Talk:Tarski's axioms

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I'm the anonymous user who wrote this article, sorry. I do actually have an account. Gene Ward Smith 8 July 2005 18:50 (UTC)

Are you sorry for not using your account or for writing the article? :)
Thanks for your work! Oleg Alexandrov 9 July 2005 02:29 (UTC)

I think there are a few typos in the Axiom schema of completeness. I am confused which variables occur in which formula and what you mean by the dots.

216.250.179.7 07:21, 3 January 2006 (UTC) Yeah, the completeness axiom is definitely typo-riddled. I would've tried to fix it myself, but I didn't want to expend the mental energy.

Contents

[edit] My extensive revisions

I am the anonymous polisher of this article, mainly because of my admiration for Alfred Tarski, and because I was fascinated to discover in middle age that Euclidian geometry, as it was (is?) taught in high school texts is logically flawed. I have hard copy of Szczerba (1986) on my desk, and urge any of you deeply interested in this entry to read him carefully. I take his statement of Tarski's axioms as the definitive one in the English language. Evidence, by the way, of Tarski's deep commitment to elementary geometry is the 40pp letter he wrote in 1978 to Schwabhäuser. That letter has been published in revised form as Tarski and Givant (1999).

I have:

  • Cleaned up the axiom schema of Completeness and renamed it Continuity;
  • Modified Pasch as per Schwabhäuser et al (1983), Tarski's swan song, so that Transitivity and Connectivity of betweenness are now theorems;
  • Altered the notation as follows. To say that ab is congruent with cd, I follow Tarski and write Cabcd but ab ≡ cd rather than Cabcd. To say that b is between a and c, I drop the predicate letter B and simply write abc. Here I deviate from Tarski;
  • Revised the verbal summaries of the axioms, and added a few of my own (I'll write description of what Five Segments does, manana);
  • I have eliminated the tedious dance of subscripts in Five Segments in favor of Tarski's prime notation;
  • Expanded 2-3 sentences in the entry to a section titled "Discussion." I invite those competent in metamathematics to revise and expand this section;
  • Added some references.

The true power of Tarski's axioms emerges when it is seen that straightforward (but tedious to express) generalizations of Upper and Lower Dimension suffice to axiomatize Euclidian geometry for any finite number of dimensions. In Hilbert's axioms, the shift from plane to solid geometry requires adding planes to the domain, and 6 new axioms.202.36.179.65 09:30, 18 March 2006 (UTC)

[edit] Attention Mathematicians

In order of increasing difficulty, perhaps...

  • How does one define betweenness from congruence? Szczerba (1986) asserts this is possible when the dimensionality > 1, but gives no details.
Given a pair of distinct points p and q, the set of all points x such that px=qx is a n-1 dimensional hyperplane. Three pairwise points are colinear points provided that any hyperplane which contains two of the points also contains the third. So we can talk about lines.
Given a points p and q, you can talk about the n-1 dimensional hypersphere centered at p and passing thru q as the set of all x such that px=pq. (The degenerate case p=q gives a point.)
The midpoint of the line segment from p to q (p, q distinct) is the unique m such that pm=qm and all three of p,q,m are colinear. (If p=q, we define m=p.)
x is between p and q provided that any line which passes thru the point x intersects the hypersphere centered at the midpoint m (of the line segment from p to q) and passing thru p and q.
--Ramsey2006 00:05, 20 October 2006 (UTC)
  • Can Five Segment be replaced by something with fewer atomic sentences? As things stand, Five Segment accounts for 9 of the 44 atomic sentences in this axiom set.
  • Can axioms in this spirit be found for affine, projective, and Riemannian geometries? The matter of how Euclidian is a special case of Riemannian should then be revisited.
  • Reflexivity of Congruence and Pasch have not been proved independent of the other axioms. The Holy Grail here is a fully independent axiom set.

202.36.179.65 19:00, 21 March 2006 (UTC)

My understanding is that affine or projective geometries can be easily described by eliminating the axioms having betweenness or congruence, respectively. I'll try to verify this. Adam 02:45, 28 April 2006 (UTC)

[edit] Diagrams of the axioms would be nice...

