Talk:Axiomatic system

Isomorphic

"An axiomatic system for which every model is isomorphic to another is called categorial" — Is this correct? I would expect "...every model is isomorphic to every other". In the original, there can be multiple, distinct isomorphy classes, while in the latter by transitivity there is only one, so I see a connection to completeness. It is hard to imagine how the property of a model being "isomorphic to another" can be meaningful in any sense, not least since isomorphy is reflexive... — Preceding unsigned comment added by 131.211.81.29 (talk) 17:17, 30 August 2004 (UTC)

Consistancy

The article says:

An axiomatic system is said to be consistent if it lacks contradiction, i.e. the ability to derive both a statement and its negation from the system's axioms.

Sigh Yet another wikipedian place where "consistant" is assumed equal to "no contradiction". Axioms systems that have no negation can never generate a contradiction. Yet such a system can be either consistant or inconsistant.

In traditional PC contradiction is bad because it allows you then prove any statement whatsoever. It is *that* property that makes contradiction fatal to the system. But it is not the only property with that sort of fatality. The system consisting of only the axiom "p" is inconsistant because you can generate (by substituting for p) any statement whatsoever.

Maybe something closer to:

An axiomatic system is said to be consistant if there are things it can prove, and things that it can not prove. Contradiction (proving something and its negation) is an example of a property that makes a system inconsistant.

This, at least, is true for systems without explicit negation. — Preceding unsigned comment added by Nahaj (talk • contribs) 14:58, 30 October 2005 (UTC)

I think you are trying to describe the property of explosion, i.e. of being able to prove every proposition. In non-paraconsistent logics, contradiction leads to explosion. However the earlier editor is still correct to say that contradiction means able to prove p and not p for some p.DesolateReality 14:07, 21 July 2007 (UTC)

But still wrong to say that consistent is the same as lack of contradiction. Even as early as the late 1940's there were logics (S0.5) that had explicit contradictions but didn't exhibit "explosion" [to use the terminology of the parconsistent crowd] and are generally called consistent. By the way, if a system can't prove "p" it is traditionally called "Hilbert consistent". Hackstaff's "Systems of Formal Logic" has a good discussion of the various notions of consistency running about.155.101.224.65 (talk) 20:38, 16 February 2011 (UTC)

I don't see those systems as "consistent". The usual definition, in any case, is for first-order theories, not for paraconsistent (but "inconsistent") theories. In Hackenstaff's terminology, we are talking about "Aristotle consistency", not "absolute consistency". But in the usual setting all four notions discussed by Hackenstaff are equivalent, of course. — Carl (CBM · talk) 02:29, 17 February 2011 (UTC)

Perhaps make it more accessible...

This page is written in very complex terms for people who do not understand high levels of mathematics. Can we edit this page (and until they are complete, tag it) so the language it is written in is more accessible to a wider group of people? — Preceding unsigned comment added by 70.20.163.69 (talk) 03:19, 4 May 2006 (UTC)

Sorry, this topic needs a math background. No easy way out. Yesterday, all my dreams... (talk) 20:12, 3 May 2025 (UTC)

"Model" as a term of art in logic

I changed the wording of the first sentence under "Models" to say "model" instead of "mathematical model" because the former is a term of art as well as standard terminology in mathematical logic. --71.246.5.61 16:22, 5 August 2006 (UTC)

No, please see Model theory. Yesterday, all my dreams... (talk) 20:12, 3 May 2025 (UTC)

Merge "Axiomatization" into "Axiomatic system"

I'm suggesting placing the content of Axiomatization under the section of Axiomatic method in the present article Axiomatic system. This will add bulk. The two articles do not seem to be very distinct from each other.--DesolateReality 03:53, 22 July 2007 (UTC)

Merger of "Axiomatic system" and "Formal system"

I suggest a merger of these two pages. I recognize a formal system as a special case of an axiomatic system. There can be axiomatic systems in political philosophy or ethics (I got this from the commentary Axiomatic system#Axiomatic method) which are not in a strict formal alphabet nor use a formal logic as means to get theorems. The section Axiomatic system#Properties such very much an appropriate description of mathematical formal systems and so the merger will make this link clearer. The commentaries Axiomatic system#Axiomatic method and Formal system#Formal proofs can also be tightened up and expanded with such a merger.

