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===Dedekind’s theory of numbers===
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''for Euclidean geometry''
  
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Dedekind’s set theory is equivalent to Peirce’s and, consequently, so is his theory of numbers. This equivalence of Peirce’s theory of natural numbers to that of Dedekind (as well as that of Giuseppe Peano’s 1889 Arithmetices principia [60]) was demonstrated by Shields (in [93] and [94]).<ref>Anellis (2012?) p. 260</ref>
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A system of axioms first proposed by D. Hilbert in 1899, and subsequently amended and made more precise by him.<ref>Hilbert (1899)</ref>
  
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There are only two significant differences between the development of number theory by Dedekind and by Pierce:<ref>Anellis (2012?) p. 259</ref>
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In Hilbert's system of axioms the primary (primitive undefined) objects are $points$, (straight) $lines$, $planes$ and the relations between these terms are those of ''belonging to'', ''being between'', and ''being congruent to''. The nature of the primary objects and of the relations between those objects are arbitrary as long as the objects and the relations satisfy the axioms.
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* Dedekind started from infinite sets rather than finite sets in defining natural numbers
 
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* Dedekind is explicitly and specifically concerned with the real number continuum, that is, with infinite sets.
 
  
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Frege and Dedekind were focused on “reducing” the natural numbers and arithmetic to “logic”. This is the main goal of Dedekind's ''Was sind und was sollen die Zahlen?''
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Hilbert's system contains 20 axioms, which are subdivided into five groups.
  
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The technical centerpiece of Dedekind’s mathematical work was in number theory, especially algebraic number theory. His primary motivation was to provide a foundation for mathematics and in particular to find a rigorous definition of real numbers and of the real-number continuum upon which to establish mathematical analysis in the style of Karl Weierstrass. This means that he sought to axiomatize the theory of numbers based upon that rigorous definition of the real numbers and the construction of the real number system and the continuum which could be employed in defining the theory of limits of a function for use in the differential and integral calculus, real analysis, and related areas of function theory. His concern, in short, was with the rigorization and arithmetization of analysis.<ref>Anellis (2012?) p. 260</ref>
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===Group I: Axioms of Incidence or Connection===
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This group comprises eight axioms which describe the relation ''belonging to''.
  
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h0474002.png" />. For any two points there exists a straight line passing through them.
  
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One goal of Dedekind’s of ''Was sind und was sollen die Zahlen?'' was to answer the question, What more can be said about the set-theoretic procedures used? For Dedekind, again like for Frege, these procedures are founded in “logic”. But then, what are the basic notions of logic?<ref>SEP</ref>
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h0474003.png" />. There exists only one straight line passing through any two distinct points.
  
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* ‘’Ordinal numbers’’ are used to count elements and place them in a succession; in such cases they correspond to Enligsh expressions such as first, second, third... and so forth.
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h0474004.png" />. At least two points lie on any straight line. There exist at least three points not lying on the same straight line.
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* ‘’Cardinal numbers’’ are used, loosely speaking, to count how many elements of some kind there are: one cat, two dogs, three horses, and so forth.
 
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For Dedekind, numbers are essentially ordinals.<ref>Azzano p. 16</ref>
 
  
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Dedekind’s efforts in foundations did not stop with the reduction of all mathematics to arithmetic, thinking that both the concepts and the rules of arithmetic itself needed clarification -- principally through logic and set theory.<ref>Azzano p. 8</ref>
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h0474005.png" />. There exists a plane passing through any three points not lying on the same straight line. At least one point lies on any given plane.
  
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The ultimate basis of a mathematician’s knowledge is, according to Dedekind, the clarification of the concept of natural numbers (viz., positive integers) in a non-mathematical fashion, which involves this twofold task:<ref>Gillies cited in Azzano p. 5</ref>
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h0474006.png" />. There exists only one plane passing through any three points not lying on the same straight line.
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# to define numerical concepts (natural numbers) through logical ones
 
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# to characterize mathematical induction (the passage from n to n+1) as a logical inference.
 
