Hilbert's axioms

2025-10-21T12:55:32Z

2A01:E0A:9B6:13F0:D9E0:A15D:FE2E:C322: /* II. Betweenness (Order) Axioms */ correction Axiom B1

{{Short description|Basis for Euclidean geometry}}
'''Hilbert's axioms''' are a set of 20 assumptions proposed by [[David Hilbert]] in 1899 in his book ''[[Grundlagen der Geometrie]]''<ref>{{cite journal|author=Sommer, Julius|title=Review: Grundlagen der Geometrie, Teubner, 1899|journal=Bull. Amer. Math. Soc.|year=1900|volume=6|issue=7|pages=287–299|url=https://www.ams.org/journals/bull/1900-06-07/S0002-9904-1900-00719-1/S0002-9904-1900-00719-1.pdf|doi=10.1090/s0002-9904-1900-00719-1|doi-access=free}}</ref><ref>{{cite journal|author=Poincaré, Henri|authorlink=Henri Poincaré|title=Poincaré's review of Hilbert's "Foundations of Geometry", translated by E. V. Huntington|journal=Bull. Amer. Math. Soc.|year=1903|volume=10|pages=1–23|url=https://www.ams.org/journals/bull/1903-10-01/S0002-9904-1903-01061-1/S0002-9904-1903-01061-1.pdf|doi=10.1090/S0002-9904-1903-01061-1|doi-access=free}}</ref><ref>{{cite journal|author=Schweitzer, Arthur Richard|title=Review: ''Grundlagen der Geometrie'', Third edition, Teubner, 1909|journal=Bull. Amer. Math. Soc.|year=1909|volume=15|issue=10|pages=510–511|url=https://www.ams.org/journals/bull/1909-15-10/S0002-9904-1909-01814-2/S0002-9904-1909-01814-2.pdf|doi=10.1090/s0002-9904-1909-01814-2|doi-access=free}}</ref><ref>{{cite journal|author=Gronwall, T. H.|authorlink=Thomas Hakon Gronwall|title=Review: ''Grundlagen der Geometrie'', Fourth edition, Teubner, 1913|journal=Bull. Amer. Math. Soc.|year=1919|volume=20|issue=6|pages=325–326|url=https://www.ams.org/journals/bull/1914-20-06/S0002-9904-1914-02492-9/S0002-9904-1914-02492-9.pdf|doi=10.1090/S0002-9904-1914-02492-9|doi-access=free}}</ref> (tr. ''The Foundations of Geometry'') as the foundation for a modern treatment of [[Euclidean geometry]]. Other well-known modern [[axiomatization]]s of Euclidean geometry are those of [[Tarski's axioms|Alfred Tarski]] and of [[Birkhoff's axioms|George Birkhoff]].

== The axioms ==
Hilbert's [[axiom system]] is constructed with six [[primitive notion]]s: three primitive terms:<ref>These axioms and their numbering are taken from the Unger translation (into English) of the 10th edition of ''Grundlagen der Geometrie''.</ref>
* [[Point (geometry)|point]];
* [[Line (geometry)|line]];
* [[Plane (mathematics)|plane]];

=== Primitive symbols ===
* Points:<math> A,B,C,\dots</math>
* Lines: <math>l,m,n,\dots</math>
* Planes: <math>\pi,\rho,\sigma,\dots</math>

=== Primitive relations ===
Hilbert's system of Euclidean geometry uses three types of primitive objects — '''points''', '''lines''', and '''planes''' — and defines the following primitive relations:

* <math>A \in l</math> or <math>A \in \pi</math> or <math>l \in \pi</math> — '''incidence''' (a point lies on a line, a point lies in a plane, or a line lies in a plane)
* <math>A * B * C</math> — '''betweenness''', meaning that <math>B</math> is between <math>A</math> and <math>C</math>
* <math>\overline{AB} \cong \overline{CD}</math> — '''segment congruence''', stating that the segments <math>AB</math> and <math>CD</math> are equal in length
* <math>\angle ABC \cong \angle DEF</math> — '''angle congruence''', stating that the angles <math>ABC</math> and <math>DEF</math> are equal in measure

Line segments, angles, and triangles may each be defined in terms of points and straight lines, using the relations of betweenness and containment. All points, straight lines, and planes in the following axioms are distinct unless otherwise stated.

