Talk:Axiomatic set theory

Early comments

I just remembered something I would absolutely love to see discussed in excrutiatingly fine detail! :-) Alternatives to Zermelo-Fraenkel set theory is a topic that has always fascinated me, but my limited mathematical skill (most of the material available on alternative theories seems to be at a graduate level -- i haven't gone beyond 1st year college mathematics) has meant I haven't found much information I could understand.

I'm especially interested in theories which allow the existence of a universal set, yet still somehow manage to get around Russell's Paradox? -- SJK

An example is New Foundations. -- Toby Bartels 07:58 12 Jun 2003 (UTC)

Formalism

A question: atleast in Europe I often hear talk about Formalism, how is this related to set theory? -- anon

This is part of the philosophy of mathematics. -- Toby Bartels 07:58 12 Jun 2003 (UTC)

Am I being dense …

Am I being very dense and missing something, or is this page inaccurate in that fails to mention the Subset Axiom (or Axiom of Separation, or Comprehension Axiom, as it is also sometimes known)? Onebyone 12:01, 16 Oct 2003 (UTC)

It seems to have been there since early on. Kudos to those who built this page and its children. Great work. Brent Gulanowski

Such as it is stated, the axiom of replacement implies the axiom of separation (simply take P(x,y) to be "x=y and Q(x)" where Q(x) is the predicate which you want to separate on). This is a classical trick. However, it doesn't do much harm to have redundant axioms (note that the axiom of infinity, properly stated, probably implies the axiom of the empty set, too). --Gro-Tsen 05:11, 16 Feb 2004 (UTC)

Moved out of the page

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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)

ZF(C) POV

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)

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)

Tarski-Grothendieck set theory

I wonder if someone could explain this. It's used in the Mizar system for automatic proof checking.

Axiomatic set theory vs. set theory

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)

Other systems of axiomatic set theory

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)

Style comments

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)

I've changed "manifold" to "many". Paul August ☎ 23:19, 22 January 2006 (UTC)

Principia Mathematica

Should Principia Mathematica be metioned here? --Salix alba (talk) 11:48, 15 February 2006 (UTC)

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)

Link Spam

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)

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)

JA: We don't use ISBN's in journals or books and they just clutter up the text. Everybody knows how to Google up a title if they want to. I used to try and keep track of them until I started finding out exactly how many of em attach to all the different versions of a single work. But keep em if you gotta. Jon Awbrey

JA: Yes, I see what you mean, I misread the history or flubbed the edit somehow on the frst try and then fixed it on the second. Sorry about that. Past my bedtime. Jon Awbrey 03:42, 25 May 2006 (UTC)

LM: Mea culpa, newbie mistake. I did not realize WP dealt with ISBNs automagically. Thanks for correcting it. -Loadmaster 21:28, 6 June 2006 (UTC)

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

Expert help

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)

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)

It's been over a year now, and the expert tag remains unexplained. It is also clear that this page has the attention of expert mathematicians. I have removed the expert tag. If anyone disagrees, then that's fine, but the tag is close to useless without comment.Trishm 11:03, 21 June 2007 (UTC)

why i deleted "set theory is a disease from which mathematics will one day recover"

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)

paradoxes as motivation for axiomatization

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)

statement that goes nowhere

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)