Mixing
Proving Hilbert's Axioms as Theorems in $ℝ^n$
$begingroup$
In KG Binmore's "Topological Ideas" he says
The geometric terms which appear in Hilbert's axioms are the words point, line, lie on, between and congruent. To show $mathbb{R}^2$ is a model for Euclidean plane geometry one has to give a precise definition of each of these words in terms of $mathbb{R}^2$ and then prove each of Hilbert's axioms for Euclidean plane geometry as a theorem in $mathbb{R}^2$... Interested readers ill find the book Elementary geometry from an advanced standpoint by E. E. Moise (Addison-Wesley, 1963) an excellent reference.
Except I recently got this book and it does not do this. It is an interesting book, but it simply accepts the primitive notions ("the geometric terms") and Hilbert's axioms. I would like to see the construction of these entities directly from $mathbb{R}^n$, Binmore defines lines, circles, points, and planes. But nowhere is there congruence (presumably for line segments this would be the usual distance between two points) or betweenness defined (presumably a point $b$ would be between $a$ and $c$ if $d(a,c) = d(a,b) + d(b,c))$. I've attempted to do this myself but I'm in over my head a little bit.
So, does anyone have resources that systematically defines each of these geometric objects as sets of $mathbb{R}^n$ and then proves hilbert's axioms as theorems in $mathbb{R}^n$? Especially Euclid's postulate.
euclidean-geometry axioms
edited Jan 19 at 6:36
Andrews
7461318
asked Jan 19 at 1:39
Remus BleysRemus Bleys
132
$endgroup$
add a comment |
$begingroup$
In KG Binmore's "Topological Ideas" he says
The geometric terms which appear in Hilbert's axioms are the words point, line, lie on, between and congruent. To show $mathbb{R}^2$ is a model for Euclidean plane geometry one has to give a precise definition of each of these words in terms of $mathbb{R}^2$ and then prove each of Hilbert's axioms for Euclidean plane geometry as a theorem in $mathbb{R}^2$... Interested readers ill find the book Elementary geometry from an advanced standpoint by E. E. Moise (Addison-Wesley, 1963) an excellent reference.
Except I recently got this book and it does not do this. It is an interesting book, but it simply accepts the primitive notions ("the geometric terms") and Hilbert's axioms. I would like to see the construction of these entities directly from $mathbb{R}^n$, Binmore defines lines, circles, points, and planes. But nowhere is there congruence (presumably for line segments this would be the usual distance between two points) or betweenness defined (presumably a point $b$ would be between $a$ and $c$ if $d(a,c) = d(a,b) + d(b,c))$. I've attempted to do this myself but I'm in over my head a little bit.
So, does anyone have resources that systematically defines each of these geometric objects as sets of $mathbb{R}^n$ and then proves hilbert's axioms as theorems in $mathbb{R}^n$? Especially Euclid's postulate.
euclidean-geometry axioms
edited Jan 19 at 6:36
Andrews
7461318
asked Jan 19 at 1:39
Remus BleysRemus Bleys
132
$endgroup$
$begingroup$
You might want to look at chapter IV of "Foundations of geometry" by Karol Borsuk and Wanda Szmielew (1960). They prove that $mathbb{R}^3$ is a model for Hilbert's axioms for Euclidean 3D geometry.(the axioms are slightly modified compared to original Hilbert's work)
$endgroup$
– Kulisty
Jan 21 at 19:31
add a comment |
$begingroup$
In KG Binmore's "Topological Ideas" he says
The geometric terms which appear in Hilbert's axioms are the words point, line, lie on, between and congruent. To show $mathbb{R}^2$ is a model for Euclidean plane geometry one has to give a precise definition of each of these words in terms of $mathbb{R}^2$ and then prove each of Hilbert's axioms for Euclidean plane geometry as a theorem in $mathbb{R}^2$... Interested readers ill find the book Elementary geometry from an advanced standpoint by E. E. Moise (Addison-Wesley, 1963) an excellent reference.
Except I recently got this book and it does not do this. It is an interesting book, but it simply accepts the primitive notions ("the geometric terms") and Hilbert's axioms. I would like to see the construction of these entities directly from $mathbb{R}^n$, Binmore defines lines, circles, points, and planes. But nowhere is there congruence (presumably for line segments this would be the usual distance between two points) or betweenness defined (presumably a point $b$ would be between $a$ and $c$ if $d(a,c) = d(a,b) + d(b,c))$. I've attempted to do this myself but I'm in over my head a little bit.