===I. Incidence===
<ol>
<li>
For every two points ''A'' and ''B'' there exists a line ''a'' that contains them both. We write ''AB'' = ''a'' or ''BA'' = ''a''. Instead of "contains", we may also employ other forms of expression; for example, we may say "''A'' lies upon ''a''", "''A'' is a point of ''a''", "''a'' goes through ''A'' and through ''B''", "''a'' joins ''A'' to ''B''", etc. If ''A'' lies upon ''a'' and at the same time upon another line ''b'', we make use also of the expression: "The lines ''a'' and ''b'' have the point ''A'' in common", etc.
For every two points there exists no more than one line that contains them both; consequently, if {{nowrap|1=''AB'' = ''a''}} and {{nowrap|1=''AC'' = ''a''}}, where {{nowrap|''B'' ≠ ''C''}}, then also {{nowrap|1=''BC'' = ''a''}}.
<math>
\begin{aligned}
(I_1)&\quad \forall A\,\forall B\,(A\neq B \rightarrow \exists! l\ (A\in l \wedge B\in l)) \\[4pt]
\end{aligned}
</math>
</li>
<li>
There exist at least two points on a line. 
<math>
\begin{aligned}
(I_2)&\quad \forall l\,\exists A\,\exists B\,(A\neq B \wedge A\in l \wedge B\in l) \\[4pt]
\end{aligned}
</math>
</li>
<li>
There exist at least three points that do not lie on the same line.
<math>
\begin{aligned}
(I_3)&\quad \exists A\,\exists B\,\exists C\ \neg \exists l\,(A\in l \wedge B\in l \wedge C\in l) \\[4pt]
\end{aligned}
</math>
</li>
<li>
For every three points ''A'', ''B'', ''C'' not situated on the same line there exists a plane α that contains all of them. For every plane there exists a point which lies on it. We write {{nowrap|1=''ABC'' = ''α''}}. We employ also the expressions: "''A'', ''B'', ''C'' lie in ''α''"; "''A'', ''B'', ''C'' are points of ''α''", etc.
<math>
\begin{aligned}
(I_4)&\quad \forall A\,\forall B\,\forall C\,[\neg \exists l\,(A\in l\wedge B\in l\wedge C\in l)\rightarrow \exists!\pi\,(A\in\pi\wedge B\in\pi\wedge C\in\pi)] \\[4pt]
\end{aligned}
</math>
</li>
<li>
For every three points ''A'', ''B'', ''C'' which do not lie in the same line, there exists no more than one plane that contains them all.
<math>
\begin{aligned}
(I_5)&\quad \forall \pi\,\exists A\,\exists B\,\exists C\,(A\in\pi\wedge B\in\pi\wedge C\in\pi\wedge \neg\exists l\,(A\in l\wedge B\in l\wedge C\in l)) \\[4pt]
\end{aligned}
</math>
</li>
<li>
If two points ''A'', ''B'' of a line ''a'' lie in a plane ''α'', then every point of ''a'' lies in ''α''. In this case we say: "The line ''a'' lies in the plane ''α''", etc.
<math>
\begin{aligned}
(I_6)&\quad \forall A\,\forall B\,\forall l\,\forall\pi\,[ (A\in l\wedge B\in l\wedge A\in\pi\wedge B\in\pi)\rightarrow l\in\pi ] \\[4pt]
\end{aligned}
</math>
</li>
<li>
If two planes ''α'', ''β'' have a point ''A'' in common, then they have at least a second point ''B'' in common.
<math>
\begin{aligned}
(I_7)&\quad \forall\pi\,\forall\rho\,\bigl(\pi\neq\rho\wedge \exists P\,(P\in\pi\wedge P\in\rho)\rightarrow \exists l\,\forall P\,(P\in l\leftrightarrow (P\in\pi\wedge P\in\rho))\bigr) \\[4pt]
\end{aligned}
</math>
</li>
<li>
There exist at least four points not lying in a plane.
<math>
\begin{aligned}
(I_8)&\quad \exists A\,\exists B\,\exists C\,\exists D\ \neg \exists \pi\,(A\in\pi\wedge B\in\pi\wedge C\in\pi\wedge D\in\pi)
\end{aligned}
</math>
</li>
</ol>

