Why is “points exist” not an axiom in geometry?A model of geometry with the negation of Pasch’s...
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Why is “points exist” 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
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I am 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
edited 23 mins ago
Peter Mortensen
559310
asked 2 hours ago
user10869858user10869858
445
$endgroup$
add a comment |
$begingroup$
I am 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
edited 23 mins ago
Peter Mortensen
559310
asked 2 hours ago
user10869858user10869858
445
$endgroup$
$begingroup$
Why not "lines exist" then points would be derived objects? Not an answer, just a thought.
$endgroup$
– Paul
1 hour ago
$begingroup$
If the question is about Euclid's Elements specifically, there's alot missing in those axioms. For example, in the very first proof, Euclid assumes that two circles draw at a certain distance from one another must intersect.
$endgroup$
– Jack M
56 mins ago
add a comment |
$begingroup$
I am 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
edited 23 mins ago
Peter Mortensen
559310
asked 2 hours ago
user10869858user10869858
445
$endgroup$
I am 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
geometry axioms
edited 23 mins ago
Peter Mortensen
559310
asked 2 hours ago
user10869858user10869858
445
edited 23 mins ago
Peter Mortensen
559310
asked 2 hours ago
user10869858user10869858
445
edited 23 mins ago
Peter Mortensen
559310
edited 23 mins ago
Peter Mortensen
559310
edited 23 mins ago
Peter Mortensen
559310
559310
asked 2 hours ago
user10869858user10869858
445
asked 2 hours ago
user10869858user10869858
445
asked 2 hours ago
user10869858user10869858
445
445
$begingroup$
Why not "lines exist" then points would be derived objects? Not an answer, just a thought.
$endgroup$
– Paul
1 hour ago
$begingroup$
If the question is about Euclid's Elements specifically, there's alot missing in those axioms. For example, in the very first proof, Euclid assumes that two circles draw at a certain distance from one another must intersect.
$endgroup$
– Jack M
56 mins ago
add a comment |
$begingroup$
Why not "lines exist" then points would be derived objects? Not an answer, just a thought.
$endgroup$
– Paul
1 hour ago
$begingroup$
If the question is about Euclid's Elements specifically, there's alot missing in those axioms. For example, in the very first proof, Euclid assumes that two circles draw at a certain distance from one another must intersect.
$endgroup$
– Jack M
56 mins ago
$begingroup$
Why not "lines exist" then points would be derived objects? Not an answer, just a thought.
$endgroup$
– Paul
1 hour ago
$begingroup$
Why not "lines exist" then points would be derived objects? Not an answer, just a thought.
$endgroup$
– Paul
1 hour ago
$begingroup$
If the question is about Euclid's Elements specifically, there's alot missing in those axioms. For example, in the very first proof, Euclid assumes that two circles draw at a certain distance from one another must intersect.
$endgroup$
– Jack M
56 mins ago
$begingroup$
If the question is about Euclid's Elements specifically, there's alot missing in those axioms. For example, in the very first proof, Euclid assumes that two circles draw at a certain distance from one another must intersect.
$endgroup$
– Jack M
56 mins ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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.
answered 1 hour 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
1 hour 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
1 hour ago
$begingroup$
I still would find it interesting to know why it's left out in so many places then.
$endgroup$
– user10869858
1 hour 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
1 hour ago
1
$begingroup$
@user10869858: What do you mean by "so many places"? Note that Euclid's Elements itself is notoriously lacking in covering all the necessary logical bases; on the other hand, Hilbert's axioms may overwhelm a new learner. It's unsurprising (though perhaps a little unfortunate) that introductory treatments (say, high school textbooks) might make certain compromises, confident that advanced courses and/or readily-available resources (like Math.SE! ;) will fill in any gaps.
$endgroup$
– Blue
1 hour ago
|
show 1 more comment
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1 Answer
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1 Answer
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$begingroup$
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.
answered 1 hour 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
1 hour 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
1 hour ago
$begingroup$
I still would find it interesting to know why it's left out in so many places then.
$endgroup$
– user10869858
1 hour 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
1 hour ago
1
$begingroup$
@user10869858: What do you mean by "so many places"? Note that Euclid's Elements itself is notoriously lacking in covering all the necessary logical bases; on the other hand, Hilbert's axioms may overwhelm a new learner. It's unsurprising (though perhaps a little unfortunate) that introductory treatments (say, high school textbooks) might make certain compromises, confident that advanced courses and/or readily-available resources (like Math.SE! ;) will fill in any gaps.
