Talk:Tarski's axioms

message: 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)
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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)

message: 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)
thread_title:  My extensive revisions 

message: 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)
thread_title:  Attention Mathematicians 

message: 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)
thread_title:  Diagrams of the axioms would be nice... 

message: 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)
thread_title:  Note about first order and finitly axiomatizable 

message: 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)
replies: text: 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)
thread_title:  Axioms 

message: 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)
thread_title:  Definition of partial order relation <= 

message: What does the term "elementary geometry" mean here, as opposed to "full geometry"?  —Jorend 23:15, 17 November 2006 (UTC)
replies: text: In this context, "elementary" is a synonym for first-order. -- EJ 13:50, 20 November 2006 (UTC)
replies: text: Thanks, EJ.  —Jorend 19:22, 20 November 2006 (UTC)
thread_title: Elementary?

message: Something seems wrong.  Let a = (0,0) in R^2  let X= \{x | \phi(x)\} = \{(\frac{1}{2},0),(\frac{3}{4},0),(\frac{7}{8},0),(\frac{15}{16},0),\ldots\}   and Y= \{y|\psi(y)\} = \{(1,0)\}.  I can't think of a b that satisfies that axiom for ALL x.  Should it be:
: \exists a \,\forall x\, \forall y\,[(\phi(x) \land \psi(y)) \rightarrow Baxy] \rightarrow  \forall x\, \forall y\,[(\phi(x) \land \psi(y)) \rightarrow \exists b\, Bxby]?
--71.3.208.120 (talk) 18:27, 7 September 2008 (UTC)
thread_title: Axiom schema of Continuity Question

message: It might be good to show how an interesting result such as the Pythagorean theorem could be expressed (never mind proved) in Tarski's system.  It seems as though we could do something like the following:
* Define scalar addition: ab + cd \equiv ef := \exists xyz (Bxyz \land ab \equiv xy \land cd \equiv yz \land ef \equiv xz)
* Define congruence of triangles: \triangle abc \equiv \triangle def := ab \equiv de \land bc \equiv ef \land ca \equiv fd
* Define "strict" in-betweenness: Sabc := Babc \land a \neq b \land b \neq c
* Define congruence of angles: \angle abc \equiv \angle def := \exists wxyz (Sawb \land Sbxc \land Sdye \land Sezf \land \triangle wbx \equiv \triangle yez)
* Then define a right angle: Rabc := \exists wxyz (Swxy \land \angle abc \equiv wxz \land \angle abc \equiv zxy)
* Define similar triangles: \triangle abc \sim \triangle def := \angle abc \equiv \angle def \land \angle bca \equiv \angle efd
* We can now define scalar multiplication, via the proportionality property of similar triangles: ab . cd \equiv ef . gh := \exist uvwxyz(\triangle uvw \sim \triangle xyz \land ab \equiv uv \land ef \equiv xy \land cd \equiv yz \land gh \equiv vw) 
* Then Pythagoras can be expressed as: Rabc \rightarrow \forall ij (i \neq j \rightarrow \exists uvwxyz (ab.ab \equiv uv.ij \land bc.bc \equiv wx.ij \land ac.ac \equiv yz.ij \land uv + wx \equiv yz)) (here ij is being used as the "unit").
Anyway, this is just me speculating.  It would be great to have input from someone who actually knows authoritative answers to these questions :)  Then maybe we could add relevant sections to the article. Cheers. Grover cleveland (talk) 19:50, 10 December 2011 (UTC)
replies: text: I think most of those definitions (or ones along those lines) are in the Schwabhäuser and Szmielew book which we cite. I haven't read that work, however, but have spent much time with one which claims to be based on it: Julien Narboux (2007), "Mechanical Theorem Proving in Tarski’s Geometry", F. Botana and T. Recio (Eds.): ADG 2006, LNAI 4869, pp. 139–156. This material is also worked out at  and following pages on wikiproofs.org. As for how much of it to include in the wikipedia article, I don't know. As your example shows, even stating a familiar result without proof takes a fair number of definitions, some of which are not really obvious (for example, your definition of congruent triangles allows for degenerate triangles, in which the points are collinear or in which two or three of the points are equal, but your definition of similar triangles does not allow the second kind of degeneracy although I guess it allows the first). Kingdon (talk) 15:01, 17 December 2011 (UTC)

text: There exists a proof of the sentence above that expresses the Pythagorean Theorem. Call this sentence  \mathsf{PT} . As Tarski’s theory is known to be complete, either  \mathsf{PT}  is provable or  \neg \mathsf{PT}  is provable. A model of the theory can be constructed in an obvious manner using  \mathbb{R}^{2}  with its standard metric, and as  \mathsf{PT}  is satisfied in this model, it follows that  \mathsf{PT}  is provable. In fact, any sentence expressed in the language of the theory that is satisfied in this model is provable. Leonard Huang (talk) 07:21, 15 October 2017 (UTC)
thread_title: Possibilities

message: In Tarski's_axioms#Discussion, it says "a dense universe of points", but the word "dense" links to Density, the physical concept. What does "dense" mean in this context? Of the items on Density (disambiguation), the most relevant-looking one is Dense-in-itself, but I don't actually know.  Thanks if anyone can clarify. :)  --DavePercy (talk) 01:48, 11 January 2015 (UTC)
replies: text: It should be Dense_set.  I'll fix it. I can't find it in the text. Briankharvey (talk) 23:47, 17 December 2023 (UTC)
thread_title:  Wrong definition of "dense" 

message: The definiotion is wrong. Let's take the model \R^2 and let the formula \phi(x) such that \{x:\phi(x)\}=\{(x,0):|x|<1\}, and the formula \psi such that \{y:\psi(y)\}=\{(y,0):|y|>1\}. then exist a=(0,0) such that if \phi(x)\land\psi(y), then Baxy, so from axiom of continuity there must be b such that \forall x\forall y(\phi(x)\land\psi(y)\Rightarrow Bxby), but I can't think about something that will do it. בנציון יעבץ (talk) 10:20, 23 July 2021 (UTC)
replies: text: When a=(0,0) and x=(1/2,0) and y=(-2,0), one has \phi(x)\land\psi(y) but not Baxy. JumpDiscont (talk) 00:51, 9 September 2021 (UTC)
thread_title:  axiom schame of continuity 

message: This article is a bit confusing because it doesn't give examples or a description of first-order sentences in Tarski's language.  I believe ancient Greek geometry had the theorem that there is a classical construction of the regular pentagon, but the regular 17-gon had to wait til Gauss.  Today we usually prove these propositions with Galois theory.  I had wondered whether they could also be handled by Tarski's decision procedure, and my suspicion is that they can't be written as first-order sentences for that procedure to handle.
:You are right: the predicate "point p can be constructed from points a and b using straightedge and compass" cannot be expressed as a first-order formula in Tarski's system, basically because you would have to quantify over the number n of steps in this construction, and Tarski's system doesn't have natural numbers. AxelBoldt (talk) 00:39, 13 December 2023 (UTC)
thread_title:  Constructibility