Also note that I'm suggesting a merger of Axiomatization into Axiomatic system, so perhaps these two issues should be considered together. --DesolateReality 04:46, 22 July 2007 (UTC)

I have removed the merger tag. read comments at Talk:Formal system#Merger of "Axiomatic system" and "Formal system" --DesolateReality 05:36, 8 August 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)

Either-or?

"An axiomatic system will be called complete if for every statement, either itself or its negation is derivable."

Wouldn't that be the definition of a "complete and consistent" axiomatic system? That's restricting the definition of "complete". Can't it be complete and inconsistent? Either-or implies non-contradiction... —Preceding unsigned comment added by 201.2.226.107 (talk) 04:40, 2 February 2009 (UTC)

Outside of mathematics

This title's content only refers to *inside* mathematics. Is this incomplete, mistitled, or ...? 76.172.28.125 (talk) 20:11, 24 June 2009 (UTC)

There are no axiom systems "outside" logic, just as there are no chemical structures outside chemmistry. Yesterday, all my dreams... (talk) 20:14, 3 May 2025 (UTC)

Completeness

I think the statement "as shown by the combined works of Kurt Gödel and Paul Cohen, impossible for axiomatic systems involving infinite sets" is somewhat misleading. Many axiom systems are complete and could be thought of as involving infinite sets (e.g. algebraically closed fields of characteristic 0). I assume the result being referred to is the independence of CH from ZFC. Maybe better to give this an explicit example. Impossibility would seem to point more toward Godel's incompleteness theorems, which I don't think Cohen's work relates to.

It might be good to qualify "impossible" as well. Complete sets of axioms exist for any model - just take the set of all statements true of that model. The incompleteness theorems talk about the impossibility of computable sets of axioms.

Jdbrody (talk) 15:34, 7 August 2009 (UTC)

Unfortunately, your comment was not noticed when you made it. You're completely correct. — Carl (CBM · talk) 12:46, 21 June 2010 (UTC)

Incomplete definition

The article addresses axioms without saying what they are about. The article should link to primitive notions. For example, William Alfred Thompson wrote in The Nature of Statistical Evidence (Springer Lecture Notes in Statistics #189), page 10:

The axiomatic method introduces primitive terms (such as point and line) and propositions concerning these terms, called axioms. The primitive terms and axioms taken together are called the axiom system Σ.

Perhaps a regular editor of this article can make an appropriate edit and note the reference or a better one.Rgdboer (talk) 01:58, 24 September 2010 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Axiomatic system/CommentsTalk:Axiomatic system/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

== Rated as stub == History section is sparse. Some cleanup done 4 years ago. Considered as math high importance (see talk history). Should add specific ref to Peano 1908 example (Russell & Whitehead cite same in Principia Mathematica), but need to brush up on Esperanto before providing adequate cite; otherwise, no cites or refs in article. This stub is a merge candidate per axiomatic system discussion as tagged. Hotfeba 23:40, 27 July 2007 (UTC)

Substituted at 21:35, 26 June 2016 (UTC)

Model as an axiomatic system in which an axiom is defined

Under "Model" it is written - "A model for an axiomatic system is a well-defined set, which assigns meaning for the undefined terms presented in the system, in a manner that is correct with the relations defined in the system."

Does this mean that an axiom can be defined?

If that is true can one cite an axiomatic system, as an example, in which an underlying foundational axiom is defined by the axiomatic system as a whole?

Furthermore, if an underlying axiom (which by definition, is "undefined primitive notion accepted as truth without evidence") is defined after the axiomatic system is developed what do we call this "defined axiom" since it is not an axiom or undefined primitive notion any more. — Preceding unsigned comment added by 197.156.86.246 (talk) 19:14, 10 March 2022 (UTC)

Can the concept of an axiomatic system be presented in the form of an axiomatic system?

