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Tarski's axioms for Euclidean geometry can also be used to axiomatize absolute geometry (by leaving out his version of the Axiom of Euclid) and hyperbolic/Lobachevskian geometry (by negating that same axiom) (see the last paragraph of "Discussion" here).

Question: Can similar subsets of Tarski's axioms be used to define axiomatizations for both:

Notes: (The tag first-order-logic is because Tarski's axioms are all written in terms of first-order logic, and to understand them well, e.g. unlike me, one would have to understand first-order logic at least somewhat well. Plus as a result the resulting axiom systems for ordered and affine geometry I am requesting would also be in terms of first-order logic. If this is an inappropriate reason for the tag, please feel free to remove the tag - I don't use it frequently so don't know its correct usage.)

The notion of betweenness which underlies ordered geometry also seems essential to Tarski's axiomatization approach. So to me it would be really surprising to me if Tarski's axioms could not be used to state an axiom system of ordered geometry.

The answer for affine geometry may be negative according to this stray quote I found:

Szmielew wished to keep the simplicity (in terms of obtaining metamathematical results) of Tarski’s system, but to gain the flexibility to be able to consider different kinds of geometry, such as affine geometry, which was not possible in Tarski’s formalization...

However, a reference for the claim nor details of a proof were forthcoming. I imagine the details might be found in the reviewed book, but I don't have access to said book and have very little knowledge of formal logic, so a dumbed-down summary of any argument why or why not it's possible would be more useful to me right now than the high-level arguments likely found therein.

Also the claim seems to possibly contradict other claims made in a paper by Tarski and Givant cited and quoted in the Wikipedia article about Tarski's axioms:

The first was the selection of the betweenness and equidistance relations as the only two primitive notions. Both notions have a clear and simple geometrical meaning; the former represents the affine, the latter the metric, aspect of geometry...

The axiom... is formulated entirely in terms of betweenness, and hence is useful in the construction of an axiom set for affine geometry.

However, the quotes might essentially just be observing that ordered geometry generalizes affine geometry (hence why betweenness as a notion can be essential for both types of geometry) and therefore not actually contradict the claim made in the review of Szmielew's book.

Still, the claim being true would be somewhat surprising to me, since Euclidean geometry is a special case of affine geometry, so why wouldn't it be possible to just drop some of Tarski's axioms (assuming that they are all independent of one another) and get affine geometry as the result?

A related question if you found this one interesting.

