Mixing
Explanation of the construction of the reals using Dedekind cuts.
$begingroup$
I have recently started studying calculus 1.
I am studying by myself (pre-academic, starting my degree next month) and have touched very lightly on the subject of constructing $mathbb{R}$.
I have seen how to construct it via Dedekind cuts, and Cauchy sequences.
I am really trying to get my head about the formationg of $mathbb{R}$ using Dedekind cuts, using Cauchy sequences is rather intuitive for me.
I'd like to receive an abstract point of view on how does it work (sort of an "explain to me like im 5" - any redditors here? lol)
I understand the basics of it. I just don't get how we actually get the real numbers from it.
calculus analysis
edited Jan 31 at 17:05
Bernard
124k741117
asked Jan 31 at 17:03
trizztrizz
235
$endgroup$
add a comment |
$begingroup$
I have recently started studying calculus 1.
I am studying by myself (pre-academic, starting my degree next month) and have touched very lightly on the subject of constructing $mathbb{R}$.
I have seen how to construct it via Dedekind cuts, and Cauchy sequences.
I am really trying to get my head about the formationg of $mathbb{R}$ using Dedekind cuts, using Cauchy sequences is rather intuitive for me.
I'd like to receive an abstract point of view on how does it work (sort of an "explain to me like im 5" - any redditors here? lol)
I understand the basics of it. I just don't get how we actually get the real numbers from it.
calculus analysis
edited Jan 31 at 17:05
Bernard
124k741117
asked Jan 31 at 17:03
trizztrizz
235
$endgroup$
1
$begingroup$
Dedekind's own explanation is very good! There is a Dover reprint. A copy can also be downloaded from Project Gutenberg.
$endgroup$
– Calum Gilhooley
Jan 31 at 17:47
add a comment |
$begingroup$
I have recently started studying calculus 1.
I am studying by myself (pre-academic, starting my degree next month) and have touched very lightly on the subject of constructing $mathbb{R}$.
I have seen how to construct it via Dedekind cuts, and Cauchy sequences.
I am really trying to get my head about the formationg of $mathbb{R}$ using Dedekind cuts, using Cauchy sequences is rather intuitive for me.
I'd like to receive an abstract point of view on how does it work (sort of an "explain to me like im 5" - any redditors here? lol)
I understand the basics of it. I just don't get how we actually get the real numbers from it.
calculus analysis
edited Jan 31 at 17:05
Bernard
124k741117
asked Jan 31 at 17:03
trizztrizz
235
$endgroup$
I have recently started studying calculus 1.
I am studying by myself (pre-academic, starting my degree next month) and have touched very lightly on the subject of constructing $mathbb{R}$.
I have seen how to construct it via Dedekind cuts, and Cauchy sequences.
I am really trying to get my head about the formationg of $mathbb{R}$ using Dedekind cuts, using Cauchy sequences is rather intuitive for me.
I'd like to receive an abstract point of view on how does it work (sort of an "explain to me like im 5" - any redditors here? lol)
I understand the basics of it. I just don't get how we actually get the real numbers from it.
calculus analysis
calculus analysis
edited Jan 31 at 17:05
Bernard
124k741117
asked Jan 31 at 17:03
trizztrizz
235
edited Jan 31 at 17:05
Bernard
124k741117
asked Jan 31 at 17:03
trizztrizz
235
edited Jan 31 at 17:05
Bernard
124k741117
edited Jan 31 at 17:05
Bernard
124k741117
edited Jan 31 at 17:05
Bernard
124k741117
124k741117
asked Jan 31 at 17:03
trizztrizz
235
asked Jan 31 at 17:03
trizztrizz
235
asked Jan 31 at 17:03
trizztrizz
235
235
1
$begingroup$
Dedekind's own explanation is very good! There is a Dover reprint. A copy can also be downloaded from Project Gutenberg.
$endgroup$
– Calum Gilhooley
Jan 31 at 17:47
add a comment |
1
$begingroup$
Dedekind's own explanation is very good! There is a Dover reprint. A copy can also be downloaded from Project Gutenberg.
$endgroup$
– Calum Gilhooley
Jan 31 at 17:47
1
1
$begingroup$
Dedekind's own explanation is very good! There is a Dover reprint. A copy can also be downloaded from Project Gutenberg.
$endgroup$
– Calum Gilhooley
Jan 31 at 17:47
$begingroup$
Dedekind's own explanation is very good! There is a Dover reprint. A copy can also be downloaded from Project Gutenberg.
