On Liouville numbers and the Continuum Hypothesis [duplicate]
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This question already has an answer here:
Is every set of measure zero countable?
3 answers
Collecting some theorems from the book Making Transcendence Transparent by its authors, there is some inconsistency, I think... :
i. $L= sum_{n=1}^{infty} 10^{-n!}$ is transcendental.
ii. Numbers of the form $sum_{n=1}^{infty} a_n 10^{-n!}$ in which $a_i in {{0,1}}$ are Liouville
numbers and thus transcendental. So, by Cantor diagonalization argument there are uncountably many Liouville numbers.
iii. The collection of all Liouville
numbers has measure zero. The set of all sequences of zeros and ones (not all zero) are in 1-1 correspondence with $(0,2)$ and this is in 1-1 correspondence with $mathbb{R}$.
How an uncountable subset of $mathbb{R}$ has measure zero?
real-analysis elementary-set-theory transcendental-numbers
edited Dec 17 '18 at 15:21
Andrés E. Caicedo
65.5k8159250
asked Dec 17 '18 at 12:55
72D72D
555116
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marked as duplicate by Dietrich Burde, Andrés E. Caicedo, Asaf Karagila♦ elementary-set-theory
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Dec 17 '18 at 15:26
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
add a comment |
$begingroup$
This question already has an answer here:
Is every set of measure zero countable?
3 answers
Collecting some theorems from the book Making Transcendence Transparent by its authors, there is some inconsistency, I think... :
i. $L= sum_{n=1}^{infty} 10^{-n!}$ is transcendental.
ii. Numbers of the form $sum_{n=1}^{infty} a_n 10^{-n!}$ in which $a_i in {{0,1}}$ are Liouville
numbers and thus transcendental. So, by Cantor diagonalization argument there are uncountably many Liouville numbers.
iii. The collection of all Liouville
numbers has measure zero. The set of all sequences of zeros and ones (not all zero) are in 1-1 correspondence with $(0,2)$ and this is in 1-1 correspondence with $mathbb{R}$.
How an uncountable subset of $mathbb{R}$ has measure zero?
real-analysis elementary-set-theory transcendental-numbers
edited Dec 17 '18 at 15:21
Andrés E. Caicedo
65.5k8159250
asked Dec 17 '18 at 12:55
72D72D
555116
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marked as duplicate by Dietrich Burde, Andrés E. Caicedo, Asaf Karagila♦ elementary-set-theory
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Dec 17 '18 at 15:26
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
3
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Which inconsistency do you see (" there is some inconsistency, I think").
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– Dietrich Burde
Dec 17 '18 at 12:56
4
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There is no inconsistency in an uncountable set of measure $0$.
$endgroup$
– Tobias Kildetoft
Dec 17 '18 at 13:02
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@TobiasKildetoft, I had never seen an uncountable set of measure $0$ that's why I am confused.
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– 72D
Dec 17 '18 at 13:03
2
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Well, you've seen one now.
$endgroup$
– Robert Israel
Dec 17 '18 at 13:31
3
$begingroup$
Incidentally, I disagree with the downvote (and have upvoted) - this is a perfectly reasonable confusion to have at first, and what is MSE for if not dealing with perfectly reasonable confusions?
$endgroup$
– Noah Schweber
Dec 17 '18 at 15:24
add a comment |
$begingroup$
This question already has an answer here:
Is every set of measure zero countable?
3 answers
Collecting some theorems from the book Making Transcendence Transparent by its authors, there is some inconsistency, I think... :
i. $L= sum_{n=1}^{infty} 10^{-n!}$ is transcendental.
ii. Numbers of the form $sum_{n=1}^{infty} a_n 10^{-n!}$ in which $a_i in {{0,1}}$ are Liouville
numbers and thus transcendental. So, by Cantor diagonalization argument there are uncountably many Liouville numbers.
iii. The collection of all Liouville
numbers has measure zero. The set of all sequences of zeros and ones (not all zero) are in 1-1 correspondence with $(0,2)$ and this is in 1-1 correspondence with $mathbb{R}$.
How an uncountable subset of $mathbb{R}$ has measure zero?
real-analysis elementary-set-theory transcendental-numbers
edited Dec 17 '18 at 15:21
Andrés E. Caicedo
65.5k8159250
asked Dec 17 '18 at 12:55
72D72D
555116
$endgroup$
This question already has an answer here:
Is every set of measure zero countable?
