Salix alba (talk) 11:48, 15 February 2006 (UTC)
replies: text: I don't think it should, but it certainly might.  I guess you can count it as a form of axiomatic set theory, indeed.  However, note that it is no longer ever used to do mathematics, and even logicians don't have much interest in it: so if it is mentioned, it should be only for its historical interest. --Gro-Tsen 16:14, 15 February 2006 (UTC)
thread_title:  Principia Mathematica 

message: JA: I see absolutely no justification for linking to a commericial site like Amazon.com in bib references, one that takes an inordinately long time to load up because of all the info it attempts to gather from the machine of the clicker thereof, and which generates popup spam.  Do you really want to open WP up to litigation that would force it to provide equal access to every other commericial site on the planet?  Jon Awbrey 03:24, 25 May 2006 (UTC)
replies: text: Paul removed the links to Amazon, replacing them with ISBNs.  Then you restored them in this edit, JA.  In a subsequent edit, you removed the Amazon links, but you failed to replace them with the ISBNs.  Paul August's revert was appropriate.  Note that ISBNs provide links which do allow equal access for all booksellers and libraries.  -lethe talk [ +] 03:28, 25 May 2006 (UTC)

text: No problem. Paul August ☎ 22:24, 6 June 2006 (UTC)
thread_title:  Link Spam 

message: The page is listed as "needing expert attention" under CAT:BACK. It would be a good idea to have a section in this talk page outlining what needs to be done. yandman 08:30, 13 November 2006 (UTC)
replies: text: The tag was added by User: Stevertigo on June 8; you could ask that user why the tag was added.  The main areas I see for immediate improvement are citation, especially in the "origins" section, and integrating some of the see also links into the main body.  The article is overly focused on ZFC; it should discuss axiomatic set theory in general, and ought to mention NF, KP, MK, and GB set theory as well. The expert tag itself seems unneeded to me.  CMummert 11:32, 13 November 2006 (UTC)
thread_title:  Expert help 

message: i believe i know whence this "quote" of Henri Poincaré comes, and this is certainly not what he meant.
In the introduction to Introduction to Set Theory, by Gaisi Takeuti and Wilson M. Zaring, the severe problems that were discovered in axiomatic set theory at the beginning of the 20th century (most notorious of which being Russell's paradox) are dicussed.
"Nevertheless set theory gained sufficient support to survive the crisis of the paradoxes. In 1908, speaking at the International Congress in Rome, the great Henri Poincare (1854-1912) urged that a remedy be sought."
a footnote then leads to the following source:
Atti del IV Congresso Internazionale dei Matematici Roma 1909, Vol I, p. 182.
Amitushtush 11:31, 31 January 2007 (UTC)
thread_title:  why i deleted "set theory is a disease from which mathematics will one day recover" 

message: The new text by Quux0r is representative of a commonly expressed view, that the purpose of Zermelo's axiomatization (let's go ahead and name him; he's the main one) was to "address paradoxes in naive set theory".
But was that really his reason? Historically it may have been; I haven't read any of his original works and am not really sure. But we know today that it's not really a very compelling reason; an intuitive view of sets based, not on Frege's extensions of properties, but on the iterative hierarchy, avoids the classic antinomies just fine without recourse to any axiomatization.
There are many and varied reasons for considering axiomatizations, among which are being able to prove independence results (pretty much impossible to prove that you can't prove something, without any limitation on the means by which you might be able to prove it), provision of a fixed point of reference against which to measure arguments and see if they need more or less than that, the ability to find models of a set of axioms and thereby know that the model also satisfies its theorems, and perhaps more. But the antinomies by themselves are not a very good reason. Were they Zermelo's reason? That I don't know; I'd like to see some evidence one way or the other. --Trovatore 08:22, 11 April 2007 (UTC)
thread_title: paradoxes as motivation for axiomatization

message: It should be mentioned that there are also mathematicians using and promoting different approaches to the foundations of mathematics.
Shouldn't this lead to links to other approaches, or the mathematics that propose them? Left on its own it seems very odd to say this without justifying or expanding on it. -- 86.154.218.67 22:54, 15 September 2007 (UTC)
thread_title:  statement that goes nowhere