  
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h0474007.png" />. If two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h0474008.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h0474009.png" /> of a straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740010.png" /> lie in a plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740011.png" />, then all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740012.png" /> lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740013.png" />.
  
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740014.png" />. If two planes have one point in common, then they have at least one more point in common.
  
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Dedekind’s methods led him to develop a “set-theoretic” style of axiomatic analysis that is quite different from the work of Peano on the natural numbers, or that of Hilbert on geometry and the real numbers…. To clarify the matter, let me remind you of Dedekind’s axioms:
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740015.png" />. There exist at least four points not lying in the same plane.
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::A simply infinite set $N$ has a distinguished element $e$ and an ordering mapping $ϕ$ such that
 
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::# $ϕ(N) ⊆ N$
 
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::# $e \notin ϕ(N)$
 
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::# $N = ϕ0(e)$, i.e. $N$ is the $ϕ$-chain of the unitary set $\{e\}$
 
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::# $ϕ$ is an injective mapping from $N$ to $N$, i.e. if $ϕ(a) = ϕ(b)$ then $a = b$.
 
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Leaving aside axioms 2. and 4., which are more easy to assimilate to Peano’s axioms, the other two axioms are characteristically set-theoretic in the intended sense, and not elementary as most of Peano’s and Hilbert’s axioms.
 
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Peano tended to impose conditions on the behaviour of his individuals, the natural numbers, and the operations on them. These are elementary conditions which most often are amenable to formalization within first-order logic. Dedekind establishes structural conditions on subsets of the (structured) sets he is defining, on the behaviour of relevant maps, or both things at a time. Axiom 1. says that $N$ is closed under the map $ϕ$, Axiom 3. says that $N$ is the minimal closure of the unitary set $\{e\}$ under $ϕ$. Such axioms are non-elementary and tend to require second-order logic for their formalisation<ref>Ferreiros “On Dedekind’s Logicism” p. 19</ref>
 
  
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A set of objects is infinite—“Dedekind-infinite”, as we now say—if it can be mapped one-to-one onto a proper subset of itself.<ref>Reck (2011) §2.2</ref>
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===Group II: Axioms of Order===
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This group comprises four axioms describing the relation ''being between''.
  
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“What are the mutually independent fundamental properties of the sequence N, that is, those properties that are not derivable from one another but from which all others follow? And how should we divest these properties of their specifically arithmetic character so that they are subsumed under more general notions and under activities of the understanding without which no thinking is possible at all, but with which a foundation is provided for the reliability and completeness of proofs and for the construction of consistent notions and definitions.”<ref>Dedekind (1888) pp. 99-100 cited in Awodey and Reck p. 8</ref>
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740017.png" />. If a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740018.png" /> lies between a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740019.png" /> and a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740023.png" /> are distinct points on the same straight line and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740024.png" /> also lies between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740026.png" />.
  
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Dedekind's intention was not to axiomatize arithmetic, but to give an "algebraic" characterization of natural numbers as a mathematical structure.<ref>Podnieks § 3.1 From Peano Axioms to First Order Arithmetic</ref>
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740027.png" />. For any two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740029.png" /> on the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740030.png" /> there exists at least one point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740031.png" /> such that the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740032.png" /> lies between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740034.png" />.
  
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What it means to be ‘’simply infinite’’ can be captured in four conditions:<ref>Reck (2011) §2.2</ref>
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740035.png" />. Out of any three points on the same straight line there exists not more than one point lying between the other two.
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:Consider a set $S$ and a subset $N$ of $S$ (possibly equal to $S$). Then $N$ is called simply infinite if there exists a function $f$ on $S$ and an element $1$ of $N$ such that
 
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:# $f$ maps $N$ into itself
 
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:# $N$ is the chain of $\{1\}$ in $S$ under $f$
 
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:# $1$ is not in the image of $N$ under $f$
 
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:# $f$ is one-to-one
 
  
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These Dedekindian conditions are a notational variant of the Peano axioms for the natural numbers. In particular, condition 2 is a version of the axiom of mathematical induction. These axioms are thus properly called the Dedekind-Peano axioms.<ref>Reck (2011) §2.2</ref>
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740036.png" /> (Pasch's axiom). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740039.png" /> be three points not lying on the same straight line, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740040.png" /> be a straight line in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740041.png" /> not passing through any of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740043.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740044.png" />. Then, if the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740045.png" /> passes through an interior point of the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740046.png" />, it also passes through an interior point of the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740047.png" /> or through an interior point of the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740048.png" />.
  
