$omega_2$ is not the countable union of countable sets [closed]
-1
I'm not sure I quite understand the form of the proof in this post '$omega_2$ is a not countable union of countable sets without AC' and similar ones. Is the idea to firstly show that there is an injection from the union into $omega_1$, and then to show there is a surjection into $omega_2$ to give a contradiction?
set-theory ordinals
asked Nov 29 '18 at 15:46
MMRMMR
267
closed as off-topic by Andrés E. Caicedo, Mostafa Ayaz, Vidyanshu Mishra, choco_addicted, ancientmathematician Nov 30 '18 at 12:19
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Andrés E. Caicedo, Mostafa Ayaz, Vidyanshu Mishra, choco_addicted, ancientmathematician
If this question can be reworded to fit the rules in the help center, please edit the question.
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-1
I'm not sure I quite understand the form of the proof in this post '$omega_2$ is a not countable union of countable sets without AC' and similar ones. Is the idea to firstly show that there is an injection from the union into $omega_1$, and then to show there is a surjection into $omega_2$ to give a contradiction?
set-theory ordinals
asked Nov 29 '18 at 15:46
MMRMMR
267
closed as off-topic by Andrés E. Caicedo, Mostafa Ayaz, Vidyanshu Mishra, choco_addicted, ancientmathematician Nov 30 '18 at 12:19
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Andrés E. Caicedo, Mostafa Ayaz, Vidyanshu Mishra, choco_addicted, ancientmathematician
If this question can be reworded to fit the rules in the help center, please edit the question.
It is better to first understand these arguments in ZFC, then try to understand them in ZF.
– Asaf Karagila♦
Nov 29 '18 at 15:53
It already makes sense to me in ZFC, since you have that the union is countable and therefore less than $omega_1$
– MMR
Nov 29 '18 at 16:05
That's a different argument. I meant this argument.
– Asaf Karagila♦
Nov 29 '18 at 16:08
1
I would suggest that you actually write down the details here, so that people can refer to specific lines rather than talking of this or that and causing misunderstandings.
– Andrés E. Caicedo
Nov 29 '18 at 17:04
1
The idea is to show that the existence of such a decomposition allows you to construct a surjection $omegatimesomega_1to omega_2$ which is a contradiction (since you can show without choice that this would imply the existence of a surjection $omega_1to omega_2$.)
– spaceisdarkgreen
Nov 29 '18 at 18:15
|
show 1 more comment
-1
-1
-1
I'm not sure I quite understand the form of the proof in this post '$omega_2$ is a not countable union of countable sets without AC' and similar ones. Is the idea to firstly show that there is an injection from the union into $omega_1$, and then to show there is a surjection into $omega_2$ to give a contradiction?
set-theory ordinals
asked Nov 29 '18 at 15:46
MMRMMR
267
I'm not sure I quite understand the form of the proof in this post '$omega_2$ is a not countable union of countable sets without AC' and similar ones. Is the idea to firstly show that there is an injection from the union into $omega_1$, and then to show there is a surjection into $omega_2$ to give a contradiction?
set-theory ordinals
set-theory ordinals
asked Nov 29 '18 at 15:46
MMRMMR
267
asked Nov 29 '18 at 15:46
MMRMMR
267
asked Nov 29 '18 at 15:46
MMRMMR
267
asked Nov 29 '18 at 15:46
MMRMMR
267
asked Nov 29 '18 at 15:46
MMRMMR
267
267
closed as off-topic by Andrés E. Caicedo, Mostafa Ayaz, Vidyanshu Mishra, choco_addicted, ancientmathematician Nov 30 '18 at 12:19
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Andrés E. Caicedo, Mostafa Ayaz, Vidyanshu Mishra, choco_addicted, ancientmathematician
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Andrés E. Caicedo, Mostafa Ayaz, Vidyanshu Mishra, choco_addicted, ancientmathematician Nov 30 '18 at 12:19
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Andrés E. Caicedo, Mostafa Ayaz, Vidyanshu Mishra, choco_addicted, ancientmathematician
If this question can be reworded to fit the rules in the help center, please edit the question.
