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A further simplification of Tarski's axioms of geometry

A slight modification to one of Tarski's axioms of plane Euclidean geometry is proposed. This modification allows another of the axioms to be omitted from the set of axioms and proven as a theorem. This change to the system of axioms simplifies the system as a whole, without sacrificing the useful modularity of some of its axioms. The new system is shown to possess all of the known independence properties of the system on which it was based; in addition, another of the axioms is shown to be independent in the new system.


Background
Alfred Tarski's axioms of geometry were first described in a course he gave at the University of Warsaw in 1926-1927. Since then, they have undergone numerous improvements, with some axioms modified, and other superfluous axioms removed; for a history of the changes, see [TG99] (especially Section 2), or for a summary, see Figure 2 in [Nar07].
The axioms rely on only one primitive notion -that of points -and two primitive relations: betweenness and congruence. Congruence is denoted ab ≡ cd, and can be interpreted as asserting that the line segment from a to b is congruent to the line segment from c to d. Betweenness is denoted Babc, and can be interpreted as asserting that b lies on the segment from a to c (and may be equal to a or c).
The version of the axioms used in [SST83] (see pages 10-14) consists of ten first-order axioms, together with either a first-order axiom schema, or a single higher-order axiom. This version has been adopted in later publications, such as [Mak12] (see Sections 2.3 and 2.4) and [Nar07] (see Figure 3), and Victor Pambuccian has called it the "most polished form" of Tarski's axioms (see [Pam06], page 122).
This semi-canonical version of the system is the result of many simplifications to the original system of twenty axioms plus one axiom schema. At least one of these simplifications appears to have taken the form of a slight alteration to one axiom in order to allow another axiom to be dropped and subsequently proven as a theorem.
Specifically, in [Tar57] (see note 18), axiom (ix) is a version of what is called the axiom of Pasch, which states (modulo notational differences): In [Tar59], this has been replaced by axiom A9, which states (again, modulo notational changes): The significant change between the two is the reversal of the order of points in the final betweenness relation. They are easily shown to be equivalent using the symmetry of betweenness, which states: The interesting thing is that in [Tar57], (SB) is an axiom (axiom (iii)), but in [Tar59], it has been removed from the list of axioms. Haragauri Gupta's proof of (SB) (see [Gup65], Theorem 2.18) relies on the precise ordering of points in the final betweenness relation in (OP ′ ).
Also, Wolfram Schwabhäuser, Wanda Szmielew, and Alfred Tarski, on page 12 of [SST83], draw attention to the ordering of points in their version of the axiom of Pasch (labelled (Pa) in this paper), noting that it is important until after the proof of (SB) (which they call Satz 3.2). Again, their proof of (SB) (which is essentially the same as this paper's proof of Lemma 4) relies on the precise ordering of points in (Pa).
Therefore, although it does not appear to be explicitly acknowledged in the published literature, it seems likely that the change from (OP) to (OP ′ ) was necessary to allow the removal of (SB) from the set of axioms. It may be the case that (SB) is a theorem even with (OP), and that this was not known when it was replaced by (OP ′ ), or that it was known, but (OP ′ ) allows a simpler proof.
In any case, it appears that Tarski was willing to reorder points in his axioms to allow the simplification of the axiom system as a whole, either by removing axioms or merely by simplifying proofs of theorems.
In the tradition of such simplifications, this paper presents one further simplification of the axiom system; one of the axioms is slightly modified, allowing another of the traditional axioms to be proven as a theorem, rather than assumed as an axiom.

