In The Mathematical Experience, Study Edition by Philip J. Davis, Reuben Hersh, and Elena Anne Marchisotto it states pp.175-176:

Just what constitutes the "straightness" of the straight line? There is undoubtedly more in this notion than we know and more than we can state in words or formulas. Here is an instance of this "more." Suppose $a$, $b$, $c$, $d$ are four points on a line. Suppose $b$ is between $a$ and $c$, and $c$ is between $b$ and $d$. Then what can we conclude about $a$, $b$, and $d$? It will not take you long to conclude that $b$ must lie between $a$ and $d$.

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This fact, surprisingly, cannot be proved from Euclid's axioms; it has to be added as an additional axiom in geometry. This omission of Euclid was first noticed 2000 years after Euclid, by M. Pasch in 1882! Moreover, there are important theorems in Euclid whose complete proof requires Pasch's axiom; without it, the proofs are not valid.

See pp.21-22 for a description of Pasch's axiom and a picture of Pasch from the linked seminar slides: StanfordLogicSeminarApril2014.pdf

From wikipedia: Pasch's axiom

A more informal version of the axiom is often seen:

If a line, not passing through any vertex of a triangle, meets one side of the triangle then it meets another side.

I wanted to know which theorems in the Elements are considered in worst shape as a consequence of Pasch's missing axiom? Also have there been any more axioms found to be missing like Paschs?


2 Answers 2


One amazing thing about Euclid's Elements is that they have held up despite not meeting the standards of modern axiomatizations. It seems Euclid had excellent intuition, and he seems to have understood Rule #1 of Mathematics: Never prove false statements.

Over the centuries there has been an increasing understanding of what's missing in Euclid, and several increasingly refined updates of Euclid's Elements. My understanding is that it was Hilbert who wrote out the first fully modern axiomatization of Euclidean planar geometry. There are excellent accounts of this in recent textbooks. For a very thorough treatment see Hartshorne's book "Geometry: Euclid and Beyond", and for a brief but pithy account see Stillwell's "Four Pillars of Geometry".

Here's a brief list of missing items, in no particular order of logical dependency.

  • An axiomatization of the order structure or "between-ness" structure along lines. Pasch's axiom is part of this. It's needed in many arguments in the Elements where the concept of "between-ness" is invoked intuitively.
  • A completeness axiom of some kind. Euclid's first result, the construction of an equilateral triangle, depends intuitively on the existence of an intersection point between two circles of equal radius passing through each others centers. This, in turn, depends on some kind of completeness (in modern coordinate geometry terms, we would say that at the least one needs solutions of quadratic equations with non-negative discriminant).
  • An axiomatization of rigid motions. Euclid makes intuitive use of reflections, translations, and rotations all over the place, for instance in setting up triangle congruence theorems such as SAS.
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    $\begingroup$ Rule #1 sums him up completely. I was aware of Hilbert's axiomatization using second-order logic, whereas Tarski did an axiomatization using first-order logic, but the definition of the real numbers is lacking in Tarski's geometry, although it is complete in the sense of the algebraic numbers. It is the Axiom V of Continuity of Hilbert's that needs the second-order logic which allows the completeness you mention. So with all the Eudoxean proportion and ratio and irrationality going on were the Greeks implicitly using second-order logic all the time, just unaware that they were, or had to? $\endgroup$ Aug 23, 2016 at 15:48
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    $\begingroup$ I think your last item is a bit misleading. Euclid actually pretty clearly tries to avoid using rigid motions intuitively, and basically only uses them to set up the needed triangle congruence theorems and thereafter avoids them. As a result, if you just turn those triangle congruence theorems (or really just one of them) into axioms then there is no need to axiomatize rigid motions, and this is indeed what Hilbert does. $\endgroup$ May 9, 2022 at 13:53
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    $\begingroup$ I know this interpretation that Euclid "tries to avoid" using rigid motions intuitively, but it's not as clear to me that this is a good description of what he is doing regarding rigid motions. An equally possible interpretation is that simultaneous motion of a set of points stretching in all directions to infinity was just not a concept in Euclid's playbook. Otherwise, I agree with everything you say, and nonetheless there is an alternative axiomatization of Euclidean geometry in which rigid motions play a central role. Hilbert did not choose to go that way, but others have. @EricWofsey $\endgroup$
    – Lee Mosher
    May 9, 2022 at 13:58

Which theorems in the Elements are considered in worst shape as a consequence of Pasch's missing axiom?

Actually, none of the theorems themselves are questioned; it's only Euclid's arguments for them. Euclid tends to assume that a given point is between two other points when this is "obvious," without explicitly proving it, that lines have two sides, and that circles have insides and outsides. All of these are correct and result from Pasch's axiom: the issue is only that the Elements don't explicitly prove it.

Even the first theorems make these assumptions, but perhaps the best example is the Exterior Angle Theorem, with the issue spelled out well here.

Of course, the steps to prove that $F$ is on the interior of $\angle ACD$ is easy, once you have Pasch's axiom. The criticism is that Euclid does not do this explicitly.

Now, it should be stated that nearly all proofs skip various steps that are "obvious". Explicitly proving each of these would make a proof very long. Proofs that skip nothing are almost always intended for computer usage, not reading: for people, they're too long and tedious to follow. Consequently, whether Euclid's omission of betweeness is a logical gap or an expository shortcut is therefore, IMO, up for debate.


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