So, does anyone have resources that systematically defines each of these geometric objects as sets of $mathbb{R}^n$ and then proves hilbert's axioms as theorems in $mathbb{R}^n$? Especially Euclid's postulate.
euclidean-geometry axioms
edited Jan 19 at 6:36
Andrews
7461318
asked Jan 19 at 1:39
Remus BleysRemus Bleys
132
$endgroup$
In KG Binmore's "Topological Ideas" he says
The geometric terms which appear in Hilbert's axioms are the words point, line, lie on, between and congruent. To show $mathbb{R}^2$ is a model for Euclidean plane geometry one has to give a precise definition of each of these words in terms of $mathbb{R}^2$ and then prove each of Hilbert's axioms for Euclidean plane geometry as a theorem in $mathbb{R}^2$... Interested readers ill find the book Elementary geometry from an advanced standpoint by E. E. Moise (Addison-Wesley, 1963) an excellent reference.
Except I recently got this book and it does not do this. It is an interesting book, but it simply accepts the primitive notions ("the geometric terms") and Hilbert's axioms. I would like to see the construction of these entities directly from $mathbb{R}^n$, Binmore defines lines, circles, points, and planes. But nowhere is there congruence (presumably for line segments this would be the usual distance between two points) or betweenness defined (presumably a point $b$ would be between $a$ and $c$ if $d(a,c) = d(a,b) + d(b,c))$. I've attempted to do this myself but I'm in over my head a little bit.
So, does anyone have resources that systematically defines each of these geometric objects as sets of $mathbb{R}^n$ and then proves hilbert's axioms as theorems in $mathbb{R}^n$? Especially Euclid's postulate.
euclidean-geometry axioms
euclidean-geometry axioms
edited Jan 19 at 6:36
Andrews
7461318
asked Jan 19 at 1:39
Remus BleysRemus Bleys
132
edited Jan 19 at 6:36
Andrews
7461318
asked Jan 19 at 1:39
Remus BleysRemus Bleys
132
edited Jan 19 at 6:36
Andrews
7461318
edited Jan 19 at 6:36
Andrews
7461318
edited Jan 19 at 6:36
Andrews
7461318
7461318
asked Jan 19 at 1:39
Remus BleysRemus Bleys
132
asked Jan 19 at 1:39
Remus BleysRemus Bleys
132
asked Jan 19 at 1:39
Remus BleysRemus Bleys
132
132
$begingroup$
You might want to look at chapter IV of "Foundations of geometry" by Karol Borsuk and Wanda Szmielew (1960). They prove that $mathbb{R}^3$ is a model for Hilbert's axioms for Euclidean 3D geometry.(the axioms are slightly modified compared to original Hilbert's work)
$endgroup$
– Kulisty
Jan 21 at 19:31
add a comment |
$begingroup$
You might want to look at chapter IV of "Foundations of geometry" by Karol Borsuk and Wanda Szmielew (1960). They prove that $mathbb{R}^3$ is a model for Hilbert's axioms for Euclidean 3D geometry.(the axioms are slightly modified compared to original Hilbert's work)
$endgroup$
– Kulisty
Jan 21 at 19:31
$begingroup$
You might want to look at chapter IV of "Foundations of geometry" by Karol Borsuk and Wanda Szmielew (1960). They prove that $mathbb{R}^3$ is a model for Hilbert's axioms for Euclidean 3D geometry.(the axioms are slightly modified compared to original Hilbert's work)
$endgroup$
– Kulisty
Jan 21 at 19:31
$begingroup$
You might want to look at chapter IV of "Foundations of geometry" by Karol Borsuk and Wanda Szmielew (1960). They prove that $mathbb{R}^3$ is a model for Hilbert's axioms for Euclidean 3D geometry.(the axioms are slightly modified compared to original Hilbert's work)
$endgroup$
– Kulisty
Jan 21 at 19:31
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
What you have is basically real linear algebra with usual metric. Define a point as an element of $Bbb R^n$, a line as the set $vec{a} + tvec{b}$, betweeness exactly like you did (using distances - in this case, the norm), and planes and other hyperplanes as you'd do with subspaces on $Bbb R^n$.