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As is also readily apparent, any simple infinity will consist of a first element $1$, a second element $f(1)$, a third $f(f(1))$, then $f(f(f(1)))$, and so on, just like any model of the Dedekind-Peano axioms.<ref>Reck (2011) §2.2</ref>
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===Group III: Axioms of Congruence===
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This group comprises five axioms that describe the relation of "being congruent to"  (Hilbert denoted this relation by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740050.png" />).
  
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Dedekind introduces the natural numbers as follows:
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740051.png" />. Given a segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740052.png" /> and a ray <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740053.png" />, there exists a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740054.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740055.png" /> such that the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740056.png" /> is congruent to the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740057.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740058.png" />.
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# he proves that every infinite set contains a simply infinite subset
 
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# he shows, in contemporary terminology, that any two simply infinite systems, or any two models of the Dedekind-Peano axioms, are isomorphic (so that the axiom system is categorical)
 
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# he notes that consequently … any truth about one of them can be translated, via the isomorphism, into a corresponding truth about the other.
 
  
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(As one may put it, all models of the Dedekind-Peano axioms are “logically equivalent”, which means that the axiom system is “semantically complete”; compare Awodey & Reck 2002 and Reck 2003a)
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740059.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740061.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740062.png" />.
  
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As Dedekind introduced them, numbers are ordinals:<ref>Azzano p. 16</ref>
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740063.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740065.png" /> be two segments on a straight line without common interior points, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740067.png" /> be two segments on the same or on a different straight line, also without any common interior points. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740069.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740070.png" />.
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* ‘’Ordinal numbers’’ are used to count objects or to place objects in a succession, in which case they correspond to expressions “first”, “second”, “third”, and so forth.
 
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* ‘’Cardinal numbers’’ are used to answer the question “how many” objects of some kind there are: one cat, two dogs, three horses, and so forth.
 
  
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Dedekind’s principles introduce finite “ordinal” numbers, but which can then be used to introduce finite “cardinal” numbers, as follows:<ref>Reck (2011) §2.2</ref>
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740071.png" />. Let there be given an angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740072.png" />, a ray <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740073.png" /> and a half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740074.png" /> bounded by the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740075.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740076.png" /> contains one and only one ray <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740077.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740078.png" />. Moreover, every angle is congruent to itself.
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::using initial segments of the number series as tallies, … we can ask which such segment, if any, can be mapped one-to-one onto it, thus measuring its “cardinality”.
 
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::a set is then finite if and only if there exists such an initial segment of the natural numbers series.
 
  
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==Notes==
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740079.png" />. If for two triangles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740081.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740082.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740083.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740084.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740085.png" />.
  
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<references/>
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===Group IV: Axioms of Continuity===
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This group comprises two continuity axioms.
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<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740087.png" /> (Archimedes' axiom). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740089.png" /> be two arbitrary segments. Then the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740090.png" /> contains a finite set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740091.png" /> such that the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740092.png" /> lies between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740094.png" />, the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740095.png" /> lies between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740097.png" />, etc., and such that the segments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740098.png" /> are congruent to the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h04740099.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h047400100.png" /> lies between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h047400101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h047400102.png" />.
 +
 
 +
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h047400103.png" /> (Cantor's axiom). Let there be given, on any straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h047400104.png" />, an infinite sequence of segments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h047400105.png" /> which satisfies two conditions: a) each segment in the sequence forms a part of the segment which precedes it; b) for each preassigned segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h047400106.png" /> it is possible to find a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h047400107.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h047400108.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h047400109.png" /> contains a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h047400110.png" /> belonging to all the segments of this sequence.
 +
 
