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Questions tagged [axiomatic-geometry]

Questions about axiomatic systems for geometry. Use this tag if you're looking for a proof starting directly from some set of axioms (e.g., Hilbert's axioms for Euclidean geometry), or if you have a question about the axioms themselves.

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Model of ordered plane which is neither isomorphic to $\mathbb{R}^2$ nor to Klein model

Let $B_{\mathbb{R}}\subset\mathbb{R}\times\mathbb{R}\times\mathbb{R}$ be standard (strict) betweenness relation on $\mathbb{R}$ i.e. $$B_{\mathbb{R}}(abc):\iff\left(a<b<c \vee c<b<a\right)$...
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Why is $\mathbb{R}^2$ endowed with the taxicab metric isomorphic to the infinity distance model of the cartesian real plane? (Hartshorne exercise 8.9) [duplicate]

My question deals with the following exercise from Hartshorne's Euclid: Geometry and Beyond: Following our general principles, we say that two models $M,M'$ of our geometry are isomorphic if there ...
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1answer
53 views

Is there a formalization of the link between geometry and analytical geometry?

Geometry and algebra/calculus can be formalized by axioms. Is there a global theory that combines both and establishes correspondences such as the equation of a straight line is $ax+by+c=0$, the ...
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2answers
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Given axioms, how do we know it defines a geometry?

It is known that besides using coordinates and algebra, there are axiomisation of geometry such as Tarski, Hilbert and Euclid. However looking at the axioms of Tarski for example: Betweeness $B(\...
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23 views

A line divides a plane into two half-planes

I am trying to learn axioms of geometry, and I can not seem to find any proof to the following theorem that doesn't use circular reasoning: If π is a plane and l is a line on that plane, then all ...
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32 views

Is this axiomatization of affine plane categorical?

First I'll give some definitions. Hilbert's plane axioms of incidence: We consider a set $P$ (plane) and a family $\mathcal{L}$ (a family of lines) with axioms: For any two distinct points $a,b$ ...
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34 views

Proof for the (second) Pasch's theorem / Hilbert's geometry axiom II, 4

In Hilbert's Foundations of Geometry, the 4th axiom of order which stays that Any four points $A, B, C, D$ of a straight line can always be so arranged that $B$ shall lie between $A$ and $C$ and ...
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2answers
49 views

Strength of languages - midpoints+betweenness vs congrunece+betweenness

Congrunece relation is a 4-ary relation between points: $\delta(x,y;z,t)$ iff $|xy|=|zt|$. Betweennes relation is a 3-ary relation between points: $\beta(x,y,z)$ iff $y \in xz$. Midpoint relation is ...
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214 views

Euclidean incidence spaces

Structure $\left<X,B\right>$ is an $n$-dimensional Euclidean incidence space iff $B$ is ternary betweenness relation associated with $n$-dimensional Euclidean geometry. That is, Euclidean ...
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1answer
112 views

Is there criticism in literature of Euclid's fifth common notion (“The whole is greater than the part”)?

In Book I of Euclid's Elements, the fifth common notion says "The whole is greater than the part". For Euclid, magnitudes are objects that can be compared, added, and subtracted, provided they are of ...
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1answer
54 views

What means ''direction'' in hyperbolic geometry?

We can define the concept of ''direction'' as the equivalence class of parallel lines, but this is a good definition only if the Euclide parallels axiom is assumed, so that there is only one ...
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1answer
63 views

How to study Euclidean geometry from axioms?

I want to know if there's a good book or any other type of guide to study Euclidean geometry by only the 5 axioms in plane geometry and prove every other theorems from them?
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1answer
54 views

Model of ordered plane with the negation of Pasch's axiom

I am interested in finding the model of particular set of geometry axioms in which Pasch's axiom fails. First I'll give the definitions. By ordered line I mean the set $L$ (line) with one three-...
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29 views

Existence of a tangent line to a circle in neutral geometry based on Hilbert's axioms. Proof with Dedekind's axiom.

I'm trying to prove the following result in neutral geometry from Hilbert's axioms: Given a circle centered at $o$ and the point $p$ lying outside the circle there exists a line passing through $p$ ...
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1answer
39 views

Where can I find a worked example of Birkhoff's Third Postulate?