===II. Betweenness (Order) Axioms ===
<ol>
<li>
 If a point ''B'' lies between points ''A'' and ''C'', ''B'' is also between ''C'' and ''A'', and there exists a line containing the distinct points ''A'', ''B'', ''C''.
<math>
\begin{aligned}
(B_1)&\quad \forall A\,\forall B\,\forall C\,(A * B * C \rightarrow (C * B * A \wedge A\neq B \wedge B \neq C \wedge A \neq C \wedge \exists l\,(A\in l\wedge B\in l\wedge C\in l))) \\[4pt]
\end{aligned}
</math>
</li>
<li>
If ''A'' and ''C'' are two points, then there exists at least one point ''B'' on the line ''AC'' such that ''C'' lies between ''A'' and ''B''.<ref>In the Townsend edition this statement differs in that it also includes the existence of at least one point ''D'' with ''C'' between ''A'' and ''D'', which became a theorem in a later edition.</ref> 
<math>
\begin{aligned}
(B_2)&\quad \forall A\,\forall C\,(A\neq C \rightarrow \exists B\,(A * C * B)) \\[4pt]
\end{aligned}
</math>
</li>
<li>
Of any three points situated on a line, there is no more than one which lies between the other two.<ref>The existence part ("there is at least one") is a theorem.</ref> 
<math>
\begin{aligned}
(B_3)&\quad \forall A,B,C\,(\text{exactly one of }A * B * C, B * C * A, C * A * B)
\end{aligned}
</math>
</li>
<li>
[[Pasch's axiom]]: Let ''A'', ''B'', ''C'' be three points not lying in the same line and let ''a'' be a line lying in the plane ''ABC'' and not passing through any of the points ''A'', ''B'', ''C''. Then, if the line ''a'' passes through a point of the segment ''AB'', it will also pass through either a point of the segment ''BC'' or a point of the segment ''AC''.

<math>
\begin{aligned}
(B_4)&\quad \forall A,B,C,l\,((A,B,C \text{ non-collinear}) \wedge (l \text{ intersects } AB) \wedge \neg(l \text{ passes through }A,B,C)) \\
&\qquad \rightarrow (l \text{ intersects } AC \lor l \text{ intersects } BC))
\end{aligned}
</math>
</li>
</ol>

===III. Congruence ===
<ol>
<li>
 If ''A'', ''B'' are two points on a line ''a'', and if ''A''′ is a point upon the same or another line ''a''′, then, upon a given side of ''A''′ on the straight line ''a''′, we can always find a point ''B''′ so that the segment ''AB'' is congruent to the segment ''A''′''B''′. We indicate this relation by writing {{nowrap|''AB'' ≅ ''A''′''B''′}}. Every segment is congruent to itself; that is, we always have {{nowrap|''AB'' ≅ ''AB''}}. We can state the above axiom briefly by saying that every segment can be ''laid off'' upon a given side of a given point of a given straight line in at least one way.

<math>
\begin{aligned}
(C_1)&\quad \forall A,B,A',B'\,\exists C'\,(\text{Ray}(A',B',C') \wedge \overline{AB}\cong\overline{A'C'})
\end{aligned}
</math>
</li>
<li>
 If a segment ''AB'' is congruent to the segment ''A''′''B''′ and also to the segment ''A''″''B''″, then the segment ''A''′''B''′ is congruent to the segment ''A''″''B''″; that is, if {{nowrap|''AB'' ≅ ''A''′''B''′}} and {{nowrap|''AB'' ≅ ''A''″''B''″}}, then {{nowrap|''A''′''B''′ ≅ ''A''″''B''″}}.

<math>
\begin{aligned}
(C_2)&\quad \forall A,B\,(\overline{AB}\cong\overline{AB})
\end{aligned}
</math> (reflexive)

<math>
\begin{aligned}
(C_2)&\quad \forall A,B,C,D\,((\overline{AB}\cong\overline{CD})\rightarrow(\overline{CD}\cong\overline{AB}))
\end{aligned}
</math> (symmetric)

<math>
\begin{aligned}
(C_2)&\quad \forall A,B,C,D,E,F\,((\overline{AB}\cong\overline{CD}\wedge\overline{CD}\cong\overline{EF})\rightarrow(\overline{AB}\cong\overline{EF}))
\end{aligned}
</math> (transitive)

</li>

<li>
 Let ''AB'' and ''BC'' be two segments of a line ''a'' which have no points in common aside from the point ''B'', and, furthermore, let ''A''′''B''′ and ''B''′''C''′ be two segments of the same or of another line ''a''′ having, likewise, no point other than ''B''′ in common. Then, if {{nowrap|''AB'' ≅ ''A''′''B''′}} and {{nowrap|''BC'' ≅ ''B''′''C''′}}, we have {{nowrap|''AC'' ≅ ''A''′''C''′}}.