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Chill2Macht
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  • I just found this paper by Tarski and Szczerba describing an axiom system for general affine geometry, although based on the introduction it is not entirely clear to me if it is the exact same thing as "affine geometry" as characterized in the Wikipedia article of the same name, or how it relates to Tarski's axiom system for Euclidean geometry (i.e. if the axioms are a subset or not). I would greatly appreciate it if someone who is better at reading sentences in first-order logic could spend thirty seconds glancing it over -- after reading for at least a half-hour already, I am still unsure – Chill2Macht Jul 02 '17 at 20:34
  • about how it addresses my question, although I will keep on attempting to read more. Here is a link to it: http://matwbn.icm.edu.pl/ksiazki/fm/fm104/fm104114.pdf The name of the paper (in case the link goes dead and it becomes necessary to find the paper somewhere else) is Metamathematical discussion of some affine geometries, by L.W. Szczerba and A. Tarski, the publishing year is 1976 I think. – Chill2Macht Jul 02 '17 at 20:36
  • Maybe the answer has something to do with the following paper, Non-axiomatizability of real general affine geometry by A. Prestel and L.W. Szczerba -- although it seems, from the introduction to that paper, that they are just saying that general real affine geometry can't be axiomatized using first-order logic, and that using $\forall \exists \forall$ one could have an axiomatization system for general real affine geometry -- so why not just use second-order logic and write it down? I've never understood the obsessive avoidance of second-order logic - why give up the extra expressive power? – Chill2Macht Jul 02 '17 at 20:44
  • It seems from this page https://books.google.de/books?id=O-bmGmCQzy0C&pg=PA106&lpg=PA106&dq=desargues+axiom+tarski&source=bl&ots=jKU7g5hO3K&sig=nB3LcYhdjTJQJz3LuFFNcQLdldo&hl=en&sa=X&redir_esc=y#v=onepage&q=desargues%20axiom%20tarski&f=false that ordered geometry might just be (affine geometry) $-$ (Desargues axiom). Since Desargues axiom is the only axiom in Szczeba and Tarski's axiom system for affine geometry which is not in Tarski's axioms for Euclidean geometry (I think), this implies probably a modular construction of axioms for ordered geometry but not affine geometry. – Chill2Macht Jul 02 '17 at 21:02
  • https://books.google.de/books?id=IMgg0Uc00I4C&pg=PA593&lpg=PA593&dq=desargues+axiom+tarski&source=bl&ots=jfYHmjuVbS&sig=38dULwRvA0RHG5uJ8B8Xh3tyzz0&hl=en&sa=X&redir_esc=y#v=onepage&q=desargues%20axiom%20tarski&f=false Never mind, it's not just Desargues axiom, also Extension axiom and (somehow) upper dimension axiom (even though something else also called the upper dimension axiom is definitely one of Tarski's axioms for Euclidean geometry -- wonder why it has to be changed for affine geometry). – Chill2Macht Jul 02 '17 at 21:05
  • Apparently the Desargues axiom is redundant in dimension $n \ge 3$ according to this source? https://books.google.de/books?id=dOTcBwAAQBAJ&pg=PA165&lpg=PA165&dq=desargues+axiom+tarski&source=bl&ots=ghX-9kNP7d&sig=svLRL12wrbDYvFUpZ0tt8xp-oHE&hl=en&sa=X&redir_esc=y#v=onepage&q=desargues%20axiom%20tarski&f=false Maybe that fact provides some room for simplification of the axioms of affine geometry in higher dimensions? – Chill2Macht Jul 02 '17 at 21:08
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    In 3D Desargues is a theorem of projective geometry, but there are non desarguian planes. Pappus implies Desargues. – Julien Narboux Jul 05 '17 at 19:35
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    Upper dimension axiom in Tarski's axiom system use segment congruence which is not available in affine geometry, that may be the reason why they have another version. – Julien Narboux Jul 05 '17 at 19:37
  • @JulienNarboux Is segment congruence only a problem with the upper dimension axiom, or do almost all of the axioms need to be reformulated then? For example, lower dimension is the negation of upper dimension, and the five-segment axioms relies on congruent segments (right?). I'm not sure exactly what you mean specifically by segment congruence, since I think all of the axioms besides A6, A7, A10, and A11 all mention congruence, and the only form of congruence defined in the axioms is segment congruence (I think). – Chill2Macht Jul 05 '17 at 21:14
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    There are different versions of lower dimension axiom, the one in wikipedia uses betweenness only. – Julien Narboux Jul 06 '17 at 10:54
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    If one remove all axioms that involve congruence of segments, I think the remaining axioms does not lead to affine geometry, nor ordered geometry. Hilbert's axioms are more modular because one can choose to keep only incidence and betweenness groups. – Julien Narboux Jul 06 '17 at 10:57
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    I think the fact that Tarski's axioms are independent does not help here. For example, there are axiom systems for the reals, which are independent and do not include the commutativity of addition (http://www.uccs.edu/Documents/goman/An%20independent%20axiom%20system%20for%20the%20real%20numbers.pdf). Hence, I think you don't have a subset of the axioms given in the paper above which give you just an abelian group for +. – Julien Narboux Jul 06 '17 at 11:05

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Given that Tarski aimed for independence of the axiom system, axioms that contained both betweenness and congruence were used to prove pure betweennness axioms, so its aim was not modularity. For axiom systems of affine geometry one would need configuration theorems, and those are absent, as they can be proved using the congruence axioms. Tarski knew this very well, and has added axioms as needed for his studies on affine geometry, such as Szczerba, L. W.; Tarski, A. Metamathematical discussion of some affine geometries. Fund. Math. 104 (1979), no. 3, 155–192.

Victor Pambuccian
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    Hi Victor! Welcome to MSE. Hope all of us here benefit further from your expertise in the history of math. You may want to see also the historical questions here. – Mikhail Katz Jul 06 '17 at 15:43
  • Thank you very much for this answer. I am familiar with the paper by Szczerba and Tarski, but I (obviously) did not understand it very well. How did they get the axioms for affine geometry from the ones for Euclidean geometry? By splitting up some of the axioms for Euclidean geometry? You explained well already why this is the case, I am just still trying to wrap my head around how the axiom system presented in Szczerba and Tarski is more "complicated" (in a stupid layman's intuitive sense) yet also less powerful than the axiom system for Euclidean geometry. – Chill2Macht Jul 07 '17 at 15:07
  • Also, for an axiomatization of ordered geometry, does it just suffice to only consider the Euclidean geometry axioms involving betweenness only? Or are those also too "entangled" with congruence axioms to be able to form an axiomatization of ordered geometry alone? I am (somewhat futilely) trying to think about the various types of (plane) geometry the way one can think of algebraic objects, with descriptions which are simple, but not necessarily minimal, and which allow one to build up less general objects with more complicated structure from more general objects with less general structure. – Chill2Macht Jul 07 '17 at 15:10