 +
===Group V: Axioms of Parallelism===
 +
This group comprises one axiom about parallels.
 +
Let there be given a straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h047400112.png" /> and a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h047400113.png" /> not on that straight line. Then there exists not more than one straight line passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h047400114.png" />, not intersecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h047400115.png" /> and lying in the plane defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h047400116.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h047400117.png" />.
 +
 
 +
(Hilbert classified the axiom about parallels in Group IV, and the continuity axioms in Group V).
 +
 
 +
All other axioms of Euclidean geometry are defined by the basic concepts of Hilbert's system of axioms, while all the statements regarding the properties of geometrical figures and not included in Hilbert's system must be logically deducible from the axioms, or from statements which are deducible from these axioms.
 +
 
 +
Hilbert's system of axioms is complete; it is consistent if the arithmetic of real numbers is consistent. If, in Hilbert's system, the axiom about parallels is replaced by its negation, the new system of axioms thus obtained is also consistent (the system of axioms of Lobachevskii geometry), which means that the axiom about parallels is independent of the other axioms in Hilbert's system. It is also possible to demonstrate that some other axioms of this system are independent of the others.
 +
 
 +
Hilbert's system of axioms was the first fairly rigorous foundation of [[Euclidean geometry|Euclidean geometry]].
 +
 
 +
====References====
 +
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Hilbert,  "Grundlagen der Geometrie" , Teubner, reprint  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.V. Efimov,  "Höhere Geometrie" , Deutsch. Verlag Wissenschaft.  (1960)  (Translated from Russian)</TD></TR></table>
 +
 
 +
* Hilbert, D. (1899). "Grundlagen der Geometrie". [Reprint (1968) Teubner.]
 +
* Efimov, N.V. (1960). "Höhere Geometrie", Deutsch. Verlag Wissenschaft. [Translated from Russian.]
 +
 
 +
 
 +
====Comments====
 +
In axiom <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h047400118.png" /> the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h047400119.png" /> is also called an interior point of the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047400/h047400120.png" /> (Pasch's axiom should be read with this in mind).
 +
 
 +
Also, Hilbert originally used different continuity axioms: the Archimedean axiom and a completeness axiom of his own.
 +
 
 +
====References====
 +
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.G. Forder,  "Foundations of Euclidean geometry" , Dover, reprint  (1958)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.D. Aleksandrov,  "Foundations of geometry"  ''Siberian Math. J.'' , July 1987, Volume 28, Issue 4, pp 523-539'''28'''  (1987)  pp. 523–539  ''Sibirsk. Mat. Zh.'' , '''28'''  (1987)  pp. 9–28</TD></TR></table>
 +
 
 +
 
 +
* Aleksandrov, A.D. "Foundations of geometry," ''Siberian Mathematical Journal'', July 1987, Vol. 28, Issue 4, pp 523-539. [Trans. ''Sibirskii Matematicheskii Zhurnal'', Vol. 28, No. 4, pp. 9–28, July–August, 1987.]
 +
* Forder, H.G. "Foundations of Euclidean geometry". [Reprint (1958) Dover.]

Revision as of 15:52, 16 September 2015

for Euclidean geometry

A system of axioms first proposed by D. Hilbert in 1899, and subsequently amended and made more precise by him.[1]

In Hilbert's system of axioms the primary (primitive undefined) objects are $points$, (straight) $lines$, $planes$ and the relations between these terms are those of belonging to, being between, and being congruent to. The nature of the primary objects and of the relations between those objects are arbitrary as long as the objects and the relations satisfy the axioms.

Hilbert's system contains 20 axioms, which are subdivided into five groups.

Group I: Axioms of Incidence or Connection

This group comprises eight axioms which describe the relation belonging to.

. For any two points there exists a straight line passing through them.

. There exists only one straight line passing through any two distinct points.

. At least two points lie on any straight line. There exist at least three points not lying on the same straight line.

. There exists a plane passing through any three points not lying on the same straight line. At least one point lies on any given plane.

. There exists only one plane passing through any three points not lying on the same straight line.

. If two points and of a straight line lie in a plane , then all points of lie in .

. If two planes have one point in common, then they have at least one more point in common.

. There exist at least four points not lying in the same plane.