The axioms
Tarski's axioms, as stated in [SST83], pages 10-14, are as follows. The names are adopted from [Mak12] (Section 2.4), which provides some diagrams and intuitive explanations for the axioms.
• Reflexivity axiom for equidistance • Transitivity axiom for equidistance ∀a, b, p, q, r, s. ab ≡ pq ∧ ab ≡ rs −→ pq ≡ rs (TE) • Identity axiom for equidistance • Identity axiom for betweenness This collection of eleven axioms, Tarski's axioms of the continuous Euclidean plane, will be denoted CE 2 . A similar collection of axioms, denoted CE ′ 2 , is obtained by removing (RE) from CE 2 and replacing (FS) with The only difference between (FS) and (FS ′ ) is the reversal of the first two points in the last congruence relation. This paper will show that CE ′ 2 is equivalent to CE 2 , and will, in fact, show a stronger result -Theorem 7 -about a smaller set of axioms.
One of the features of Tarski's axiom system is its modularity: some texts omit or delay the introduction of (Co) (see, for example, [Ehr97], page 61); (Eu) can be replaced by another axiom in order to investigate hyperbolic geometry, or omitted entirely, for absolute geometry (see [Pam02], pages 331-333); (Lo 2 ) and (Up 2 ) can be replaced by other axioms that characterize other dimensions (see [Tar59], footnote 5). The stronger result that this paper shows is that A ′ is equivalent to A. Thus, the modularity of axioms (Lo 2 ), (Up 2 ), (Eu), and (Co) is unaffected by the proposed change to Tarski's axiom system.
3 Proof of equivalence Lemma 1. If (TE) and (SC) hold, then given any points a and b, we have ab ≡ ab.
Proof. Given a and b, (SC) lets us obtain a point x such that ax ≡ ab. Using this twice in (TE) gives us ab ≡ ab.
Lemma 2. If (TE) and (SC) hold and a, b, c, and d are points such that ab ≡ cd, then cd ≡ ab.
Proof. By Lemma 1, we have ab ≡ ab. Using ab ≡ cd and ab ≡ ab, (TE) tells us that cd ≡ ab.  Proof. Suppose we are given points a, b, and c such that Babc. We also have Bbcc, by Lemma 3. Then (Pa) lets us obtain a point x such that Bbxb and Bcxa. According to (IB), the former implies b = x, so the latter tells us that Bcba.
Proof. Given arbitrary points a and b, (SC) lets us obtain a point x such that Bbax and ax ≡ ba. We consider two cases: x = a and x = a.
If x = a, then aa ≡ ba. By Lemma 2, we have ba ≡ aa, so by (IE), we have b = a. Substituting this back into the congruence as necessary gives us ab ≡ ba, as desired.
Suppose, on the other hand, that x = a. Lemma 4 and Bbax tell us that Bxab. Lemma 1 tells us that xa ≡ xa, ab ≡ ab, and aa ≡ aa. We make the following substitutions in (FS ′ ): a, a ′ → x; b, b ′ , d, d ′ → a; and c, c ′ → b. Then all of the hypotheses of (FS ′ ) are satisfied, and its conclusion is that ab ≡ ba. Proof. Because the hypotheses of (FS) and (FS ′ ) are identical, we need only show that their conclusions are equivalent.
(RE) tells us that cd ≡ dc and dc ≡ cd.
If cd ≡ c ′ d ′ , then cd ≡ dc together with this fact and (TE) let us conclude that dc ≡ c ′ d ′ .
Similarly, if dc ≡ c ′ d ′ , then dc ≡ cd together with this fact and (TE) let us conclude that cd ≡ c ′ d ′ .
Theorem 7. A ′ is equivalent to A.

Proof. By Lemmas 5 and 6, A ′ implies (RE) and (FS).
A ′ contains all of the other axioms of A, so A ′ implies A.

By Lemma 6, A implies (FS ′ ), and it contains all of the other axioms of
Therefore A ′ is equivalent to A.
As an immediate corollary, we have the following:

