Proof that an ordinal number does not contain itself [duplicate]
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This question already has an answer here:
Prove that no ordinal is an element of itself
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Hrbacek and Jech gives the following definition for ordinal numbers:
However, the following proof seems to rely on the fact that an ordinal number does not contain itself (argument circled in red).
It isn't clear to me why the red circle is true. Why is it clear that an ordinal does not contain itself?
elementary-set-theory ordinals
edited Nov 23 at 5:03
Andrés E. Caicedo
64.6k8158246
asked Nov 23 at 4:07
Barycentric_Bash
31528
marked as duplicate by Eric Wofsey, Andrés E. Caicedo, user10354138, Rebellos, José Carlos Santos Nov 23 at 11:26
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 |
up vote
0
down vote
favorite
This question already has an answer here:
Prove that no ordinal is an element of itself
1 answer
Hrbacek and Jech gives the following definition for ordinal numbers:
However, the following proof seems to rely on the fact that an ordinal number does not contain itself (argument circled in red).
It isn't clear to me why the red circle is true. Why is it clear that an ordinal does not contain itself?
elementary-set-theory ordinals
edited Nov 23 at 5:03
Andrés E. Caicedo
64.6k8158246
asked Nov 23 at 4:07
Barycentric_Bash
31528
marked as duplicate by Eric Wofsey, Andrés E. Caicedo, user10354138, Rebellos, José Carlos Santos Nov 23 at 11:26
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.
The circled fact is trivially true regardless of choice of alpha. There is no need to invoke the fact that no ordinal contains itself.
– Ben W
Nov 23 at 4:22
Is the question about the fact that the containment is proper? Are you actually using this?
– Andrés E. Caicedo
Nov 23 at 5:04
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
This question already has an answer here:
Prove that no ordinal is an element of itself
1 answer
Hrbacek and Jech gives the following definition for ordinal numbers:
However, the following proof seems to rely on the fact that an ordinal number does not contain itself (argument circled in red).
It isn't clear to me why the red circle is true. Why is it clear that an ordinal does not contain itself?
elementary-set-theory ordinals
edited Nov 23 at 5:03
Andrés E. Caicedo
64.6k8158246
asked Nov 23 at 4:07
Barycentric_Bash
31528
This question already has an answer here:
Prove that no ordinal is an element of itself
1 answer
Hrbacek and Jech gives the following definition for ordinal numbers:
However, the following proof seems to rely on the fact that an ordinal number does not contain itself (argument circled in red).
It isn't clear to me why the red circle is true. Why is it clear that an ordinal does not contain itself?
This question already has an answer here:
Prove that no ordinal is an element of itself
1 answer
elementary-set-theory ordinals
elementary-set-theory ordinals
edited Nov 23 at 5:03
Andrés E. Caicedo
64.6k8158246
asked Nov 23 at 4:07
Barycentric_Bash
31528
edited Nov 23 at 5:03
Andrés E. Caicedo
64.6k8158246
asked Nov 23 at 4:07
Barycentric_Bash
31528
edited Nov 23 at 5:03
Andrés E. Caicedo
64.6k8158246
edited Nov 23 at 5:03
Andrés E. Caicedo
64.6k8158246
edited Nov 23 at 5:03
Andrés E. Caicedo
64.6k8158246
64.6k8158246
asked Nov 23 at 4:07
Barycentric_Bash
31528
asked Nov 23 at 4:07
Barycentric_Bash
31528
asked Nov 23 at 4:07
Barycentric_Bash
31528
31528
marked as duplicate by Eric Wofsey, Andrés E. Caicedo, user10354138, Rebellos, José Carlos Santos Nov 23 at 11:26
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 Eric Wofsey, Andrés E. Caicedo, user10354138, Rebellos, José Carlos Santos Nov 23 at 11:26
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.
The circled fact is trivially true regardless of choice of alpha. There is no need to invoke the fact that no ordinal contains itself.
– Ben W
Nov 23 at 4:22
Is the question about the fact that the containment is proper? Are you actually using this?
– Andrés E. Caicedo
Nov 23 at 5:04
add a comment |
The circled fact is trivially true regardless of choice of alpha. There is no need to invoke the fact that no ordinal contains itself.