<math>
\begin{aligned}
(C_3)&\quad \forall A,B,C,A',B',C'\,((\overline{AB}\cong\overline{A'B'}\wedge\overline{BC}\cong\overline{B'C'})\rightarrow\overline{AC}\cong\overline{A'C'})
\end{aligned}
</math>
</li>

<li>
 Let an angle {{nowrap|∠ (''h'',''k'')}} be given in the plane ''α'' and let a line ''a''′ be given in a plane ''α''′. Suppose also that, in the plane ''α''′, a definite side of the straight line ''a''′ be assigned. Denote by ''h''′ a ray of the straight line ''a''′ emanating from a point ''O''′ of this line. Then in the plane ''α''′ there is one and only one ray ''k''′ such that the angle {{nowrap|∠ (''h'', ''k'')}}, or {{nowrap|∠ (''k'', ''h'')}}, is congruent to the angle {{nowrap|1=∠ (''h''′, ''k''′)}} and at the same time all interior points of the angle {{nowrap|1=∠ (''h''′, ''k''′)}} lie upon the given side of ''a''′. We express this relation by means of the notation {{nowrap|∠ (''h'', ''k'') ≅ ∠ (''h''′, ''k''′)}}.

<math>
\begin{aligned}
(C_4)&\quad \forall A,B,C,A',B'\,\exists !\,C'\,(\angle ABC\cong\angle A'B'C')
\end{aligned}
</math>
</li>

<li>
 If the angle {{nowrap|∠ (''h'', ''k'')}} is congruent to the angle {{nowrap|∠ (''h''′, ''k''′)}} and to the angle {{nowrap|∠ (''h''″, ''k''″)}}, then the angle {{nowrap|∠ (''h''′, ''k''′)}} is congruent to the angle {{nowrap|∠ (''h''″, ''k''″)}}; that is to say, if {{nowrap|∠ (''h'', ''k'') ≅ ∠ (''h''′, ''k''′)}} and {{nowrap|∠ (''h'', ''k'') ≅ ∠ (''h''″, ''k''″)}}, then {{nowrap|∠ (''h''′, ''k''′) ≅ ∠ (''h''″, ''k''″)}}.

<math>
\begin{aligned}
(C_5)&\quad \forall A,B,C\,(\angle ABC\cong\angle ABC)
\end{aligned}
</math>

<math>
\begin{aligned}
(C_5)&\quad \forall A,B,C,D,E,F\,((\angle ABC\cong\angle DEF)\rightarrow(\angle DEF\cong\angle ABC))
\end{aligned}
</math>

<math>
\begin{aligned}
(C_5)&\quad \forall A,B,C,D,E,F,G,H,I\,((\angle ABC\cong\angle DEF\wedge\angle DEF\cong\angle GHI)\rightarrow\angle ABC\cong\angle GHI)
\end{aligned}
</math>
</li>
<li>
 If, in the two triangles ''ABC'' and ''A''′''B''′''C''′ the congruences {{nowrap|''AB'' ≅ ''A''′''B''′}}, {{nowrap|''AC'' ≅ ''A''′''C''′}}, {{nowrap|∠''BAC'' ≅ ∠''B''′''A''′''C''′}} hold, then the congruence {{nowrap|∠''ABC'' ≅ ∠''A''′''B''′''C''′}} holds (and, by a change of notation, it follows that {{nowrap|∠''ACB'' ≅ ∠''A''′''C''′''B''′}} also holds).

