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Why “Points exist” is not an axiom in Geometry

A model of geometry with the negation of Pasch’s axiom?Why is the Generalization Axiom considered a Pure Axiom?Tarski-like axiomatization of spherical or elliptic geometryHilbert's Foundations of Geometry Axiom II, 1 : Why is this relevant?Why is “lies between” a primitive notion in Hilbert's Foundations of Geometry?Alternatives to Fano's Axiom in Projective SpaceAxiom of Choice — Why is it an axiom and not a theorem?Replacing axiom SAS by AAS in neutral geometry.Redunduncy of Pasch's Axiom of Hilbert's Foundations of GeometryModel of ordered plane with the negation of Pasch's axiom

3

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Not sure why "Points exist" is not an axiom in Geometry, given that the other axioms are likewise primitive and seemingly as obvious.

geometry axioms

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asked 56 mins ago

user10869858user10869858

344

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  $begingroup$
  Why not "lines exist" then points would be derived objects? Not an answer, just a thought.
  $endgroup$
  – Paul
  17 mins ago
  

  
  
  
  

add a comment | 

3

$begingroup$

Not sure why "Points exist" is not an axiom in Geometry, given that the other axioms are likewise primitive and seemingly as obvious.

geometry axioms

share|cite|improve this question

asked 56 mins ago

user10869858user10869858

344

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Why not "lines exist" then points would be derived objects? Not an answer, just a thought.
  $endgroup$
  – Paul
  17 mins ago
  

  
  
  
  

add a comment | 

$begingroup$

Not sure why "Points exist" is not an axiom in Geometry, given that the other axioms are likewise primitive and seemingly as obvious.

geometry axioms

share|cite|improve this question

asked 56 mins ago

user10869858user10869858

344

$endgroup$

share|cite|improve this question

asked 56 mins ago

user10869858user10869858

344

share|cite|improve this question

asked 56 mins ago

user10869858user10869858

344

asked 56 mins ago

user10869858user10869858

344

asked 56 mins ago

user10869858user10869858

344

2 Answers
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4

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In the presentation I have most convenient to me (Hilbert's axioms), the axioms for plane geometry start with a trio of "axioms of incidence". One of those axioms is "There exist three non-collinear points". That axiom certainly includes the existence of points.

share|cite|improve this answer

answered 29 mins ago

jmerryjmerry

11.1k1225

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  +1. For a specific reference, see Wikipedia's "Hilbert's Axioms" entry. In that listing, the third Incidence axiom reads "There exist at least two points on a line. There exist at least three points that do not lie on the same line." Further, the eighth Incidence axiom reads "There exist at least four points not lying in a plane."
  $endgroup$
  – Blue
  27 mins ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The version I'm working from (Hartshorne's textbook) organizes it a little differently. The other two incidence axioms in that version are "For any two points $A,B$, there exists a unique line $l$ containing them", and "Every line contains at least two points". (The axioms from #4 on in the Wikipedia version are axioms for three-dimensional geometry)
  $endgroup$
  – jmerry
  16 mins ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I still would find it interesting to know why it's left out in so many places then.
  $endgroup$
  – user10869858
  10 mins ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @jmerry: Probably every text organizes things a little differently. :) I just thought OP would benefit from access to some version of a "complete" reference, in case there are additional questions about what a comprehensive axiomatic system may-or-may-not cover.
  $endgroup$
  – Blue
  10 mins ago
  

  
  
  
  

add a comment | 

1

$begingroup$

Not versed in that topic, but I believe that this is implicit. Otherwise, there would be no geometry at all.

The axioms that have a clause "there exists" refer to an entity with specific properties, which could have had the option of not existing. ("There exists a line by two points" is an axiom, while "there exists a line by three points" is not.)

share|cite|improve this answer

answered 37 mins ago

Yves DaoustYves Daoust

129k675227

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add a comment | 

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4

$begingroup$

In the presentation I have most convenient to me (Hilbert's axioms), the axioms for plane geometry start with a trio of "axioms of incidence". One of those axioms is "There exist three non-collinear points". That axiom certainly includes the existence of points.

