Fhyujk

How can I deliver in-universe written lore to players without it being dry exposition?

What happens to output if OPAMP its supply is tweaked?

How can my powered armor quickly replace its ceramic plates?

Why avoid shared user accounts?

Writing a character who is going through a civilizing process without overdoing it?

Intern applicant asking for compensation equivalent to that of permanent employee

Dilemma of explaining to interviewer that he is the reason for declining second interview

Using only 1s, make 29 with the minimum number of digits

Roman Numerals equation 1

Why "Points exist" is not an axiom in Geometry

How to limit sight distance to 1 km

Pronunciation of umlaut vowels in the history of German

Why did other German political parties disband so fast when Hitler was appointed chancellor?

Is a debit card dangerous in my situation?

Why has the mole been redefined for 2019?

Can an insurance company drop you after receiving a bill and refusing to pay?

Difference between `vector<int> v;` and `vector<int> v = vector<int>();`

Can we use the stored gravitational potential energy of a building to produce power?

Unable to list services in AWS EKS

What's a good word to describe a public place that looks like it wouldn't be rough?

Avoiding morning and evening handshakes

Why do members of Congress in committee hearings ask witnesses the same question multiple times?

Why did the villain in the first Men in Black movie care about Earth's Cockroaches?

Why do neural networks need so many training examples to perform?

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

$begingroup$

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 1 hour ago

user10869858user10869858

344

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Why not "lines exist" then points would be derived objects? Not an answer, just a thought.
  $endgroup$
  – Paul
  21 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 1 hour ago

user10869858user10869858

344

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Why not "lines exist" then points would be derived objects? Not an answer, just a thought.
  $endgroup$
  – Paul
  21 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 1 hour ago

user10869858user10869858

344

$endgroup$

share|cite|improve this question

asked 1 hour ago

user10869858user10869858

344

share|cite|improve this question

asked 1 hour ago

user10869858user10869858

344

asked 1 hour ago

user10869858user10869858

344

asked 1 hour ago

user10869858user10869858

344

2 Answers
2

active

oldest

votes

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 33 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
  30 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
  19 mins ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I still would find it interesting to know why it's left out in so many places then.
  $endgroup$
  – user10869858
  14 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
  13 mins 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
  2 mins ago
  
  
  

  
  
  
  

add a comment | 

0

$begingroup$

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 40 mins ago

Yves DaoustYves Daoust

129k675227

$endgroup$

add a comment | 

Your Answer

StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name

Email

Required, but never shown

StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3131186%2fwhy-points-exist-is-not-an-axiom-in-geometry%23new-answer', 'question_page');
}
);

Post as a guest

Name

Email

Required, but never shown

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 33 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
  30 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
  19 mins ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I still would find it interesting to know why it's left out in so many places then.
  $endgroup$
  – user10869858
  14 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
  13 mins 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
  2 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 33 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
  30 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
  19 mins ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I still would find it interesting to know why it's left out in so many places then.
  $endgroup$
  – user10869858
  14 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
  13 mins 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
  2 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 33 mins ago

jmerryjmerry

11.1k1225

$endgroup$

share|cite|improve this answer

answered 33 mins ago

jmerryjmerry

11.1k1225

answered 33 mins ago

jmerryjmerry

11.1k1225

answered 33 mins ago

jmerryjmerry

11.1k1225

0

$begingroup$

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 40 mins ago

Yves DaoustYves Daoust

129k675227

$endgroup$

add a comment | 

0

$begingroup$

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 40 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 40 mins ago

Yves DaoustYves Daoust

129k675227

$endgroup$

share|cite|improve this answer

answered 40 mins ago

Yves DaoustYves Daoust

129k675227

answered 40 mins ago

Yves DaoustYves Daoust

129k675227

answered 40 mins ago

Yves DaoustYves Daoust

129k675227