Independence results
The first part of Section 5 of [TG99] (see pages 199 and 200) concerns the independence of Tarski's axioms. One problem seen there is that the various historical changes to Tarski's axioms often force a reconsideration of previously established independence results. This paper's suggested simplification of Tarski's axioms is no exception, so this section aims to establish which of the known independence results apply to the specific set of axioms CE ′ 2 . Because CE 2 and CE ′ 2 differ only in their subsets A and A ′ , Theorem 7 tells us that the axioms in CE ′ 2 but not in A ′ are independent if and only if they are independent in CE 2 . In fact, we can go further than this.
Proof. Note that (Ax) is not (TE), because this was explicitly excluded; nor is it (RE) or (FS), because these are not axioms of CE ′ 2 , from which (Ax) was chosen. Therefore All of the axioms of CE ′ 2 other than (FS ′ ) are also axioms of CE 2 , so Taking the contrapositive of this statement, we see that if (Ax) is independent in CE 2 then it is independent in CE ′ 2 . This allows us to adopt almost all of the independence results known for CE 2 . Proof. Schwabhäuser and his colleagues note the independence of each of these axioms in CE 2 ; see [SST83], page 26.
The remaining independence result noted in [SST83] can also be adapted to CE ′ 2 : Proof. Schwabhäuser and his co-authors note the existence of a model M demonstrating the independence of (FS) in CE 2 ; see [SST83], page 26. Because M satisfies (RE) and (TE), but violates (FS), we can conclude, using Lemma 6, that M also violates (FS ′ ). M satisfies every other axiom of CE ′ 2 , because all such axioms are also axioms of CE 2 (and are not equal to (FS)); therefore, M demonstrates the independence of (FS ′ ) in CE ′ 2 . Because of the absence of (RE) from CE ′ 2 , we can easily show the independence of (TE) in that axiom system: Proof. We proceed as is usual in independence proofs, by defining a model M of every axiom of CE ′ 2 except (TE). As in the real Cartesian plane (the standard model of CE 2 , and hence of CE ′ 2 ), we take R 2 to be the set of points, and define betweenness so that Babc if and only if b is a convex combination of a and c. Departing from the standard model, we define congruence so that ab ≡ cd if and only if a = b.
Because the real Cartesian plane is a model of CE ′ 2 , and M differs from the real Cartesian plane only in its definition of congruence, we can conclude that M is a model of all of the axioms of CE ′ 2 that make no mention of congruence. Those axioms are (IB), (Pa), (Lo 2 ), (Eu), and (Co).
The definition of congruence ensures that M trivially satisfies (IE). Choosing x = a ensures that (SC) is satisfied.
The hypotheses of (FS ′ ) include a = b and ab ≡ a ′ b ′ , which implies a = b; this contradiction in the hypotheses means that (FS ′ ) is vacuously true.
The hypotheses of (Up 2 ) imply that a = b = c = p; it is always the case that Bppp, so (Up 2 ) is satisfied.
Because M satisfies every axiom of CE ′ 2 except (TE), we can conclude that (TE) is independent in CE ′ 2 . Notice that M in the proof of Theorem 12 also violates (RE) (because it is not the case that (0, 0)(0, 1) ≡ (0, 1)(0, 0)), so this model would not demonstrate the independence of (TE) in CE 2 , which is, as far as the author is aware, still an open question.
We now have independence results for all of the axioms of CE ′ 2 except (IE) and (Pa). As far as the author knows, the independence of these in CE 2 is still an open question (see also [TG99], pages 199 and 200), although there are independence results relating to these axioms in other versions of Tarski's axiom system.
Gupta shows the independence of (IE) (his axiom A5; see [Gup65], pages 41 and 41a), but only in the context of a particular variant of Tarski's axiom system. This variant uses a weaker but more complicated form of the upper 2-dimensional axiom, Gupta's A ′ 12; 1 his independence model for (IE) also violates (Up 2 ), which is trivially equivalent to his original axiom A12.
Les law Szczerba established the independence of a version of the axiom of Pasch within a certain variant of Tarski's axiom system (see [Szc70]). This variant used, instead of (Eu), an axiom essentially asserting that any three non-collinear points have a circumcentre. 2

Conclusions
Although this paper has not answered the question of whether (RE) is independent in CE 2 , it has demonstrated that (RE) is superfluous to Tarski's 1 It seems that A ′ 12 in Gupta's thesis ought to include u = v among the hypotheses, as A12 does. Taken exactly as it is printed, A ′ 12 is violated by the real Cartesian plane whenever x, y, and z are any non-collinear points and u = v.
2 There appears to be a typographical error in the statement of this axiom in [Szc70] (page 492). His axiom A8 ′ states (in our notation): It seems that the second congruence relation in the consequent should state bp ≡ cp; see also [TG99], pages 199 and 184. axioms of geometry. A slight modification to (FS) allows (RE) to be proven as a theorem, and therefore removed from the set of axioms. This simplification of the axiom system (to CE ′ 2 ) does not diminish its deductive power or the important ways in which it exhibits modularity.
Besides removing one of the axioms not known to be independent in CE 2 , CE ′ 2 allows an easy proof of the independence of (TE), which was also not known to be independent in CE 2 . The two remaining independence questions for CE ′ 2 are, as far as the author knows, still open questions for CE 2 ; furthermore, if either axiom is shown to be independent in CE 2 , then Theorem 9 would immediately establish its independence in CE ′ 2 . As well as trying to resolve these remaining independence questions, future work might seek other slight modifications to the axioms that may allow even known independent axioms to be dropped. That this may be possible can be seen by considering Gupta's fully independent version of Tarski's axioms for plane Euclidean geometry (see [Gup65], pages 41-41c).
His version had eleven axioms, 3 some of which were deliberately made more complicated in order to allow easy proofs of the independence of other axioms (see, for example, his note on the independence of his axiom A7 on page 41b). CE 2 already has simpler axioms than Gupta's system (which he used only for the demonstration that a fully independent system is possible), but CE ′ 2 now shows that a reduction in the number of axioms is also possible, without making any of them more complex, despite Gupta's system being fully independent.