– Ben W
Nov 23 at 4:22
Is the question about the fact that the containment is proper? Are you actually using this?
– Andrés E. Caicedo
Nov 23 at 5:04
The circled fact is trivially true regardless of choice of alpha. There is no need to invoke the fact that no ordinal contains itself.
– Ben W
Nov 23 at 4:22
The circled fact is trivially true regardless of choice of alpha. There is no need to invoke the fact that no ordinal contains itself.
– Ben W
Nov 23 at 4:22
Is the question about the fact that the containment is proper? Are you actually using this?
– Andrés E. Caicedo
Nov 23 at 5:04
Is the question about the fact that the containment is proper? Are you actually using this?
– Andrés E. Caicedo
Nov 23 at 5:04
add a comment |
1 Answer
1
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votes
up vote
0
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No set (including an ordinal) is permitted to include itself due to the Axiom of regularity.
answered Nov 23 at 4:46
Q the Platypus
2,754933
1
Yes. But this is not the point. The result is true without assuming foundation. The point is that ordinals are well-founded regardless of whether the axiom is assumed. This is useful information to know, and simply appealing to foundation hides its relevance.
– Andrés E. Caicedo
Nov 23 at 5:02
That's true but the definition that is given for ordinal should be state that should be should be strict well ordered rather then just well ordered. As in an anti-foundational FAS there is a distinction between well founded and strictly well founded.
– Q the Platypus
Nov 23 at 5:22
There is no distinction in the text: well-ordered is in the strict sense. What is FAS?
– Andrés E. Caicedo
Nov 23 at 5:28
FAS = formal axiomatic system
– Q the Platypus
Nov 23 at 5:47
How does the text define ”transitive set”? In particular, are Quine atoms ($a={a}$) transitive in that definition?
– celtschk
Nov 23 at 7:01
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
No set (including an ordinal) is permitted to include itself due to the Axiom of regularity.
answered Nov 23 at 4:46
Q the Platypus
2,754933
1
Yes. But this is not the point. The result is true without assuming foundation. The point is that ordinals are well-founded regardless of whether the axiom is assumed. This is useful information to know, and simply appealing to foundation hides its relevance.
– Andrés E. Caicedo
Nov 23 at 5:02
That's true but the definition that is given for ordinal should be state that should be should be strict well ordered rather then just well ordered. As in an anti-foundational FAS there is a distinction between well founded and strictly well founded.
– Q the Platypus
Nov 23 at 5:22
There is no distinction in the text: well-ordered is in the strict sense. What is FAS?
– Andrés E. Caicedo
Nov 23 at 5:28
FAS = formal axiomatic system
– Q the Platypus
Nov 23 at 5:47
How does the text define ”transitive set”? In particular, are Quine atoms ($a={a}$) transitive in that definition?
– celtschk
Nov 23 at 7:01
add a comment |
up vote
0
down vote
No set (including an ordinal) is permitted to include itself due to the Axiom of regularity.
answered Nov 23 at 4:46
Q the Platypus
2,754933
1
Yes. But this is not the point. The result is true without assuming foundation. The point is that ordinals are well-founded regardless of whether the axiom is assumed. This is useful information to know, and simply appealing to foundation hides its relevance.
– Andrés E. Caicedo
Nov 23 at 5:02
That's true but the definition that is given for ordinal should be state that should be should be strict well ordered rather then just well ordered. As in an anti-foundational FAS there is a distinction between well founded and strictly well founded.
– Q the Platypus
Nov 23 at 5:22
There is no distinction in the text: well-ordered is in the strict sense. What is FAS?
– Andrés E. Caicedo
Nov 23 at 5:28
FAS = formal axiomatic system
– Q the Platypus
Nov 23 at 5:47
How does the text define ”transitive set”? In particular, are Quine atoms ($a={a}$) transitive in that definition?
