Talk:Axiom.html



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One of the 500 most frequently viewed mathematics articles.
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"True" and "universal" hopeless dream?

In the twentieth century, the grand goal of finding a "true" and "universal" set of axioms was shown to be a hopeless dream by Gödel and others.

That's not what Gödel showed.  :-) --LMS

Well, that's a reasonable paraphrase of what he showed, which was that no set of axioms sufficiently large for ordinary mathematics could be both (1) complete, i.e., capable of proving every truth; and (2) consistent, i.e., never proving an untruth. Or to put it yet another way, there must exist some assertions that are true but unprovable. --LDC

Not even that... He showed that any formal system strong enough to have equality, addition, and multiplication, had those properties. (It didn't have to encompass all of ordinary mathematics.) [He also had a loophole that might have allowed non Omega consistant logics to bypass the problems, but Rosser closed that hole. Nahaj 01:57:56, 2005-09-08 (UTC)

Hey... I was just following the style guidelines that said that I should leave something hanging! (I still stand be the statement that Incompleteness can be colloquially said to imply that there is no universal and true set of axioms. There are definitely complete and consistent systems such as real arithmetic, but they lack the power of, say, integer arithmetic and thus can be said not to be universal. Another way to read what I was saying is that Principia was a hopeless task and not just because of a few paradoxes that might someday be weaseled around. -- TedDunning

What do you mean by real arithmetic? Not arithmetic of real numbers, surely, because that includes integer arithmetic as a subset and so is just as powerful. -- Josh Grosse

Actually, real arithmetic does not include integer arithmetic as a subset. The reals include the integers, but logical systems built on the two fields are not equivalent. In particular, real arithmetic is generally taken as not including comparison while integer arithmetic has comparison. The exclusion of comparison is generally due to the complexity of the definitions of the reals. The completeness of the real system was proved (I think) by either Banach or Tarski in the middle of the twentieth century.

My own personal view is that Incompleteness is just a guise of the Halting problem. Since you can solve the Halting problem with real arithmetic where the reals are defined using bit-strings and you are allowed to look at and compare a finite prefix of any real. The trick is that the algorithm requires an initial condition that is not a computable real (TANSTAAFL!) -- TedDunning

Illegible

The Greek word in the etymology in this article is illegible on this browser (Netscape) and looks like a sequence of question marks. Contrast this:

αβ&gamma

and this:

αβγ

the first is also illegible on Netscape, but you can tell what was intended; the second is perfectly legible. Michael Hardy 18:45 Mar 10, 2003 (UTC)

Note also that the first isn't valid HTML. Character entities *MUST* have the closing ";" to be valid. (I.E. It should be "γ" instead of "&gamma") Since browser behavior is (as far as the standards are concerned) undefined if the HTML is invalid, one ought not to expect that the first case do anything reasonable. That said, the use of the math markup is preferable anyway, in my opinion. Nahaj 01:54:53, 2005-09-08 (UTC)

Self-evident?

"As the word axiom is understood in mathematics, an axiom is not a proposition that is self-evident."

The Liddell and Scott entry for (axioma) says the exact opposite --Dwight 15:36, 12 Apr 2004 (UTC)

That remark seems very silly. The only "Liddell and Scott" I've been able to find is a lexicon translating ancient Greek words into English. They would therefore be expected to write about what the word meant in Ancient Greek, not about what it means in the usage of modern mathematicians. Liddell and Scott are probably right, and the statement you quote above about use in mathematics is also right. They do not contradict each other; they are about two different things. Liddell and Scott do not appear to be mathematicians and cannot be supposed to have expertise in that area. I, on the other hand, am a mathematician, and I am quite familiar with both usages. I suggest you read the whole Wikipedia article, and you will see that there is no contradiction between these points. Michael Hardy 22:33, 12 Apr 2004 (UTC)

Defined by Websters as a "self evident truth." It is one of those things that you think up while sitting on the can, or when when you can't sleep at 3:30 in the morning and you have some huge presentation to give the next day. You know, it just sort of hits you, but you knew it all along. Not to be confused with an epiphany. —The preceding unsigned comment was added by 172.198.219.243 (talk • contribs) 07:53, June 23, 2004 (UTC)

Nor with the trivial and obvious, which are theorems =) 142.177.126.230 21:19, 4 Aug 2004 (UTC)

...except that there is are technical definitions of self-evident in epistemology. See self-evidence. Michael Hardy 01:41, 5 Aug 2004 (UTC)

Axiom and postulate are different things. Axioms are taken as self evident. Postulates are accepted because the theory that is derived from them is proven to be correct. Manuel, march 2008.