For example: 1. An axiomatic system contains a set of axioms. 2. Any axiom of the axiomatic system only has validity as a whole. 3. Any axiom of the axiomatic system is always valid within that system. etc. 2001:983:A334:1:C45E:732D:F639:D9C4 (talk) 07:14, 25 March 2022 (UTC)

Move discussion in progress

There is a move discussion in progress on Talk:Hilbert system which affects this page. Please participate on that page and not in this talk page section. Thank you. —RMCD bot 02:48, 27 August 2024 (UTC)

Theory

A clearer delineation of 'theory' (see uses of 'theorem', 'theory', and 'formal theory' in the lead of Axiomatic system) is found at Axiom#Role_in_mathematical_logic, which distinguishes logical and non-logical theories (there called 'axioms'). Tule-hog (talk) 05:55, 16 December 2024 (UTC)

Axiomatization

I think axiomatization merits an article of its own. Charles Matthews (talk) 08:21, 24 September 2025 (UTC)

First, please let me note that you have done very well in expanding the development path for axiomatic systems. But you know that already. My 3 minor comments are below.

1. I do not think we need a new page on Axiomatization but do ned a page on History and development of axiomatic systems

based on the material you have added as timeline. At the moment that is too large in this article. If you look at the article on Physics you will see it has a subsection called History of physics. Most people want to know what physics is about, and a smaller portion are interested in the history.

2. The statement "The reduction of a body of propositions to a particular collection of axioms underlies mathematical research." is not true, nor seriously sourced. I know a good number of mathematicians who may pay lip service to axiomatization but could not care less about it but are focused on proving other results. it needs to be watered down.

3. I do not see "postulational analysis" as a significant of notable item these days. Yes, the Stanford Enc. plays it up with Susan Stebbing, but a simple Goog scholar search will tell you that most people these days could not care less. And note that

Stebbing wrote before WWII and hence had never heard of Tarki's early start on model theroy.

As an aside, I should say that I learned something interesting by reading your mention of Stebbing. I think her main impact on the modern logic and model theory was that she effectively started a logic group at Bedford College. She got hired there because she was probably the best woman in logic at the time. The existence of that group at Bedford then provided Willy Hodges with his first job, when math jobs were hard to get in the UK. He then became the major force in model theory for a while, given that the then master of the field had stopped math research and focused on "personal development" so to speak. So Stebbing's arrival at Bedford had more impact than it may have been realized.

Finally, thank you for your work. Yesterday, all my dreams... (talk) 18:02, 26 September 2025 (UTC)

@Yesterday, all my dreams...: I'm pretty much done now with the timelines, and the article is at 50K. I'm open to reorganisation of the material, under WP:SUMMARY. Charles Matthews (talk) 08:33, 16 October 2025 (UTC)

The best characterization of what you did is: first class. But you knew that, I am sure. I am just wondering if Norbert Wiener is necessary. That is all. A move of the timeline material to another page would be appropriate given the page size you mentioned. It would make an excellent article on its own. Better than many/most math logic pages on the Wiki. Yesterday, all my dreams... (talk) 11:46, 16 October 2025 (UTC)

Thanks for the compliment. Answering on Wiener: Einstein's work on Brownian motion was an instant classic in physics. Wiener measure fits very well into Hilbert's project of axiomatisation, and also is an example for why Kolmogorov's work is of major importance: Brownian motion qua stochastic process qua measure on a function space with a principled construction was and is a big deal. Charles Matthews (talk) 05:01, 17 October 2025 (UTC)

Well, I beg to differ on that. And very strongly so. But now I understand why you wrote that "The reduction of a body of propositions to a particular collection of axioms underlies mathematical research." The Wiener issue is a manifestation of the fact that we differ on the more basic issue of axiomatization. You seem to view the definition of a process as a form of axiom definition. I do not.