– celtschk
Nov 23 at 7:01
add a comment |
up vote
0
down vote
up vote
0
down vote
No set (including an ordinal) is permitted to include itself due to the Axiom of regularity.
answered Nov 23 at 4:46
Q the Platypus
2,754933
No set (including an ordinal) is permitted to include itself due to the Axiom of regularity.
answered Nov 23 at 4:46
Q the Platypus
2,754933
answered Nov 23 at 4:46
Q the Platypus
2,754933
answered Nov 23 at 4:46
Q the Platypus
2,754933
answered Nov 23 at 4:46
Q the Platypus
2,754933
2,754933
1
Yes. But this is not the point. The result is true without assuming foundation. The point is that ordinals are well-founded regardless of whether the axiom is assumed. This is useful information to know, and simply appealing to foundation hides its relevance.
– Andrés E. Caicedo
Nov 23 at 5:02
That's true but the definition that is given for ordinal should be state that should be should be strict well ordered rather then just well ordered. As in an anti-foundational FAS there is a distinction between well founded and strictly well founded.
– Q the Platypus
Nov 23 at 5:22
There is no distinction in the text: well-ordered is in the strict sense. What is FAS?
– Andrés E. Caicedo
Nov 23 at 5:28
FAS = formal axiomatic system
– Q the Platypus
Nov 23 at 5:47
How does the text define ”transitive set”? In particular, are Quine atoms ($a={a}$) transitive in that definition?
– celtschk
Nov 23 at 7:01
add a comment |
1
Yes. But this is not the point. The result is true without assuming foundation. The point is that ordinals are well-founded regardless of whether the axiom is assumed. This is useful information to know, and simply appealing to foundation hides its relevance.
– Andrés E. Caicedo
Nov 23 at 5:02
That's true but the definition that is given for ordinal should be state that should be should be strict well ordered rather then just well ordered. As in an anti-foundational FAS there is a distinction between well founded and strictly well founded.
– Q the Platypus
Nov 23 at 5:22
There is no distinction in the text: well-ordered is in the strict sense. What is FAS?
– Andrés E. Caicedo
Nov 23 at 5:28
FAS = formal axiomatic system
– Q the Platypus
Nov 23 at 5:47
How does the text define ”transitive set”? In particular, are Quine atoms ($a={a}$) transitive in that definition?
– celtschk
Nov 23 at 7:01
1
1
Yes. But this is not the point. The result is true without assuming foundation. The point is that ordinals are well-founded regardless of whether the axiom is assumed. This is useful information to know, and simply appealing to foundation hides its relevance.
– Andrés E. Caicedo
Nov 23 at 5:02
Yes. But this is not the point. The result is true without assuming foundation. The point is that ordinals are well-founded regardless of whether the axiom is assumed. This is useful information to know, and simply appealing to foundation hides its relevance.
– Andrés E. Caicedo
Nov 23 at 5:02
That's true but the definition that is given for ordinal should be state that should be should be strict well ordered rather then just well ordered. As in an anti-foundational FAS there is a distinction between well founded and strictly well founded.
– Q the Platypus
Nov 23 at 5:22
That's true but the definition that is given for ordinal should be state that should be should be strict well ordered rather then just well ordered. As in an anti-foundational FAS there is a distinction between well founded and strictly well founded.
– Q the Platypus
Nov 23 at 5:22
There is no distinction in the text: well-ordered is in the strict sense. What is FAS?
– Andrés E. Caicedo
Nov 23 at 5:28
There is no distinction in the text: well-ordered is in the strict sense. What is FAS?
– Andrés E. Caicedo
Nov 23 at 5:28
FAS = formal axiomatic system
– Q the Platypus
Nov 23 at 5:47
FAS = formal axiomatic system
– Q the Platypus
Nov 23 at 5:47
How does the text define ”transitive set”? In particular, are Quine atoms ($a={a}$) transitive in that definition?
– celtschk
Nov 23 at 7:01
How does the text define ”transitive set”? In particular, are Quine atoms ($a={a}$) transitive in that definition?
– celtschk
Nov 23 at 7:01
add a comment |
The circled fact is trivially true regardless of choice of alpha. There is no need to invoke the fact that no ordinal contains itself.
– Ben W
Nov 23 at 4:22
Is the question about the fact that the containment is proper? Are you actually using this?
– Andrés E. Caicedo
Nov 23 at 5:04