$endgroup$
– Blue
1 hour ago
|
show 1 more 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.
answered 1 hour 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
1 hour 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
1 hour ago
$begingroup$
I still would find it interesting to know why it's left out in so many places then.
$endgroup$
– user10869858
1 hour 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
1 hour ago
1
$begingroup$
@user10869858: What do you mean by "so many places"? Note that Euclid's Elements itself is notoriously lacking in covering all the necessary logical bases; on the other hand, Hilbert's axioms may overwhelm a new learner. It's unsurprising (though perhaps a little unfortunate) that introductory treatments (say, high school textbooks) might make certain compromises, confident that advanced courses and/or readily-available resources (like Math.SE! ;) will fill in any gaps.
$endgroup$
– Blue
1 hour ago
|
show 1 more 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.
answered 1 hour ago
jmerryjmerry
11.1k1225
$endgroup$
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.
answered 1 hour ago
jmerryjmerry
11.1k1225
answered 1 hour ago
jmerryjmerry
11.1k1225
answered 1 hour ago
jmerryjmerry
11.1k1225
answered 1 hour ago
jmerryjmerry
11.1k1225
11.1k1225
$begingroup$
+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
1 hour 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
1 hour ago
$begingroup$
I still would find it interesting to know why it's left out in so many places then.
$endgroup$
– user10869858
1 hour 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
1 hour ago
1
$begingroup$
@user10869858: What do you mean by "so many places"? Note that Euclid's Elements itself is notoriously lacking in covering all the necessary logical bases; on the other hand, Hilbert's axioms may overwhelm a new learner. It's unsurprising (though perhaps a little unfortunate) that introductory treatments (say, high school textbooks) might make certain compromises, confident that advanced courses and/or readily-available resources (like Math.SE! ;) will fill in any gaps.
$endgroup$
– Blue
1 hour ago
|
show 1 more comment
$begingroup$
+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
1 hour 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
1 hour ago
$begingroup$
I still would find it interesting to know why it's left out in so many places then.
$endgroup$
– user10869858
1 hour 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
1 hour ago
1
$begingroup$
@user10869858: What do you mean by "so many places"? Note that Euclid's Elements itself is notoriously lacking in covering all the necessary logical bases; on the other hand, Hilbert's axioms may overwhelm a new learner. It's unsurprising (though perhaps a little unfortunate) that introductory treatments (say, high school textbooks) might make certain compromises, confident that advanced courses and/or readily-available resources (like Math.SE! ;) will fill in any gaps.
$endgroup$
– Blue
1 hour ago
$begingroup$
+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
1 hour ago
$begingroup$
+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
1 hour 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
1 hour 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
1 hour ago
$begingroup$
I still would find it interesting to know why it's left out in so many places then.
$endgroup$
– user10869858
1 hour ago
$begingroup$
I still would find it interesting to know why it's left out in so many places then.
$endgroup$
– user10869858
1 hour 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
1 hour 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
1 hour ago
1
1
$begingroup$
@user10869858: What do you mean by "so many places"? Note that Euclid's Elements itself is notoriously lacking in covering all the necessary logical bases; on the other hand, Hilbert's axioms may overwhelm a new learner. It's unsurprising (though perhaps a little unfortunate) that introductory treatments (say, high school textbooks) might make certain compromises, confident that advanced courses and/or readily-available resources (like Math.SE! ;) will fill in any gaps.
$endgroup$
– Blue
1 hour ago
$begingroup$
@user10869858: What do you mean by "so many places"? Note that Euclid's Elements itself is notoriously lacking in covering all the necessary logical bases; on the other hand, Hilbert's axioms may overwhelm a new learner. It's unsurprising (though perhaps a little unfortunate) that introductory treatments (say, high school textbooks) might make certain compromises, confident that advanced courses and/or readily-available resources (like Math.SE! ;) will fill in any gaps.
$endgroup$
– Blue
1 hour ago
|
show 1 more comment
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$begingroup$
Why not "lines exist" then points would be derived objects? Not an answer, just a thought.
$endgroup$
– Paul
1 hour ago
$begingroup$
If the question is about Euclid's Elements specifically, there's alot missing in those axioms. For example, in the very first proof, Euclid assumes that two circles draw at a certain distance from one another must intersect.
$endgroup$
– Jack M
56 mins ago