$endgroup$
– Calum Gilhooley
Jan 31 at 17:47
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
I'll just describe the geometric heuristics here. Reading the analytical rewriting would be easier after:
Dedekind noticed that the concept of a rational number can be thought of as representing a separation, partition, or cut (schnitt, to use the German he used) of the rational line into two sets, say L and R, so that every point of L is to the left of every point of R.
After this realisation, and the fact that some cuts of $mathbf Q$ are not rational (Dedekind used the cut defined by all $x$ separated as to whether $x^2$ is less than or greater than two, which has become the classic non-example ever since), the completion of the cuts of the rational numbers so that every such cut is represented by a number is how Dedekind understood the reals.
That's the basic idea, and you'll find the detailed and precise development of this idea (and how to define the usual arithmetic operations on these cuts consistently with those operations we are used to) in your favourite analysis text, namely Rudin. Or better still, read Dedekind himself. If I recall, I think it's called (in translation): What are numbers and what are they for? or something similar.
Good luck.
answered Jan 31 at 17:18
AllawonderAllawonder
2,251616
$endgroup$
$begingroup$
"Dedekind noticed that the concept of a rational number can be thought of"... surely you meant "real number".
$endgroup$
– David C. Ullrich
Jan 31 at 21:21
$begingroup$
@DavidC.Ullrich No, rational indeed, as Dedekind himself explained in the article I cited. I see someone has produced a link (the one to PG) to the one I read myself. See to confirm what led Dedekind into this construction in the first place. Then from the context the reason for first thinking of the rationals as cuts should become clear.
$endgroup$
– Allawonder
Feb 3 at 6:37
add a comment |
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$begingroup$
I'll just describe the geometric heuristics here. Reading the analytical rewriting would be easier after:
Dedekind noticed that the concept of a rational number can be thought of as representing a separation, partition, or cut (schnitt, to use the German he used) of the rational line into two sets, say L and R, so that every point of L is to the left of every point of R.
After this realisation, and the fact that some cuts of $mathbf Q$ are not rational (Dedekind used the cut defined by all $x$ separated as to whether $x^2$ is less than or greater than two, which has become the classic non-example ever since), the completion of the cuts of the rational numbers so that every such cut is represented by a number is how Dedekind understood the reals.
That's the basic idea, and you'll find the detailed and precise development of this idea (and how to define the usual arithmetic operations on these cuts consistently with those operations we are used to) in your favourite analysis text, namely Rudin. Or better still, read Dedekind himself. If I recall, I think it's called (in translation): What are numbers and what are they for? or something similar.
Good luck.
answered Jan 31 at 17:18
AllawonderAllawonder
2,251616
$endgroup$
$begingroup$
"Dedekind noticed that the concept of a rational number can be thought of"... surely you meant "real number".
$endgroup$
– David C. Ullrich
Jan 31 at 21:21
$begingroup$
@DavidC.Ullrich No, rational indeed, as Dedekind himself explained in the article I cited. I see someone has produced a link (the one to PG) to the one I read myself. See to confirm what led Dedekind into this construction in the first place. Then from the context the reason for first thinking of the rationals as cuts should become clear.
$endgroup$
– Allawonder
Feb 3 at 6:37
add a comment |
$begingroup$
I'll just describe the geometric heuristics here. Reading the analytical rewriting would be easier after:
Dedekind noticed that the concept of a rational number can be thought of as representing a separation, partition, or cut (schnitt, to use the German he used) of the rational line into two sets, say L and R, so that every point of L is to the left of every point of R.
After this realisation, and the fact that some cuts of $mathbf Q$ are not rational (Dedekind used the cut defined by all $x$ separated as to whether $x^2$ is less than or greater than two, which has become the classic non-example ever since), the completion of the cuts of the rational numbers so that every such cut is represented by a number is how Dedekind understood the reals.
That's the basic idea, and you'll find the detailed and precise development of this idea (and how to define the usual arithmetic operations on these cuts consistently with those operations we are used to) in your favourite analysis text, namely Rudin. Or better still, read Dedekind himself. If I recall, I think it's called (in translation): What are numbers and what are they for? or something similar.
Good luck.
answered Jan 31 at 17:18
AllawonderAllawonder
2,251616
$endgroup$
$begingroup$
"Dedekind noticed that the concept of a rational number can be thought of"... surely you meant "real number".
$endgroup$
– David C. Ullrich
Jan 31 at 21:21
$begingroup$
@DavidC.Ullrich No, rational indeed, as Dedekind himself explained in the article I cited. I see someone has produced a link (the one to PG) to the one I read myself. See to confirm what led Dedekind into this construction in the first place. Then from the context the reason for first thinking of the rationals as cuts should become clear.