3 answers
Collecting some theorems from the book Making Transcendence Transparent by its authors, there is some inconsistency, I think... :
i. $L= sum_{n=1}^{infty} 10^{-n!}$ is transcendental.
ii. Numbers of the form $sum_{n=1}^{infty} a_n 10^{-n!}$ in which $a_i in {{0,1}}$ are Liouville
numbers and thus transcendental. So, by Cantor diagonalization argument there are uncountably many Liouville numbers.
iii. The collection of all Liouville
numbers has measure zero. The set of all sequences of zeros and ones (not all zero) are in 1-1 correspondence with $(0,2)$ and this is in 1-1 correspondence with $mathbb{R}$.
How an uncountable subset of $mathbb{R}$ has measure zero?
This question already has an answer here:
Is every set of measure zero countable?
3 answers
real-analysis elementary-set-theory transcendental-numbers
real-analysis elementary-set-theory transcendental-numbers
edited Dec 17 '18 at 15:21
Andrés E. Caicedo
65.5k8159250
asked Dec 17 '18 at 12:55
72D72D
555116
edited Dec 17 '18 at 15:21
Andrés E. Caicedo
65.5k8159250
asked Dec 17 '18 at 12:55
72D72D
555116
edited Dec 17 '18 at 15:21
Andrés E. Caicedo
65.5k8159250
edited Dec 17 '18 at 15:21
Andrés E. Caicedo
65.5k8159250
edited Dec 17 '18 at 15:21
Andrés E. Caicedo
65.5k8159250
65.5k8159250
asked Dec 17 '18 at 12:55
72D72D
555116
asked Dec 17 '18 at 12:55
72D72D
555116
asked Dec 17 '18 at 12:55
72D72D
555116
555116
marked as duplicate by Dietrich Burde, Andrés E. Caicedo, Asaf Karagila♦ elementary-set-theory
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Dec 17 '18 at 15:26
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by Dietrich Burde, Andrés E. Caicedo, Asaf Karagila♦ elementary-set-theory
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Dec 17 '18 at 15:26
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
3
$begingroup$
Which inconsistency do you see (" there is some inconsistency, I think").
$endgroup$
– Dietrich Burde
Dec 17 '18 at 12:56
4
$begingroup$
There is no inconsistency in an uncountable set of measure $0$.
$endgroup$
– Tobias Kildetoft
Dec 17 '18 at 13:02
$begingroup$
@TobiasKildetoft, I had never seen an uncountable set of measure $0$ that's why I am confused.
$endgroup$
– 72D
Dec 17 '18 at 13:03
2
$begingroup$
Well, you've seen one now.
$endgroup$
– Robert Israel
Dec 17 '18 at 13:31
3
$begingroup$
Incidentally, I disagree with the downvote (and have upvoted) - this is a perfectly reasonable confusion to have at first, and what is MSE for if not dealing with perfectly reasonable confusions?
$endgroup$
– Noah Schweber
Dec 17 '18 at 15:24
add a comment |
3
$begingroup$
Which inconsistency do you see (" there is some inconsistency, I think").
$endgroup$
– Dietrich Burde
Dec 17 '18 at 12:56
4
$begingroup$
There is no inconsistency in an uncountable set of measure $0$.
$endgroup$
– Tobias Kildetoft
Dec 17 '18 at 13:02
$begingroup$
@TobiasKildetoft, I had never seen an uncountable set of measure $0$ that's why I am confused.
$endgroup$
– 72D
Dec 17 '18 at 13:03
2
$begingroup$
Well, you've seen one now.
$endgroup$
– Robert Israel
Dec 17 '18 at 13:31
3
$begingroup$
Incidentally, I disagree with the downvote (and have upvoted) - this is a perfectly reasonable confusion to have at first, and what is MSE for if not dealing with perfectly reasonable confusions?
$endgroup$
– Noah Schweber
Dec 17 '18 at 15:24
3
3
$begingroup$
Which inconsistency do you see (" there is some inconsistency, I think").
$endgroup$
– Dietrich Burde
Dec 17 '18 at 12:56
$begingroup$
Which inconsistency do you see (" there is some inconsistency, I think").
$endgroup$
– Dietrich Burde
Dec 17 '18 at 12:56
4
4
$begingroup$
There is no inconsistency in an uncountable set of measure $0$.
$endgroup$
– Tobias Kildetoft
Dec 17 '18 at 13:02
$begingroup$
There is no inconsistency in an uncountable set of measure $0$.
$endgroup$
– Tobias Kildetoft
Dec 17 '18 at 13:02
$begingroup$
@TobiasKildetoft, I had never seen an uncountable set of measure $0$ that's why I am confused.