It is better to first understand these arguments in ZFC, then try to understand them in ZF.
– Asaf Karagila♦
Nov 29 '18 at 15:53
It already makes sense to me in ZFC, since you have that the union is countable and therefore less than $omega_1$
– MMR
Nov 29 '18 at 16:05
That's a different argument. I meant this argument.
– Asaf Karagila♦
Nov 29 '18 at 16:08
1
I would suggest that you actually write down the details here, so that people can refer to specific lines rather than talking of this or that and causing misunderstandings.
– Andrés E. Caicedo
Nov 29 '18 at 17:04
1
The idea is to show that the existence of such a decomposition allows you to construct a surjection $omegatimesomega_1to omega_2$ which is a contradiction (since you can show without choice that this would imply the existence of a surjection $omega_1to omega_2$.)
– spaceisdarkgreen
Nov 29 '18 at 18:15
|
show 1 more comment
It is better to first understand these arguments in ZFC, then try to understand them in ZF.
– Asaf Karagila♦
Nov 29 '18 at 15:53
It already makes sense to me in ZFC, since you have that the union is countable and therefore less than $omega_1$
– MMR
Nov 29 '18 at 16:05
That's a different argument. I meant this argument.
– Asaf Karagila♦
Nov 29 '18 at 16:08
1
I would suggest that you actually write down the details here, so that people can refer to specific lines rather than talking of this or that and causing misunderstandings.
– Andrés E. Caicedo
Nov 29 '18 at 17:04
1
The idea is to show that the existence of such a decomposition allows you to construct a surjection $omegatimesomega_1to omega_2$ which is a contradiction (since you can show without choice that this would imply the existence of a surjection $omega_1to omega_2$.)
– spaceisdarkgreen
Nov 29 '18 at 18:15
It is better to first understand these arguments in ZFC, then try to understand them in ZF.
– Asaf Karagila♦
Nov 29 '18 at 15:53
It is better to first understand these arguments in ZFC, then try to understand them in ZF.
– Asaf Karagila♦
Nov 29 '18 at 15:53
It already makes sense to me in ZFC, since you have that the union is countable and therefore less than $omega_1$
– MMR
Nov 29 '18 at 16:05
It already makes sense to me in ZFC, since you have that the union is countable and therefore less than $omega_1$
– MMR
Nov 29 '18 at 16:05
That's a different argument. I meant this argument.
– Asaf Karagila♦
Nov 29 '18 at 16:08
That's a different argument. I meant this argument.
– Asaf Karagila♦
Nov 29 '18 at 16:08
1
1
I would suggest that you actually write down the details here, so that people can refer to specific lines rather than talking of this or that and causing misunderstandings.
– Andrés E. Caicedo
Nov 29 '18 at 17:04
I would suggest that you actually write down the details here, so that people can refer to specific lines rather than talking of this or that and causing misunderstandings.
– Andrés E. Caicedo
Nov 29 '18 at 17:04
1
1
The idea is to show that the existence of such a decomposition allows you to construct a surjection $omegatimesomega_1to omega_2$ which is a contradiction (since you can show without choice that this would imply the existence of a surjection $omega_1to omega_2$.)
– spaceisdarkgreen
Nov 29 '18 at 18:15
The idea is to show that the existence of such a decomposition allows you to construct a surjection $omegatimesomega_1to omega_2$ which is a contradiction (since you can show without choice that this would imply the existence of a surjection $omega_1to omega_2$.)
– spaceisdarkgreen
Nov 29 '18 at 18:15
|
show 1 more comment
1 Answer
1
active
oldest
votes
1
There is no surjection from $omega_1$ onto $omega_2$. So if a set can be mapped injectively into $omega_1$, then it cannot be mapped surjectively onto $omega_2$.
answered Nov 29 '18 at 15:52
Asaf Karagila♦Asaf Karagila
302k32427757
What is the purpose of the surjection, if we already know that the union is equal to $omega_2$?
