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We have many different systems of axioms in geometry and from observations we most often use Euclidean ones. Euclid's postulates are insufficient, but the Hilbert system seems overloaded and redundant. (For example, the axiom of two triangles is very similar to the theorem of the equality of triangles, and it is the same with the axioms of the congruence of angles and segments.) There is a school system of postulates (where some can be derived from others, but for the sake of simplicity they are shown as axioms) and in general Birkhoff's system, in which there are only four axioms, but clarity is lost). When studying these, there is a feeling of loss of theorems, when there are already 20 axioms.

Which system of axioms is used most often in modern geometry?

Is there a system of axioms in algebra and in other parts of mathematics?

philipxy
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Nikolai Vorobiev
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  • Modern geometry has three spaces of constant curvature $0$,$1$ and $-1$, the Euclidean $\Bbb R^n$, the spherical $\Bbb S^n$ and the hyperbolic space $\Bbb H^n$. – Dietrich Burde May 31 '25 at 18:27
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    First,of all, Euclid's geometry was not complete. There were hidden axioms that he was using about "betweenness," as in, what does it mean for one point to be between two others on a line. – Thomas Andrews May 31 '25 at 18:27
  • Hilbert came up with a fixed version of the axiom system, but some of it is overkill, essentially trying to ensure the line is "complete" in some sense. – Thomas Andrews May 31 '25 at 18:31
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    But there is no "this is geometry now" axiom set. There are lots of different geometries The "real" geometry of the universe appears to have properties that are hard to axiomatize. We aren't close to understanding it, but we are closer than Euclid. – Thomas Andrews May 31 '25 at 18:33
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    Another thing to note is that the modern usage of "geometry" is so vast that it involves things like number theory (arithmetic geometry), algebra (algebraic geometry), logic (the geometry of toposes) and much more. – Trebor Jun 01 '25 at 06:34
  • Although a lot of mathematicians say "ZFC" when asked what axioms they use, the ironic fact is that most of them cannot even write down a correct description of ZFC. Even when they can, they almost always use many notations beyond plain ZFC. If you actually want to do 100% rigorous formal mathematics, you need more than plain ZFC, such as this system. Most also do not ever use certain things ZFC allows such as unbounded Specification/Replacement, which violate the iterative conception of sets. – user21820 Jun 01 '25 at 08:36
  • @user21820, It seemed to me more like ZFC was a list of definitions rather than axioms. – Nikolai Vorobiev Jun 01 '25 at 09:12
  • The version of this post prior to my edit was a lot of sentence parts that did not fully make sense, and they were joined in ways that did not make sense. My edit has tried to make some sense. Anyway it's not clear what any of the post other than the question sentence is trying to add to it. PS Please ask 1 (specific researched non-duplicate) question. – philipxy Jun 01 '25 at 09:30
  • @ThomasAndrews This seems useful information, but a comment is not the right place to put it; would you like to write an answer based on that text? – Federico Poloni Jun 01 '25 at 12:03
  • @FedericoPoloni I'm waiting for the question to be clearer. – Thomas Andrews Jun 01 '25 at 12:28
  • @NikolaiVorobiev: Well that just implies that you do not understand set theory at all. – user21820 Jun 01 '25 at 13:08
  • @ThomasAndrews It might be useful for OP if you specify what kind of clarification you are looking for then; thanks! – Federico Poloni Jun 01 '25 at 13:39

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The axioms of Euclidean geometry are rarely used by mathematicians nowadays, outside of teaching purposes or historical purposes. The importance of Hilbert's axioms for Euclidean geometry (from his Grundlagen around 1899) is that he used them to present the first example of the modern concept of an axiomatic system. Hilbert improved these axioms in later editions of the book, removing redundancies. And while other writers have toyed with variations on the axioms of Euclidean geometry, with various different pedagogical goals, the object is always to have axiom systems for the same theory, that is, axiom systems that are equivalent to each other. So there's not really any way to say "these axioms are better than those ones" without some subjective criterion as a guide. Personally, I like axioms of the Euclidean plane which incorporate rigid motions, rather than axioms with two different flavors of congruence as Hilbert has (I've taught this in my classes).

On the other hand, the axiomatic method --- of which the Euclid book is the founding example (however imperfect), and the Hilbert book is the first modern example --- now pervades much of modern mathematics. Yes, there are systems of axioms throughout algebra, topology, linear algebra, set theory, analysis (including axioms of the real numbers), and so on.

Perhaps the most important axiom system nowadays is the ZFC set theory axioms. Many important definitions of mathematical structures are formulated in an axiomatic manner within set theory. For example, a vector space is usually defined as a model --- within set theory --- of the axioms of vector spaces.

Another example of an axiomatic system, one that is taught in the college math curriculum, are the axioms of the real numbers, meaning the axioms for a complete, ordered field. For this example (as for many others), one usually learns at some point how to model the axioms of the real numbers within ZFC. This is usually done in a series of steps: first one models (the axioms of) the natural numbers within ZFC; then one models the integers within the natural numbers; then one models the rational numbers within the integers; and finally one models the real numbers within the rationals, using for example the Dedekind cut method.

And one can then model the axioms of the Euclidean plane within the real numbers: a point is an ordered pair of real numbers $(x,y)$; a line is the solution set of an equation of the form $Ax+By=C$ where $A,B,C$ are real numbers and $A,B$ are not both zero; one proves that two points determine a unique line; and so on...

So with that as an example, one might say that the ZFC axioms of set theory form the "main" axiom system of mathematics (including geometry, algebra, analysis, ...).

Lee Mosher
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    Thank you for your detailed answer! Could you explain why even with an equal number of axioms, for example, the axioms of the school course and Hilbert's axioms, the former will be understood much faster, this is due to the gradual replacement with simpler equivalent formulas or with simplifications allowed during study (because I remember that one of the 21 axioms in the course, for example, was definitely for simplification of the material and could be proven). And aren't the ZFC axioms definitions? – Nikolai Vorobiev May 31 '25 at 20:35
  • I don't know what axioms you might have learned in "the school course". But I do know that Euclid's postulate (axioms) are not equivalent to Hilbert's axioms. In fact it was already known in antiquity that Euclid's postulates have holes: Euclid's arguments do not all follow from the stated postulates, and as the centuries and millenia passed others began to fill in the holes with new axioms. Hilbert's axioms were the culmination of that multi-millenium process. – Lee Mosher May 31 '25 at 21:50
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    ZFC axioms are just that --- axioms. But you're on the right track: sometimes the border between axioms and definitions is very narrow. One could write down the laws of group theory and call those the axioms of a group or the definition of a group. One could even write down the axioms of Euclidean geometry and call the the definition of a Euclidean plane. – Lee Mosher May 31 '25 at 21:52