Completeness?

Gödel's completeness theorem establishes that every deductive system with a consistent set of non-logical axioms is complete

Does it? Doesn't it apply to a certain set of logical axioms and rules of deduction? Take a typical deductive system and remove a logical axiom schema or modus ponens. You still have a deductive system with a consistent set of non-logical axioms--albeit one that would not ordinarily be used (except perhaps by an intuitionist)--but it's not complete. Josh Cherry 02:07, 24 Oct 2004 (UTC)

Uniqueness of the reals?

We are fortunate enough to have that the standard model of "real analysis", described by the axioms of a complete ordered commutative field, is unique up to isomorphism.

This seems to say that there is a set of axioms that picks out the reals uniquely (up to isomorphism). What about the Löwenheim-Skolem theorem and such? I presume that although the reals are the unique complete ordered commutative field, completeness can not be expressed axiomatically, at least in systems to which the L-S theorems apply. Josh Cherry 02:58, 24 Oct 2004 (UTC)

L-S is about first-order theories, and this axiomization isn't one. You can come up with a first-order theory of all first-order sentences true of the real numbers using the ring functions of addition and multiplication, as well as the order relation, and then L-S would apply, and would tell us it has a countable model. We get the theory of formally real fields in this way, but not the real numbers uniquely. Gene Ward Smith 08:01, 1 Dec 2004 (UTC)

OK, I've changed the article to discuss this point. Josh Cherry 23:50, 1 Dec 2004 (UTC)

Examples

This editorial text was removed from the end of the examples page and is reproduced here:

[OK. The later two are being presumed to actually be logical axioms, i.e. valid formulas. It would better be to say "valid formulas, as follows..." The proofs of these facts are definitely a technical issue, but interesting enough on their own.]

Hu 20:36, 2004 Nov 22 (UTC)

Link suggestions

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When are axioms used?

The article now claims:

• Basic theories, such as arithmetic, real analysis (sometimes referred to as the theory of functions of one real variable), linear algebra, and complex analysis (a.k.a. complex variables), are often introduced non-axiomatically in mostly technical studies, but any rigorous course in these subjects always begins by presenting its axioms.

It seems to me this just isn't true. Most often, what is done instead is to present definitions inside of set theory, and the set theory used is normally naive set theory. Linear algebra uses axioms when it wants to talk about vector spaces over arbitary fields, but that is quite different than using axioms to define the integers, real numbers, or complex numbers. If this sentence isn't given a good defense I'll remove it. Gene Ward Smith 19:34, 29 April 2006 (UTC)

Galois and geometry

While Galois theory was successfully applied to classic questions of geometry, the names here should be Gauss and Pierre Wantzel, not Galois. Gene Ward Smith 02:02, 2 May 2006 (UTC)

I have a complaint.

I probably shouldn't be saying this. And this maybe should be deleted. But I'm a little annoyed how every equation in wiki makes absolutely no sense. I would think there should be easy and hard ones to demonstrate how it works. —The preceding unsigned comment was added by 208.186.255.18 (talk • contribs) 05:54, June 3, 2006 (UTC)

Firstly, could you identify yourself by user name so we don't need to do detective work with the edit history to find out who wrote this?

Secondly, your complaint is horribly vague. Please explain what in the world you mean and cite examples. Michael Hardy 00:31, 5 June 2006 (UTC)

Lack of sources

Sorry, I forgot to post here after adding {{unreferenced|article}}. There's not a single reference in the entire article, so I think the tag is warranted until the problem can be addressed. Simões (^talk/_contribs) 01:17, 22 October 2006 (UTC)

abstracted 'truth' out of introductory paragraph

Assuming it is not controversial to make the point that an 'axiom' does not necessarily connote a notion of "truth" or actuality, (except perhaps in the realms of epistemology, deontology, etc.) and therefore axioms are subject to whatever motivation is deemed appropriate under the circumstances, intro paragraph should reflect this. drefty.mac 07:00, 28 October 2006 (UTC)

Given

I came here looking for the goalkeeper Shay Given, and typed in "Given". Was redirected here. Obviously the disambiguation page for "axiom" was no use for me. Somebody might want to look into this. —Preceding unsigned comment added by 212.64.98.189 (talk • contribs) 22:41, March 7, 2007 (UTC)

Misleading Statement

The article states that

"...for nearly two millennia the fifth (parallel) postulate ("through a point outside a line there is exactly one parallel") was suspected of being derivable from the first four. Ultimately, the fifth postulate was found to be independent of the first four. Indeed, one can assume that no parallels through a point outside a line exist, that exactly one exists, or that infinitely many exist."

