Talk:Axiom
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[edit] 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.
[edit] "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
[edit] 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)
[edit] 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)
[edit] 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)
[edit] 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)
[edit] 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)
[edit] Link suggestions
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[edit] 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)
[edit] 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)
[edit] 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)
[edit] 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)
[edit] 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)