– MMR
Nov 29 '18 at 16:05
Your comment makes no sense to me. Can you clarify it?
– Asaf Karagila♦
Nov 29 '18 at 16:08
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
1
There is no surjection from $omega_1$ onto $omega_2$. So if a set can be mapped injectively into $omega_1$, then it cannot be mapped surjectively onto $omega_2$.
answered Nov 29 '18 at 15:52
Asaf Karagila♦Asaf Karagila
302k32427757
What is the purpose of the surjection, if we already know that the union is equal to $omega_2$?
– MMR
Nov 29 '18 at 16:05
Your comment makes no sense to me. Can you clarify it?
– Asaf Karagila♦
Nov 29 '18 at 16:08
add a comment |
1
There is no surjection from $omega_1$ onto $omega_2$. So if a set can be mapped injectively into $omega_1$, then it cannot be mapped surjectively onto $omega_2$.
answered Nov 29 '18 at 15:52
Asaf Karagila♦Asaf Karagila
302k32427757
What is the purpose of the surjection, if we already know that the union is equal to $omega_2$?
– MMR
Nov 29 '18 at 16:05
Your comment makes no sense to me. Can you clarify it?
– Asaf Karagila♦
Nov 29 '18 at 16:08
add a comment |
1
1
1
There is no surjection from $omega_1$ onto $omega_2$. So if a set can be mapped injectively into $omega_1$, then it cannot be mapped surjectively onto $omega_2$.
answered Nov 29 '18 at 15:52
Asaf Karagila♦Asaf Karagila
302k32427757
There is no surjection from $omega_1$ onto $omega_2$. So if a set can be mapped injectively into $omega_1$, then it cannot be mapped surjectively onto $omega_2$.
answered Nov 29 '18 at 15:52
Asaf Karagila♦Asaf Karagila
302k32427757
answered Nov 29 '18 at 15:52
Asaf Karagila♦Asaf Karagila
302k32427757
answered Nov 29 '18 at 15:52
Asaf Karagila♦Asaf Karagila
302k32427757
answered Nov 29 '18 at 15:52
Asaf Karagila♦Asaf Karagila
302k32427757
302k32427757
What is the purpose of the surjection, if we already know that the union is equal to $omega_2$?
– MMR
Nov 29 '18 at 16:05
Your comment makes no sense to me. Can you clarify it?
– Asaf Karagila♦
Nov 29 '18 at 16:08
add a comment |
What is the purpose of the surjection, if we already know that the union is equal to $omega_2$?
– MMR
Nov 29 '18 at 16:05
Your comment makes no sense to me. Can you clarify it?
– Asaf Karagila♦
Nov 29 '18 at 16:08
What is the purpose of the surjection, if we already know that the union is equal to $omega_2$?
– MMR
Nov 29 '18 at 16:05
What is the purpose of the surjection, if we already know that the union is equal to $omega_2$?
– MMR
Nov 29 '18 at 16:05
Your comment makes no sense to me. Can you clarify it?
– Asaf Karagila♦
Nov 29 '18 at 16:08
Your comment makes no sense to me. Can you clarify it?
– Asaf Karagila♦
Nov 29 '18 at 16:08
add a comment |
It is better to first understand these arguments in ZFC, then try to understand them in ZF.
– Asaf Karagila♦
Nov 29 '18 at 15:53
It already makes sense to me in ZFC, since you have that the union is countable and therefore less than $omega_1$
– MMR
Nov 29 '18 at 16:05
That's a different argument. I meant this argument.
– Asaf Karagila♦
Nov 29 '18 at 16:08
1
I would suggest that you actually write down the details here, so that people can refer to specific lines rather than talking of this or that and causing misunderstandings.
– Andrés E. Caicedo
Nov 29 '18 at 17:04
1
The idea is to show that the existence of such a decomposition allows you to construct a surjection $omegatimesomega_1to omega_2$ which is a contradiction (since you can show without choice that this would imply the existence of a surjection $omega_1to omega_2$.)
– spaceisdarkgreen
Nov 29 '18 at 18:15