<math>
\begin{aligned}
(C_6)&\quad \forall A,B,C,A',B',C'\,((\overline{AB}\cong\overline{A'B'}\wedge\overline{AC}\cong\overline{A'C'}\wedge\angle BAC\cong\angle B'A'C')\rightarrow(\angle{ACB}\cong\angle{A'C'B'}))
\end{aligned}
</math>
</li>
</ol>

===IV. Parallels ===
# [[Playfair's axiom]]:<ref>This is Hilbert's terminology. This statement is more familiarly known as [[Playfair's axiom]].</ref> Let ''a'' be any line and ''A'' a point not on it. Then there is at most one line in the plane, determined by ''a'' and ''A'', that passes through ''A'' and does not intersect ''a''.
<math>
\begin{aligned}
(P)\quad \forall A\,\forall l\ \bigl(A\notin l \rightarrow \exists! m\ &(A\in m\wedge \neg\exists P\,(P\in l\wedge P\in m))\bigr).
\end{aligned}
</math>

===V. Continuity ===
# [[Axiom of Archimedes]]: If ''AB'' and ''CD'' are any segments then there exists a number ''n'' such that ''n'' segments ''CD'' constructed contiguously from ''A'', along the ray from ''A'' through ''B'', will pass beyond the point ''B''.
# ''Axiom of line completeness'': An extension (An extended line from a line that already exists, usually used in geometry) of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follows from Axioms I-III and from V-1 is impossible.
<math>
\begin{aligned}
(Ct)\quad &\forall l\ \forall X\subseteq\{P: P\in l\}\ \forall Y\subseteq\{P: P\in l\}\ \Bigl[\, X\neq\varnothing\wedge Y\neq\varnothing\wedge X\cup Y=\{P:P\in l\}\\
&\qquad\wedge X\cap Y=\varnothing\wedge (\forall x\in X\ \forall y\in Y\ \neg\exists z\,(z\in l\wedge y * z * x))\\
&\qquad\longrightarrow \exists c\,(c\in l\wedge \forall x\in X\ \forall y\in Y\ (x * c * y\vee x=c\vee y=c))\,\Bigr]
\end{aligned}
</math>

== Hilbert's discarded axiom ==
Hilbert (1899) included a 21st axiom that read as follows:
:II.4. Any four points ''A'', ''B'', ''C'', ''D'' of a line can always be labeled so that ''B'' shall lie between ''A'' and ''C'' and also between ''A'' and ''D'', and, furthermore, that ''C'' shall lie between ''A'' and ''D'' and also between ''B'' and ''D''.

This statement is also known as [[Pasch's theorem]].

[[E. H. Moore]] and [[R. L. Moore]] independently proved that this axiom is redundant, and the former published this result in an article appearing in the ''Transactions of the American Mathematical Society'' in 1902.<ref>{{citation|first=E. H.|last=Moore|title=On the projective axioms of geometry|journal=Transactions of the American Mathematical Society|year=1902|volume=3|issue=1 |pages=142–158|url=https://www.ams.org/journals/tran/1902-003-01/S0002-9947-1902-1500592-8/S0002-9947-1902-1500592-8.pdf|doi=10.2307/1986321|jstor=1986321 |doi-access=free}}</ref>

Before this, [[Pasch's axiom]], now listed as II.4, was numbered II.5.

== Editions and translations of ''Grundlagen der Geometrie'' ==
The original monograph, based on his own lectures, was organized and written by Hilbert for a memorial address given in 1899. This was quickly followed by a French translation, in which Hilbert added V.2, the Completeness Axiom. An English translation, authorized by Hilbert, was made by E.J. Townsend and copyrighted in 1902. This translation incorporated the changes made in the French translation and so is considered to be a translation of the 2nd edition. Hilbert continued to make changes in the text and several editions appeared in German. The 7th edition was the last to appear in Hilbert's lifetime. In the Preface of this edition Hilbert wrote:
:"The present Seventh Edition of my book ''Foundations of Geometry'' brings considerable improvements and additions to the previous edition, partly from my subsequent lectures on this subject and partly from improvements made in the meantime by other writers. The main text of the book has been revised accordingly."
New editions followed the 7th, but the main text was essentially not revised. The modifications in these editions occur in the appendices and in supplements. The changes in the text were large when compared to the original and a new English translation was commissioned by Open Court Publishers, who had published the Townsend translation. So, the 2nd English Edition was translated by Leo Unger from the 10th German edition in 1971. This translation incorporates several revisions and enlargements of the later German editions by Paul Bernays.