share|cite|improve this answer

answered 29 mins ago

jmerryjmerry

11.1k1225

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  +1. For a specific reference, see Wikipedia's "Hilbert's Axioms" entry. In that listing, the third Incidence axiom reads "There exist at least two points on a line. There exist at least three points that do not lie on the same line." Further, the eighth Incidence axiom reads "There exist at least four points not lying in a plane."
  $endgroup$
  – Blue
  27 mins ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The version I'm working from (Hartshorne's textbook) organizes it a little differently. The other two incidence axioms in that version are "For any two points $A,B$, there exists a unique line $l$ containing them", and "Every line contains at least two points". (The axioms from #4 on in the Wikipedia version are axioms for three-dimensional geometry)
  $endgroup$
  – jmerry
  16 mins ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I still would find it interesting to know why it's left out in so many places then.
  $endgroup$
  – user10869858
  10 mins ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @jmerry: Probably every text organizes things a little differently. :) I just thought OP would benefit from access to some version of a "complete" reference, in case there are additional questions about what a comprehensive axiomatic system may-or-may-not cover.
  $endgroup$
  – Blue
  10 mins ago
  

  
  
  
  

add a comment | 

4

$begingroup$

In the presentation I have most convenient to me (Hilbert's axioms), the axioms for plane geometry start with a trio of "axioms of incidence". One of those axioms is "There exist three non-collinear points". That axiom certainly includes the existence of points.

share|cite|improve this answer

answered 29 mins ago

jmerryjmerry

11.1k1225

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  +1. For a specific reference, see Wikipedia's "Hilbert's Axioms" entry. In that listing, the third Incidence axiom reads "There exist at least two points on a line. There exist at least three points that do not lie on the same line." Further, the eighth Incidence axiom reads "There exist at least four points not lying in a plane."
  $endgroup$
  – Blue
  27 mins ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The version I'm working from (Hartshorne's textbook) organizes it a little differently. The other two incidence axioms in that version are "For any two points $A,B$, there exists a unique line $l$ containing them", and "Every line contains at least two points". (The axioms from #4 on in the Wikipedia version are axioms for three-dimensional geometry)
  $endgroup$
  – jmerry
  16 mins ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I still would find it interesting to know why it's left out in so many places then.
  $endgroup$
  – user10869858
  10 mins ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @jmerry: Probably every text organizes things a little differently. :) I just thought OP would benefit from access to some version of a "complete" reference, in case there are additional questions about what a comprehensive axiomatic system may-or-may-not cover.
  $endgroup$
  – Blue
  10 mins ago
  

  
  
  
  

add a comment | 

$begingroup$

In the presentation I have most convenient to me (Hilbert's axioms), the axioms for plane geometry start with a trio of "axioms of incidence". One of those axioms is "There exist three non-collinear points". That axiom certainly includes the existence of points.

share|cite|improve this answer

answered 29 mins ago

jmerryjmerry

11.1k1225

$endgroup$

share|cite|improve this answer

answered 29 mins ago

jmerryjmerry

11.1k1225

answered 29 mins ago

jmerryjmerry

11.1k1225

answered 29 mins ago

jmerryjmerry

11.1k1225

1

$begingroup$

Not versed in that topic, but I believe that this is implicit. Otherwise, there would be no geometry at all.

The axioms that have a clause "there exists" refer to an entity with specific properties, which could have had the option of not existing. ("There exists a line by two points" is an axiom, while "there exists a line by three points" is not.)

share|cite|improve this answer

answered 37 mins ago

Yves DaoustYves Daoust

129k675227

$endgroup$

add a comment | 

1

$begingroup$

Not versed in that topic, but I believe that this is implicit. Otherwise, there would be no geometry at all.

The axioms that have a clause "there exists" refer to an entity with specific properties, which could have had the option of not existing. ("There exists a line by two points" is an axiom, while "there exists a line by three points" is not.)

share|cite|improve this answer

answered 37 mins ago

Yves DaoustYves Daoust

129k675227

$endgroup$

add a comment | 

$begingroup$

Not versed in that topic, but I believe that this is implicit. Otherwise, there would be no geometry at all.

The axioms that have a clause "there exists" refer to an entity with specific properties, which could have had the option of not existing. ("There exists a line by two points" is an axiom, while "there exists a line by three points" is not.)

share|cite|improve this answer

answered 37 mins ago

Yves DaoustYves Daoust

129k675227

$endgroup$

share|cite|improve this answer

answered 37 mins ago

Yves DaoustYves Daoust

129k675227

answered 37 mins ago

Yves DaoustYves Daoust

129k675227

answered 37 mins ago

Yves DaoustYves Daoust

129k675227