I've been reviewing unpackings of Birkhoff's postulates and I may be out of my depth, at least as far as terminology is concerned. I think that I understand the general concept of being able to ...
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1answer
101 views

Neutral geometry: If one triangle has angle-sum $180^\circ$, then all triangles have angle-sum $180^\circ$

One of the more interesting things we can say in neutral geometry (that is, without assuming the parallel postulate, but assuming e.g. the rest of Hilbert's axioms) is the following: Suppose that a ...
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1answer
143 views

How Hilbert's axioms of incidence ensure the existence of a space?

My book asks two questions. Any of the axioms of incidence ensures the existence of space? I know that For every two points A and B there exists a line a that contains them both. For every two ...
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1answer
84 views

What's the need for Hilbert's 7th axiom of incidence?

If two planes α, β have a point A in common, then they have at least a second point B in common. I perfectly understand the axiom, but i don't see why it's necessary, and it's kind of ...
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1answer
120 views

How can affine plane extended of projective plane?

In the book of Euclidean and Non-Euclidean Geometry by Greenberg, it is given that DEFINITION: Lines l and m are parallel if they are distinct lines and no point is incident with both of them. ...
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2answers
237 views

A confusion about the second connection axiom of Euclidean Geometry

In the book of Foundations of Geometry by Hilbert, at page 2, it is stated that I, 1. Two distinct points A and B always completely determine a straight line a. We write AB = a or BA = a. I, ...
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1answer
99 views

Independency of Hilbert system's axioms [closed]

How to prove that the 20 axioms of geometry, in Hilbert's axiomatic system, are independent of each other? In other words, based on a logic-theory statement, which are the worlds in which 19 of the ...
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2answers
68 views

Does the Archimedean axiom guarantee a monotone and additive metric?

I read here: http://www.evlm.stuba.sk/databasemenu/Geometry/Geometric%20spaces/Axioms/Theory/axioms5.xml that The Archimedean axiom guarantees the existence of a metric of line segments m(U) as ...
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1answer
101 views

Hilbert axioms (groups I and II) and first theorem

I'm having some trouble proving the first theorem from Hilbert's Foundations of Geometry, which says «two straight lines of a plane have either one point or no point in common; two planes have no ...
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1answer
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It is possible to prove that there are infinitely many points in space in Hilber'ts axiomatization of geometry? [closed]

Dumb question, i know, there's no explicit axiom about it, but it's somehow possible? Thanks in advance.
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1answer
279 views

Playfair's Axiom for parallel planes instead of lines

In three-dimensional space, can Playfair's Axiom: Given a line $a$ and a point $P$ not in $a$, there is at most one line in $P$ parallel to $a$. be “replaced by“ the following axiom? Given a ...
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0answers
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Is there a finite axiomatization of Tarski's geometry axioms?

Tarski's geometry axioms include an axiom schema, the axiom schema of continuity. Let $\phi(x)$ be a first order formula not containing $y$, $a$, or $b$ as free variables. Let $\psi(y)$ be a first ...
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0answers
471 views

Show that any line through the origin of a circle has two point on the circle and that a circle has infinite points.

Given two distinct points O,A we define the circle with center O and radius OA to be the set $\Gamma$ of all points B such that $ OA \cong OB$ . (a) Show that any line through a meets the circle in ...
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1answer
110 views

taxicab geometry

Consider the real Cartesian plane $\mathbb{R}$, with lines and betweenness but define a different notion of congruence of line segments using the distance function given by the sum of the absolute ...
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1answer
100 views

half of two equal line segments are equal.

Show that "halves of equals are equal" in the following sense: if $AB \cong CD$, and if E is a midpoint of AB in the sense that $A * E * B $ and $AE \cong EB$, and if F is a midpoint of CD, then $AE \...
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1answer
60 views

Transitivity of containment of angles.

Suppose the ray AD is in the interior of the $\angle BAC$, and the ray AE is in the interior of the $ \angle DAC$. Show that AE is also in the interior of $ \angle BAC$. Using Hilbert's axioms this is ...
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How does Hilbert's axiomatization relate to set theory?

I'm studying Hilbert's axiomatization of Euclidean geometry, and I'm trying to combine my current understanding into my knowledge on mathematical logic (not very much). At the beginning of this ...
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44 views

How to define orientation of ordered plane?