Group II: Axioms of Order

This group comprises four axioms describing the relation being between.

. If a point lies between a point and a point , then , and are distinct points on the same straight line and also lies between and .

. For any two points and on the straight line there exists at least one point such that the point lies between and .

. Out of any three points on the same straight line there exists not more than one point lying between the other two.

(Pasch's axiom). Let , and be three points not lying on the same straight line, and let be a straight line in the plane not passing through any of the points , or . Then, if the straight line passes through an interior point of the segment , it also passes through an interior point of the segment or through an interior point of the segment .

Group III: Axioms of Congruence

This group comprises five axioms that describe the relation of "being congruent to" (Hilbert denoted this relation by the symbol ).

. Given a segment and a ray , there exists a point on such that the segment is congruent to the segment , i.e. .

. If and , then .

. Let and be two segments on a straight line without common interior points, and let and be two segments on the same or on a different straight line, also without any common interior points. If and , then .

. Let there be given an angle , a ray and a half-plane bounded by the straight line . Then contains one and only one ray such that . Moreover, every angle is congruent to itself.

. If for two triangles and one has , , , then .

Group IV: Axioms of Continuity

This group comprises two continuity axioms.

(Archimedes' axiom). Let and be two arbitrary segments. Then the straight line contains a finite set of points such that the point lies between and , the point lies between and , etc., and such that the segments are congruent to the segment , and lies between and .

(Cantor's axiom). Let there be given, on any straight line , an infinite sequence of segments which satisfies two conditions: a) each segment in the sequence forms a part of the segment which precedes it; b) for each preassigned segment it is possible to find a natural number such that . Then contains a point belonging to all the segments of this sequence.

Group V: Axioms of Parallelism

This group comprises one axiom about parallels. Let there be given a straight line and a point not on that straight line. Then there exists not more than one straight line passing through , not intersecting and lying in the plane defined by and .

(Hilbert classified the axiom about parallels in Group IV, and the continuity axioms in Group V).

All other axioms of Euclidean geometry are defined by the basic concepts of Hilbert's system of axioms, while all the statements regarding the properties of geometrical figures and not included in Hilbert's system must be logically deducible from the axioms, or from statements which are deducible from these axioms.

Hilbert's system of axioms is complete; it is consistent if the arithmetic of real numbers is consistent. If, in Hilbert's system, the axiom about parallels is replaced by its negation, the new system of axioms thus obtained is also consistent (the system of axioms of Lobachevskii geometry), which means that the axiom about parallels is independent of the other axioms in Hilbert's system. It is also possible to demonstrate that some other axioms of this system are independent of the others.

Hilbert's system of axioms was the first fairly rigorous foundation of Euclidean geometry.

References

[1] D. Hilbert, "Grundlagen der Geometrie" , Teubner, reprint (1968)
[2] N.V. Efimov, "Höhere Geometrie" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian)
  • Hilbert, D. (1899). "Grundlagen der Geometrie". [Reprint (1968) Teubner.]
  • Efimov, N.V. (1960). "Höhere Geometrie", Deutsch. Verlag Wissenschaft. [Translated from Russian.]


Comments

In axiom the point is also called an interior point of the segment (Pasch's axiom should be read with this in mind).

Also, Hilbert originally used different continuity axioms: the Archimedean axiom and a completeness axiom of his own.

References

[a1] H.G. Forder, "Foundations of Euclidean geometry" , Dover, reprint (1958)
[a2] A.D. Aleksandrov, "Foundations of geometry" Siberian Math. J. , July 1987, Volume 28, Issue 4, pp 523-53928 (1987) pp. 523–539 Sibirsk. Mat. Zh. , 28 (1987) pp. 9–28


  • Aleksandrov, A.D. "Foundations of geometry," Siberian Mathematical Journal, July 1987, Vol. 28, Issue 4, pp 523-539. [Trans. Sibirskii Matematicheskii Zhurnal, Vol. 28, No. 4, pp. 9–28, July–August, 1987.]
  • Forder, H.G. "Foundations of Euclidean geometry". [Reprint (1958) Dover.]
  • ↑ Hilbert (1899)
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