$endgroup$
– 72D
Dec 17 '18 at 13:03
$begingroup$
@TobiasKildetoft, I had never seen an uncountable set of measure $0$ that's why I am confused.
$endgroup$
– 72D
Dec 17 '18 at 13:03
2
2
$begingroup$
Well, you've seen one now.
$endgroup$
– Robert Israel
Dec 17 '18 at 13:31
$begingroup$
Well, you've seen one now.
$endgroup$
– Robert Israel
Dec 17 '18 at 13:31
3
3
$begingroup$
Incidentally, I disagree with the downvote (and have upvoted) - this is a perfectly reasonable confusion to have at first, and what is MSE for if not dealing with perfectly reasonable confusions?
$endgroup$
– Noah Schweber
Dec 17 '18 at 15:24
$begingroup$
Incidentally, I disagree with the downvote (and have upvoted) - this is a perfectly reasonable confusion to have at first, and what is MSE for if not dealing with perfectly reasonable confusions?
$endgroup$
– Noah Schweber
Dec 17 '18 at 15:24
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
How about the Cantor subset of $mathbb{R}$: it is compact, uncountable, with no isolated points yet it has measure zero.
answered Dec 17 '18 at 13:13
MindlackMindlack
4,740210
$endgroup$
$begingroup$
You beat me to it. It is probably the easiest to understand example of an uncountable set of measure zero.
$endgroup$
– badjohn
Dec 17 '18 at 13:18
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
How about the Cantor subset of $mathbb{R}$: it is compact, uncountable, with no isolated points yet it has measure zero.
answered Dec 17 '18 at 13:13
MindlackMindlack
4,740210
$endgroup$
$begingroup$
You beat me to it. It is probably the easiest to understand example of an uncountable set of measure zero.
$endgroup$
– badjohn
Dec 17 '18 at 13:18
add a comment |
$begingroup$
How about the Cantor subset of $mathbb{R}$: it is compact, uncountable, with no isolated points yet it has measure zero.
answered Dec 17 '18 at 13:13
MindlackMindlack
4,740210
$endgroup$
$begingroup$
You beat me to it. It is probably the easiest to understand example of an uncountable set of measure zero.
$endgroup$
– badjohn
Dec 17 '18 at 13:18
add a comment |
$begingroup$
How about the Cantor subset of $mathbb{R}$: it is compact, uncountable, with no isolated points yet it has measure zero.
answered Dec 17 '18 at 13:13
MindlackMindlack
4,740210
$endgroup$
How about the Cantor subset of $mathbb{R}$: it is compact, uncountable, with no isolated points yet it has measure zero.
answered Dec 17 '18 at 13:13
MindlackMindlack
4,740210
answered Dec 17 '18 at 13:13
MindlackMindlack
4,740210
answered Dec 17 '18 at 13:13
MindlackMindlack
4,740210
answered Dec 17 '18 at 13:13
MindlackMindlack
4,740210
4,740210
$begingroup$
You beat me to it. It is probably the easiest to understand example of an uncountable set of measure zero.
$endgroup$
– badjohn
Dec 17 '18 at 13:18
add a comment |
$begingroup$
You beat me to it. It is probably the easiest to understand example of an uncountable set of measure zero.
$endgroup$
– badjohn
Dec 17 '18 at 13:18
$begingroup$
You beat me to it. It is probably the easiest to understand example of an uncountable set of measure zero.
$endgroup$
– badjohn
Dec 17 '18 at 13:18
$begingroup$
You beat me to it. It is probably the easiest to understand example of an uncountable set of measure zero.
$endgroup$
– badjohn
Dec 17 '18 at 13:18
add a comment |
3
$begingroup$
Which inconsistency do you see (" there is some inconsistency, I think").
$endgroup$
– Dietrich Burde
Dec 17 '18 at 12:56
4
$begingroup$
There is no inconsistency in an uncountable set of measure $0$.
$endgroup$
– Tobias Kildetoft
Dec 17 '18 at 13:02
$begingroup$
@TobiasKildetoft, I had never seen an uncountable set of measure $0$ that's why I am confused.
$endgroup$
– 72D
Dec 17 '18 at 13:03
2
$begingroup$
Well, you've seen one now.
$endgroup$
– Robert Israel
Dec 17 '18 at 13:31
3
$begingroup$
Incidentally, I disagree with the downvote (and have upvoted) - this is a perfectly reasonable confusion to have at first, and what is MSE for if not dealing with perfectly reasonable confusions?
$endgroup$
– Noah Schweber
Dec 17 '18 at 15:24