By orientation I mean roughly speaking whether we rotate clockwise or anti-clockwise. Formally I want to define relation $\sim$ between triangles (contained in the same plane) such that $\triangle abc\...
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2answers
166 views

Hilbert's foundations of geometry theorem 17

I am reading Hilbert's foundations of geometry, translated by Townsend. The theorem 17 and 18 don't have proofs, probably because they are too obvious, I don't know. And I haven't been able to see why ...
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Model of Hilbert's plane ordered geometry in which the elliptic parallel property holds

Elliptic parallel property: Given a line $L$ and a point $a\notin L$, there exist no lines parallel to $L$ passing through $a$. Euclidean parallel property: Given a line $L$ and a point $a\notin L$, ...
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Can these axioms serve as an alternative to Hilbert's axioms?

Axioms: (1) Hilbert's axioms of incidence (2) Hilbert's axioms of order (3) Definition of length for a smooth curve (4) Given a plane and a line lying in the plane, we can find at ...
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How can we derive ruler postulate from Hilbert's axioms?

Ruler postulate: For every pair of points $P$ and $Q$ there exists a real number PQ, called the distance from $P$ to $Q$. For each line $l$ there is a one-to-one correspondence from $l$ to $R$ such ...
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1answer
84 views

Saccheri quadrilaterals in Hartshorne's “Euclid and Beyond”

In proposition 34.1, Hartshorne proves: In a Hilbert plane, suppose that two equal perpendiculars $AC$, $BD$ stand at the ends of an interval $AB$, and we join $CD$. Then the angles at $C$ and $D$ ...
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Can we really intersect circles?

Euclid's Elements were a great work, but in modern standards it is not totally rigorous. One of its biggest flaws is right on Proposition 1, where he doesn't prove that the two circles intersect, in ...
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1answer
93 views

Axioms of one-dimension geometry in Hilbert's style

Do somewhere full axiomatic system axioms of one-dimensional geometry presented in Hilbert's "Foundations of Geometry" style? (With incidence, order, congruence and continuity axioms, as described in ...
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2answers
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Do we have to define natural numbers in geometry?

I have been thinking about the axiomatization of geometry, and I don't know one thing. Imagine you are defining triangles: Definition [Triangle]: A triangle is a polygon with 3 sides. In this ...
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1answer
83 views

Axiomatic geometry - How do you define the measurement of lengths, areas, angles, etc.?

In an axiomatic approach to geometry (i.e., excluding explicit construction from $\mathbb{R}^2$), what is the best way to define numerical concepts, like length, area or angle measuring? I've ...
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0answers
151 views

Categorical axiomatization of projective spaces

In their book “Foundations of Geometry” (1960), Karol Borsuk and Wanda Szmielew give a categorical axiomatization of the projective space (three dimensional), using axioms analogous to those used by ...
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1answer
279 views

Equivalence relation between vectors in Euclidean geometry

I'm working in Hilbert's axioms of Euclidean plane geometry. I have problems with proving one thing concerning vectors. My definition of the vector is as follows: Vector $\overrightarrow{ab}$ is an ...
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1answer
452 views

Formal definition of trigonometric functions

I want to define trigonometric function (say sine) formally with the definition that the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. But there is ...
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4answers
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Why is “lies between” a primitive notion in Hilbert's Foundations of Geometry?

I read this question: Hilbert's Foundations of Geometry Axiom II, 1 : Why is this relevant? and its answer by Eric Wofsey. In this answer, it is stated that "lies between" a primitive notion in ...
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1answer
173 views

Hilbert's Foundations of Geometry Axiom II, 1 : Why is this relevant?

Please refer the following book for this question: David Hilbert, The Foundations of Geometry, English Translation by E. J. Townsend, Reprint Edition, The Open Court Publishing Company, 1950. In this ...
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1answer
607 views

Angles in Hilbert's axioms for geometry

In Hilbert's axioms for geometry, the following elements are presented as undefined (meaning "to be defined in a specific model"): point, line, incidence, betweenness, congruence. In the euclidean ...
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3answers
533 views

Axioms of Geometry?

I have taken the generic low level undergraduate classes, such as Calculus 1-3, Differential Equations, and Linear algebra. Since I never learned Geometry past a basic high school level, I thought it ...
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1answer
75 views

Cardinality of Sets of Points in Neutral Geometry

I recently proved the following result: (1) Let $M$ be a model of Neutral (Absolute) Geometry with set of points $\mathbb{P}$. Then $|\mathbb{P}|= \mathfrak{c}$. However, the proof relies upon the ...
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1answer
205 views

On a Simple Theorem from Hilbert's *The Foundations of Geometry*

I want my proof writing skills to get better. I am trying to do this through proving theorems from Hilbert's axioms for Euclidean Geometry. I found Hilbert's The Foundations of Geometry here, a ...