I think it might be a good idea to change part of this statement. It is true that the fifth postulate is independent of the first four, but the assumption that no parallels exist is not - it is a much stronger statement than the negation of Euclid's fifth and is inconsistent with the first four postulates. The difference between the two kinds of geometry with the first four postulates already given is that in non-euclidean geometry parallels are not unique, whereas in euclidean geometry they are. See "Euclidean and non-euclidean geometries" by Greenberg. Stephen Thompson 01:14, 29 April 2007 (UTC)

φ, ψ, χ or φ, χ, ψ?

Under "Mathematical logic" the article says: "...φ, ψ, and χ can be any formulae of the language...". Are these letters in the correct order? The article "Greek alphabet" says the alphabetical order of these letters is φ, χ, ψ. (Complex Buttons 20:03, 4 July 2007 (UTC))

WikiProject class rating

This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 03:48, 10 November 2007 (UTC)

acceptation?

Is this a word??? maybe Bush wrote it? —Preceding unsigned comment added by 128.232.238.250 (talk) 22:58, 24 November 2007 (UTC)

Yes, "acceptation" is a word. It means the generally recognized meaning or sense attributed to a word. It is a term frequently used in both philosophy (especially logic and epistemology) and linguistics. (I think the confusion here may be originating from a failure to maintain the distinction between words that do not exist and words that one simply does not know.) Mardiste (talk) 12:58, 28 January 2008 (UTC)

External Links

I added a link to http://www.allmathwords.org/axiom.html to this article. It was removed. Wikipedia guidelines allow links to sites that provide something the Wikipedia article does not. The Wikipedia article 'Axiom' is written at a college level. The All Math Words Encyclopedia is written at a level for grades 7-10 (U.S.). It is much more accessable to middle school and high school students than the Wikipedia article. I feel this is sufficient reason to include the link to All Math Words Encyclopedia. —Preceding unsigned comment added by DEMcAdams (talk • contribs) 15:22, 8 October 2008 (UTC)

I don't think this external link should be included. According to WP:LINKSTOAVOID, links that duplicate content that the article would contain if it were a featured article should be avoided. The allmathwords page has very little content, and apart from different choice of examples its content is already included in the present article. — Carl (CBM · talk) 16:04, 8 October 2008 (UTC)

I'm not familiar with the EL guidelines for simple.wikipedia.com, but it may be more appropriate there. — Arthur Rubin (talk) 16:07, 8 October 2008 (UTC)

Should we write about "bad axiom"??

formally, axiom is just anything you want to call an axiom... you can define any system you want and call any random sequence of symbols the "axioms" of the system...

Just because every system we ACTUALLY USE are nice and useful and consistent and have nice little axioms as starting points doesn't mean we HAVE TO have axioms like that... we can just as well have an inconsistent axiom that totally screw up the entire system. Alternatively we can have an axiom that doesn't imply anything (for example in systems where there are no inference rules that can be used to derive theorems from the axiom)..

Basically it is quite possible to have "non-ideal" axioms... but the thing is, i'm not entirely sure if it'd be useful to point out that we can have these "non-ideal" axioms... most wikipedia readers are probably not going to find the comment useful... and it's probably confusing to non-specialists Philosophy.dude (talk) 01:19, 2 December 2008 (UTC)

btw, i think the comment that logical axioms are universally "true" is not entirely correct. .. there are plenty of formal axiomatic systems that does not assume truth at all... Philosophy.dude (talk) 01:26, 2 December 2008 (UTC)

The article already says, in the lede, "In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived." Does it say somewhere that axioms, in the contemporary sense, have to be nice in some way?

Regarding the use of the word "truth", I think the article seems reasonable. The contemporary definition of a "logic" involves both syntactic and semantical parts, and thus includes a notion of truth. Systems of mathematical interest with no semantics are extremely rare. — Carl (CBM · talk) 02:42, 2 December 2008 (UTC)