You’ll might be interested in this question.
answered Jan 19 at 1:49
Lucas HenriqueLucas Henrique
1,026414
$endgroup$
$begingroup$
So, I get that you would define two lines to be parallel if the intersection is zero, how do you show that given a point and a line on a plane there is one and only one parallel line containing that point? How would you define angles? Dot products?
$endgroup$
– Remus Bleys
Jan 19 at 1:54
$begingroup$
Angles are actually not cited by Hilber, but you can use angles from the dot product definition and Cauchy-Schwarz, while $cos$ and $sin$ being solely algebraic functions (in the sense that they don't depend on geometric reasoning). Dot product is usual coordinate product and sum. Say that a plane is a subspace of dimension 2; for a line on a plane spanned by $vec{u}, vec{v}$ you can do a change of basis to get this plane to $xy$ (in terms of Cartesian coordinates) and then the proof is very simple.
$endgroup$
– Lucas Henrique
Jan 19 at 2:02
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
});
}
});
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3078931%2fproving-hilberts-axioms-as-theorems-in-%25e2%2584%259dn%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
What you have is basically real linear algebra with usual metric. Define a point as an element of $Bbb R^n$, a line as the set $vec{a} + tvec{b}$, betweeness exactly like you did (using distances - in this case, the norm), and planes and other hyperplanes as you'd do with subspaces on $Bbb R^n$.
You’ll might be interested in this question.
answered Jan 19 at 1:49
Lucas HenriqueLucas Henrique
1,026414
$endgroup$
$begingroup$
So, I get that you would define two lines to be parallel if the intersection is zero, how do you show that given a point and a line on a plane there is one and only one parallel line containing that point? How would you define angles? Dot products?
$endgroup$
– Remus Bleys
Jan 19 at 1:54
$begingroup$
Angles are actually not cited by Hilber, but you can use angles from the dot product definition and Cauchy-Schwarz, while $cos$ and $sin$ being solely algebraic functions (in the sense that they don't depend on geometric reasoning). Dot product is usual coordinate product and sum. Say that a plane is a subspace of dimension 2; for a line on a plane spanned by $vec{u}, vec{v}$ you can do a change of basis to get this plane to $xy$ (in terms of Cartesian coordinates) and then the proof is very simple.
$endgroup$
– Lucas Henrique
Jan 19 at 2:02
add a comment |
$begingroup$
What you have is basically real linear algebra with usual metric. Define a point as an element of $Bbb R^n$, a line as the set $vec{a} + tvec{b}$, betweeness exactly like you did (using distances - in this case, the norm), and planes and other hyperplanes as you'd do with subspaces on $Bbb R^n$.
You’ll might be interested in this question.
answered Jan 19 at 1:49
Lucas HenriqueLucas Henrique
1,026414
$endgroup$
$begingroup$
So, I get that you would define two lines to be parallel if the intersection is zero, how do you show that given a point and a line on a plane there is one and only one parallel line containing that point? How would you define angles? Dot products?
$endgroup$
– Remus Bleys
Jan 19 at 1:54
$begingroup$
Angles are actually not cited by Hilber, but you can use angles from the dot product definition and Cauchy-Schwarz, while $cos$ and $sin$ being solely algebraic functions (in the sense that they don't depend on geometric reasoning). Dot product is usual coordinate product and sum. Say that a plane is a subspace of dimension 2; for a line on a plane spanned by $vec{u}, vec{v}$ you can do a change of basis to get this plane to $xy$ (in terms of Cartesian coordinates) and then the proof is very simple.
$endgroup$
– Lucas Henrique
Jan 19 at 2:02
add a comment |
$begingroup$
What you have is basically real linear algebra with usual metric. Define a point as an element of $Bbb R^n$, a line as the set $vec{a} + tvec{b}$, betweeness exactly like you did (using distances - in this case, the norm), and planes and other hyperplanes as you'd do with subspaces on $Bbb R^n$.
You’ll might be interested in this question.
answered Jan 19 at 1:49
Lucas HenriqueLucas Henrique
1,026414
$endgroup$
What you have is basically real linear algebra with usual metric. Define a point as an element of $Bbb R^n$, a line as the set $vec{a} + tvec{b}$, betweeness exactly like you did (using distances - in this case, the norm), and planes and other hyperplanes as you'd do with subspaces on $Bbb R^n$.
