# Talk:Axiom

## "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)
That's first incompleteness theorem. There was also a second one, which is what LMS refers to — Kallikanzaridtalk 08:57, 6 March 2011 (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

αβ&gamma

and this:

${\displaystyle \alpha \beta \gamma }$

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 "&gamma;" 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 (talkcontribs) 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.

I'm a layman in logic, but from what I read in other aricles, treating axioms as self-evident is an outdated approach, they are rather considered starting points of theories. In fact, there are many different logics (classical logic, intuitionist logic etc.) so treating axioms as self-evident becomes philosophically moot. — Kallikanzaridtalk 08:54, 6 March 2011 (UTC)

The article has a logicistic attitude. It suggests non-logical axioms are not assumed to be true, but mathematical axioms are just as self-evidenct as logical axioms. — Preceding unsigned comment added by 72.238.115.40 (talk) 02:20, 11 December 2013 (UTC)

## 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)

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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 (talkcontribs) 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)

I think what this persons complaint was trying to convey is the language and explanations provided assume the average person knows as much as you do, I ended up here in the process of reading about a prescription drug which in the study cited, refers to percentages from (n=(some number), so in my search to find what specifically they were referring to I wiki'd statistics, proceeded to standard deviation, than on to algebraic symbols, than epsilon, than summation, harmonic numbers (though that was out of curiosity). I appreciate that knowledge especially mathematical is built on a chain of previous knowledge and that people take the time to share this. I did find what I was looking for as well and maybe better off for the journey, however I have been discouraged by other articles which seem more technical and maybe partly driven by the types of debates I've read. I think accessibility and relevance should be priority over precision as these topic's tend to mirror the tangent's of the contributors who may be focused on something more technical than required for a basic understanding and maybe a little less accessible for most people. However if I had a sample I could show you a graph now. Thanks again. —Preceding unsigned comment added by 64.53.203.180 (talk) 17:23, 31 August 2010 (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 (talkcontribs) 22:41, March 7, 2007 (UTC)

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))

## 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)

Snork Mardiste! :) 85.210.15.219 (talk) 23:29, 6 May 2011 (UTC)