$endgroup$
– Allawonder
Feb 3 at 6:37
add a comment |
$begingroup$
I'll just describe the geometric heuristics here. Reading the analytical rewriting would be easier after:
Dedekind noticed that the concept of a rational number can be thought of as representing a separation, partition, or cut (schnitt, to use the German he used) of the rational line into two sets, say L and R, so that every point of L is to the left of every point of R.
After this realisation, and the fact that some cuts of $mathbf Q$ are not rational (Dedekind used the cut defined by all $x$ separated as to whether $x^2$ is less than or greater than two, which has become the classic non-example ever since), the completion of the cuts of the rational numbers so that every such cut is represented by a number is how Dedekind understood the reals.
That's the basic idea, and you'll find the detailed and precise development of this idea (and how to define the usual arithmetic operations on these cuts consistently with those operations we are used to) in your favourite analysis text, namely Rudin. Or better still, read Dedekind himself. If I recall, I think it's called (in translation): What are numbers and what are they for? or something similar.
Good luck.
answered Jan 31 at 17:18
AllawonderAllawonder
2,251616
$endgroup$
I'll just describe the geometric heuristics here. Reading the analytical rewriting would be easier after:
Dedekind noticed that the concept of a rational number can be thought of as representing a separation, partition, or cut (schnitt, to use the German he used) of the rational line into two sets, say L and R, so that every point of L is to the left of every point of R.
After this realisation, and the fact that some cuts of $mathbf Q$ are not rational (Dedekind used the cut defined by all $x$ separated as to whether $x^2$ is less than or greater than two, which has become the classic non-example ever since), the completion of the cuts of the rational numbers so that every such cut is represented by a number is how Dedekind understood the reals.
That's the basic idea, and you'll find the detailed and precise development of this idea (and how to define the usual arithmetic operations on these cuts consistently with those operations we are used to) in your favourite analysis text, namely Rudin. Or better still, read Dedekind himself. If I recall, I think it's called (in translation): What are numbers and what are they for? or something similar.
Good luck.
answered Jan 31 at 17:18
AllawonderAllawonder
2,251616
answered Jan 31 at 17:18
AllawonderAllawonder
2,251616
answered Jan 31 at 17:18
AllawonderAllawonder
2,251616
answered Jan 31 at 17:18
AllawonderAllawonder
2,251616
2,251616
$begingroup$
"Dedekind noticed that the concept of a rational number can be thought of"... surely you meant "real number".
$endgroup$
– David C. Ullrich
Jan 31 at 21:21
$begingroup$
@DavidC.Ullrich No, rational indeed, as Dedekind himself explained in the article I cited. I see someone has produced a link (the one to PG) to the one I read myself. See to confirm what led Dedekind into this construction in the first place. Then from the context the reason for first thinking of the rationals as cuts should become clear.
$endgroup$
– Allawonder
Feb 3 at 6:37
add a comment |
$begingroup$
"Dedekind noticed that the concept of a rational number can be thought of"... surely you meant "real number".
$endgroup$
– David C. Ullrich
Jan 31 at 21:21
$begingroup$
@DavidC.Ullrich No, rational indeed, as Dedekind himself explained in the article I cited. I see someone has produced a link (the one to PG) to the one I read myself. See to confirm what led Dedekind into this construction in the first place. Then from the context the reason for first thinking of the rationals as cuts should become clear.
$endgroup$
– Allawonder
Feb 3 at 6:37
$begingroup$
"Dedekind noticed that the concept of a rational number can be thought of"... surely you meant "real number".
$endgroup$
– David C. Ullrich
Jan 31 at 21:21
$begingroup$
"Dedekind noticed that the concept of a rational number can be thought of"... surely you meant "real number".
$endgroup$
– David C. Ullrich
Jan 31 at 21:21
$begingroup$
@DavidC.Ullrich No, rational indeed, as Dedekind himself explained in the article I cited. I see someone has produced a link (the one to PG) to the one I read myself. See to confirm what led Dedekind into this construction in the first place. Then from the context the reason for first thinking of the rationals as cuts should become clear.
$endgroup$
– Allawonder
Feb 3 at 6:37
$begingroup$
@DavidC.Ullrich No, rational indeed, as Dedekind himself explained in the article I cited. I see someone has produced a link (the one to PG) to the one I read myself. See to confirm what led Dedekind into this construction in the first place. Then from the context the reason for first thinking of the rationals as cuts should become clear.
$endgroup$
– Allawonder
Feb 3 at 6:37
add a comment |
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$begingroup$
Dedekind's own explanation is very good! There is a Dover reprint. A copy can also be downloaded from Project Gutenberg.
$endgroup$
– Calum Gilhooley
Jan 31 at 17:47