The Unger translation differs from the Townsend translation with respect to the axioms in the following ways:
* Old axiom II.4 is renamed as Theorem 5 and moved.
* Old axiom II.5 (Pasch's Axiom) is renumbered as II.4.
* V.2, the Axiom of Line Completeness, replaced:
:: ''Axiom of completeness''. To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner that the system thus generalized shall form a new geometry obeying all of the five groups of axioms. In other words, the elements of geometry form a system which is not susceptible of extension, if we regard the five groups of axioms as valid.
* The old axiom V.2 is now Theorem 32.

The last two modifications are due to P. Bernays.

Other changes of note are:
* The term ''straight line'' used by Townsend has been replaced by ''line'' throughout.
* The ''Axioms of Incidence'' were called ''Axioms of Connection'' by Townsend.

== Application ==
These axioms [[axiom]]atize Euclidean [[solid geometry]]. Removing five axioms mentioning "plane" in an essential way, namely I.4–8, and modifying III.4 and IV.1 to omit mention of planes, yields an axiomatization of [[Euclidean plane geometry]].

Hilbert's axioms, unlike [[Tarski's axioms]], do not constitute a [[first-order logic|first-order theory]] because the axioms V.1–2 cannot be expressed in [[first-order logic]].

The value of Hilbert's ''Grundlagen'' was more methodological than substantive or pedagogical. Other major contributions to the axiomatics of geometry were those of [[Moritz Pasch]], [[Mario Pieri]], [[Oswald Veblen]], [[Edward Vermilye Huntington]], [[Gilbert de Beauregard Robinson|Gilbert Robinson]], and [[Henry George Forder]]. The value of the ''Grundlagen'' is its pioneering approach to [[metamathematics|metamathematical]] questions, including the use of models to prove axioms independent; and the need to prove the consistency and completeness of an axiom system.

Mathematics in the twentieth century evolved into a network of axiomatic [[formal system]]s. This was, in considerable part, influenced by the example Hilbert set in the ''Grundlagen''. A 2003 effort (Meikle and Fleuriot) to formalize the ''Grundlagen'' with a computer, though, found that some of Hilbert's proofs appear to rely on diagrams and geometric intuition, and as such revealed some potential ambiguities and omissions in his definitions.<ref>On page 334: ''"By formalizing the ''Grundlagen'' in Isabelle/Isar we showed that Hilbert's work glossed over subtle points of reasoning and relied heavily, in some cases, on diagrams which allowed implicit assumptions to be made. For this reason it can be argued that Hilbert interleaved his axioms with geometric intuition in order to prove many of his theorems."''</ref>

==See also==

* [[Euclidean space]]
* [[Foundations of geometry]]

==Notes==
{{reflist|2}}

==References==
* [[Howard Eves]], 1997 (1958). ''Foundations and Fundamental Concepts of Mathematics''. Dover. Chpt. 4.2 covers the Hilbert axioms for plane geometry.
*[[Ivor Grattan-Guinness]], 2000. ''In Search of Mathematical Roots''. Princeton University Press.
*[[David Hilbert]], 1980 (1899). ''[http://www.gutenberg.org/files/17384/17384-pdf.pdf The Foundations of Geometry]'', 2nd ed. Chicago: Open Court.
*Laura I. Meikle and Jacques D. Fleuriot (2003), [http://homepages.inf.ed.ac.uk/s9811254/papers/TPHOLs2003.ps Formalizing Hilbert's Grundlagen in Isabelle/Isar] {{Webarchive|url=https://web.archive.org/web/20160304000229/http://homepages.inf.ed.ac.uk/s9811254/papers/TPHOLs2003.ps |date=2016-03-04 }}, Theorem Proving in Higher Order Logics, Lecture Notes in Computer Science, Volume 2758/2003, 319-334, {{doi|10.1007/10930755_21}}

==External links==
* {{springer|title=Hilbert system of axioms|id=p/h047400}}
* [http://www.math.umbc.edu/~campbell/Math306Spr02/Axioms/Hilbert.html "Hilbert's Axioms" at the UMBC Math Department]
* [http://mathworld.wolfram.com/HilbertsAxioms.html "Hilbert's Axioms" at Mathworld]
* {{librivox book | title=Foundations of Geometry| author=Hilbert}}

{{Mathematical logic}}

[[Category:David Hilbert|Axioms]]
[[Category:Foundations of geometry]]

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Hilbert's axioms