The axioms are not illustrated because I do not know how to upload images (scanned from Tarski and Givant 1999) to Wikipedia.202.36.179.65 19:49, 26 March 2006 (UTC)

[edit] Note about first order and finitly axiomatizable

I deleted a remark saying that hilbert aximatic is not first order because of quantification over lines. I think it is false because lines are nto defined as a set of points in hilbert, lines could be anything. I also deleted a comment saying that euclidean geometry is not finitly axiomatizable beacause the result of Tarski says that epsilon2 is not equivalent to a finite axiomatic system expressed *in the language of epsilon2*. The question of the language is important because othewise it can be finitly axiomtized. So i think the comment is misleading. —The preceding unsigned comment was added by Jnarboux (talk • contribs) 15:06, 23 May 2006 (UTC)

I've reinstated both. Hilbert's axioms are heavily non-first-order for reasons much more important than quantification over lines. Finite non-axiomatizability of Euclidean geometry is an important property and should be mentioned, though you are right that care with the language is in order. Nevertheless, I'd love to see a complete finite FO axiomatization of geometry in any nontrivial language. -- EJ 23:51, 7 August 2006 (UTC)

[edit] Axioms

The schema of continuity given in the article is nonsense, it is logically equivalent to the single formula

\forall a\,(axy)\to\exists b\,(xby),

which moreover follows from the dimension axioms. I'll try to lookup the correct statement. -- EJ 00:16, 8 August 2006 (UTC)

Done. I also made the axioms more in line with Tarski's formulation, undoing the omission of B and prenexation of the formulas, which only served to make it unreadable. -- EJ 01:42, 8 August 2006 (UTC)

[edit] Definition of partial order relation <=

What does "ywuv" mean in xy≤zu↔∀v(zv≡uv→∃w(xw≡yw∧ywuv)) ? Is this definition a wff? Otto 17:51, 27 August 2006 (UTC)

[edit] Congruence is not an equivalence relation

Now it says that "congruence is an equivalence relation", but that cannot be the case, because congruence is a quaternery relation and an equivalence relation is binary. I do not see how you can construct an equivalence relation out of congruence. It should be possible if you can construct an ordered pair of points and so define a binary relation of congruence between two ordered pairs of points. I do not see how to get there. Otto 20:49, 27 August 2006 (UTC)

And where exactly do you want to get? Congruence is formally a quaternary relation on points, but it can be considered a binary relation on pairs of points (or on line segments, if you wish). Indeed, the notation xy\equiv zw strongly suggests such interpretation, and it is the only way to make any intuitive sense of the predicate. Now, as a relation on pairs, congruence is reflexive, transitive, and symmetric, hence an equivalence relation. In fact, congruence is a prototypic equivalence: it was the first equivalence relation recorded in history, and its name was thus adopted for other equivalence relations in modular arithmetic, universal algebra, etc., see congruence relation. What more do you expect? -- EJ 19:02, 29 August 2006 (UTC)
EJ, what you ask me is a rhetoric question. It is easy to talk here about "pairs of points", but that doesn't solve the problem. I asked for a definition of that concept within the language of Tarksi's axioms. Or alternatively, a definition from congruence as a binary relation. The text as it was before I changed it, was confusing, because it didn't comply with the ordinary meaning of an equivalence relation, which is binary. Otto 20:17, 29 August 2006 (UTC)
But there is no problem to solve, that's the point. Equivalence is a reflexive, symmetric, and transitive relation, period, there is no requirement in the definition that its domain contains only points, not pairs. I don't understand what you mean by "within the language of Tarski's axioms". Sure, pairs of points are not a primitive notion in Tarski's language, but we can talk about pairs in the theory because concepts involving pairs are trivially definable in the language. We have an infinite supply of variables for points, so we just use two of them when we need to speak about a pair. That's the usual modus operandi in axiomatic theories: we keep the basic language minimal, and we introduce other concepts by definitions. For example, read the descriptions of the axioms as given in the article: it mentions "distance", "line segment", "triangle", "intersect", "collinear", "midpoint", "angle", "circle", none of which are primitive concepts, but are easily definable.
So back to the "problem". The original formulation could do with specification of the domain of the equivalence (the original author presumably omitted it because it is obvious), but blurring it by insertion of words like "suggest" and "similar" only makes it more confusing. -- EJ 21:06, 29 August 2006 (UTC)

[edit] Elementary?

What does the term "elementary geometry" mean here, as opposed to "full geometry"? —Jorend 23:15, 17 November 2006 (UTC)

In this context, "elementary" is a synonym for first-order. -- EJ 13:50, 20 November 2006 (UTC)
Thanks, EJ. —Jorend 19:22, 20 November 2006 (UTC)
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