You’ll might be interested in this question.
answered Jan 19 at 1:49
Lucas HenriqueLucas Henrique
1,026414
answered Jan 19 at 1:49
Lucas HenriqueLucas Henrique
1,026414
answered Jan 19 at 1:49
Lucas HenriqueLucas Henrique
1,026414
answered Jan 19 at 1:49
Lucas HenriqueLucas Henrique
1,026414
1,026414
$begingroup$
So, I get that you would define two lines to be parallel if the intersection is zero, how do you show that given a point and a line on a plane there is one and only one parallel line containing that point? How would you define angles? Dot products?
$endgroup$
– Remus Bleys
Jan 19 at 1:54
$begingroup$
Angles are actually not cited by Hilber, but you can use angles from the dot product definition and Cauchy-Schwarz, while $cos$ and $sin$ being solely algebraic functions (in the sense that they don't depend on geometric reasoning). Dot product is usual coordinate product and sum. Say that a plane is a subspace of dimension 2; for a line on a plane spanned by $vec{u}, vec{v}$ you can do a change of basis to get this plane to $xy$ (in terms of Cartesian coordinates) and then the proof is very simple.
$endgroup$
– Lucas Henrique
Jan 19 at 2:02
add a comment |
$begingroup$
So, I get that you would define two lines to be parallel if the intersection is zero, how do you show that given a point and a line on a plane there is one and only one parallel line containing that point? How would you define angles? Dot products?
$endgroup$
– Remus Bleys
Jan 19 at 1:54
$begingroup$
Angles are actually not cited by Hilber, but you can use angles from the dot product definition and Cauchy-Schwarz, while $cos$ and $sin$ being solely algebraic functions (in the sense that they don't depend on geometric reasoning). Dot product is usual coordinate product and sum. Say that a plane is a subspace of dimension 2; for a line on a plane spanned by $vec{u}, vec{v}$ you can do a change of basis to get this plane to $xy$ (in terms of Cartesian coordinates) and then the proof is very simple.
$endgroup$
– Lucas Henrique
Jan 19 at 2:02
$begingroup$
So, I get that you would define two lines to be parallel if the intersection is zero, how do you show that given a point and a line on a plane there is one and only one parallel line containing that point? How would you define angles? Dot products?
$endgroup$
– Remus Bleys
Jan 19 at 1:54
$begingroup$
So, I get that you would define two lines to be parallel if the intersection is zero, how do you show that given a point and a line on a plane there is one and only one parallel line containing that point? How would you define angles? Dot products?
$endgroup$
– Remus Bleys
Jan 19 at 1:54
$begingroup$
Angles are actually not cited by Hilber, but you can use angles from the dot product definition and Cauchy-Schwarz, while $cos$ and $sin$ being solely algebraic functions (in the sense that they don't depend on geometric reasoning). Dot product is usual coordinate product and sum. Say that a plane is a subspace of dimension 2; for a line on a plane spanned by $vec{u}, vec{v}$ you can do a change of basis to get this plane to $xy$ (in terms of Cartesian coordinates) and then the proof is very simple.
$endgroup$
– Lucas Henrique
Jan 19 at 2:02
$begingroup$
Angles are actually not cited by Hilber, but you can use angles from the dot product definition and Cauchy-Schwarz, while $cos$ and $sin$ being solely algebraic functions (in the sense that they don't depend on geometric reasoning). Dot product is usual coordinate product and sum. Say that a plane is a subspace of dimension 2; for a line on a plane spanned by $vec{u}, vec{v}$ you can do a change of basis to get this plane to $xy$ (in terms of Cartesian coordinates) and then the proof is very simple.
$endgroup$
– Lucas Henrique
Jan 19 at 2:02
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3078931%2fproving-hilberts-axioms-as-theorems-in-%25e2%2584%259dn%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
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
Required, but never shown
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
Required, but never shown
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
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
You might want to look at chapter IV of "Foundations of geometry" by Karol Borsuk and Wanda Szmielew (1960). They prove that $mathbb{R}^3$ is a model for Hilbert's axioms for Euclidean 3D geometry.(the axioms are slightly modified compared to original Hilbert's work)
$endgroup$
– Kulisty
Jan 21 at 19:31