That is why I wrote that a good number of mathematicians could not care less about axioms, eg those studying fluid dynamics. Many of those would probably not even pass a final exam in an advanced logic course if they just walked into it. And Wiener, who died in 1964, was most probably unaward of the emergence of model theory in the 1960s. He was working on "processes" not axioms, in my view. In my experience most (but not all) mathematicians live in different corners of the mathematical jungle, and more and more so as fields get more advanced. I doubt if Wiener would have passed an advanced course in logic, and I am certain that Dana Scott would not have passed an advanced course in fluid dynamics. In my view axiomatics is a different corner of the jungle.

But I do not think we will agree on that. So let us leave it there. Yesterday, all my dreams... (talk) 09:30, 17 October 2025 (UTC)

For the record, "More generally, the reduction of a body of propositions to a particular collection of axioms underlies the mathematician's research program" was already in the article. The sentence you mention came from that. Obviously it could be improved. Charles Matthews (talk) 13:30, 17 October 2025 (UTC)

I think that sentence should be deleted. But I will leave it in your hands. I should also mention one other item that was making me very uncomfortable about your argument. Your reference to Einstein and Brownian motion was just baffling. Einstein wrote that around 1904/1905 a few years before Russell had published Principia. And also before Zermelo had published on axiomatic sets. I hazard a guess that Einstein knew very very little about formal logic and axiomatization, if anything. Not only was logic before his time, but because he was from ETH. In 1990 or so I talked to a young man who had a PhD in mathematical physics from ETH and (are you sitting down?) he had no idea at all that the field of proof theory exists. But he had respectable publications in high energy physics. So I am pretty sure Einstein had no idea about axiomatics. He would been called an "einstein" in discussions about that issue.Yesterday, all my dreams... (talk) 18:09, 17 October 2025 (UTC)

I believe you are missing my point. Certainly Einstein was a physicist through and through, guided by "gut feeling" and a type of geometric algebra. But he was in close connection with both Minkowski, and then Hilbert, as can be seen by the history of general relativity around 1915. It was Hilbert's view that his type of axiomatic thinking was of value in the ideas thrown up by physics. Wiener was influenced by Hilbert (in 1914).

It's not as if Hilbert was right about everything (in a sense proof theory, since you mention it, is not as central to mathematical logic as recursion theory, and as everyone knows, he had a blind spot about incompleteness). But Hilbert's thinking has left a big footprint in mathematical physics. I'm no expert in stochastic differential equations, which is where Brownian motion leads. Both Itô calculus and Malliavin calculus are pure mathematical theories dealing with those. Physicists tend to use empirical computational methods.

Anyway, post-Gödel, there is indeed a tendency to equate (the good part of) logic with mathematical logic. I would say that's an artefact, and historically - say meaning the genetic method implied by using timelines - it is misleading. What Bourbaki/Dieudonné does with history is pretty much history is written by the winners. Axiomatic quantum field theory has not been a winner, so far, but that does not mean it doesn't have a place here. Kolmogorov came out of the descriptive set theory tradition, as did Cantor for that matter. Thank you for the feedback. Charles Matthews (talk) 04:13, 18 October 2025 (UTC)

This discussion about the minor issue of Weiner being in the article is going to be too long. Obviously, we are not likely to agree on the basic issue of the nature of logic used in the article. But then no one agrees on a definition of what logic is anyway, so let it be. I do not come to London that often any more, but if I do, we should have a discussion over a drink, or three. I should say in passing that perhaps it should be mentioned that the discussion in the article is all about exact logic. Zadeh is no longer with us, but I am sure he would have wanted a ref to the 1968 work of Goguen on inexact logic, which he saw as a key step in the field. In their world two valued logic is simply a small island in the world of logic. I will leave that for you to consider. Yesterday, all my dreams... (talk) 08:02, 18 October 2025 (UTC)

Create the paragraph: experimental axiomatic system (-s)

. ~2025-43464-99 (talk) 16:38, 31 December 2025 (UTC)

Create the paragraph: nullomniaxiomaticity

nullomniaxiomaticity (null + omni- + axiomaticity < [axiomatic + -ity]):

The universal axiomatic system doesn't exist due to mutual exclusions and it cannot function as a single axiomatic system for various other reasons. ~2025-43464-99 (talk) 16:41, 31 December 2025 (UTC)