Making sense of a bijection between proper classes? [duplicate]
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Can proper classes also have cardinality?
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Bijection between collection C and proper class PC makes C a proper class?
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I am trying to solve the following problem in ZF: "Show that the collection ${kappa | kappa=aleph_kappa}$ is a proper class. My thinking is that in the construction any element of this collection, one takes an arbitrary ordinal $alpha$, and that therefore one might be able to show it is a proper class by showing that there is a bijection between the class of all ordinals and the class given above, since the class of all ordinals is a proper class.
However, I'm not sure how to fully make sense of the concept of a function between proper classes, since a function is normally defined as being between sets. Is this a valid line of reasoning, and if so, how does one formalise this using the axioms of ZF?
set-theory ordinals
asked Nov 25 at 13:48
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marked as duplicate by Asaf Karagila♦ ordinals
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Nov 25 at 13:59
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.
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This question already has an answer here:
Can proper classes also have cardinality?
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Bijection between collection C and proper class PC makes C a proper class?
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I am trying to solve the following problem in ZF: "Show that the collection ${kappa | kappa=aleph_kappa}$ is a proper class. My thinking is that in the construction any element of this collection, one takes an arbitrary ordinal $alpha$, and that therefore one might be able to show it is a proper class by showing that there is a bijection between the class of all ordinals and the class given above, since the class of all ordinals is a proper class.
However, I'm not sure how to fully make sense of the concept of a function between proper classes, since a function is normally defined as being between sets. Is this a valid line of reasoning, and if so, how does one formalise this using the axioms of ZF?
set-theory ordinals
asked Nov 25 at 13:48
MMR
267
marked as duplicate by Asaf Karagila♦ ordinals
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Nov 25 at 13:59
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.
How about showing it's a proper class by showing it isn't a set?
– Lord Shark the Unknown
Nov 25 at 13:59
This is truly a diagonal argument kind of thing, though. Show that for any set, there is an element of the class not in that set.
– Asaf Karagila♦
Nov 25 at 14:01
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This question already has an answer here:
Can proper classes also have cardinality?
2 answers
Bijection between collection C and proper class PC makes C a proper class?
3 answers
I am trying to solve the following problem in ZF: "Show that the collection ${kappa | kappa=aleph_kappa}$ is a proper class. My thinking is that in the construction any element of this collection, one takes an arbitrary ordinal $alpha$, and that therefore one might be able to show it is a proper class by showing that there is a bijection between the class of all ordinals and the class given above, since the class of all ordinals is a proper class.
However, I'm not sure how to fully make sense of the concept of a function between proper classes, since a function is normally defined as being between sets. Is this a valid line of reasoning, and if so, how does one formalise this using the axioms of ZF?
set-theory ordinals
asked Nov 25 at 13:48
MMR
267
This question already has an answer here:
Can proper classes also have cardinality?
2 answers
Bijection between collection C and proper class PC makes C a proper class?
3 answers
I am trying to solve the following problem in ZF: "Show that the collection ${kappa | kappa=aleph_kappa}$ is a proper class. My thinking is that in the construction any element of this collection, one takes an arbitrary ordinal $alpha$, and that therefore one might be able to show it is a proper class by showing that there is a bijection between the class of all ordinals and the class given above, since the class of all ordinals is a proper class.
However, I'm not sure how to fully make sense of the concept of a function between proper classes, since a function is normally defined as being between sets. Is this a valid line of reasoning, and if so, how does one formalise this using the axioms of ZF?
This question already has an answer here:
Can proper classes also have cardinality?
2 answers
Bijection between collection C and proper class PC makes C a proper class?
3 answers
set-theory ordinals
set-theory ordinals
asked Nov 25 at 13:48
MMR
267
asked Nov 25 at 13:48
MMR
267
asked Nov 25 at 13:48
MMR
267
asked Nov 25 at 13:48
MMR
267
asked Nov 25 at 13:48
MMR
267
267
marked as duplicate by Asaf Karagila♦ ordinals
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Nov 25 at 13:59
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 Asaf Karagila♦ ordinals
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Nov 25 at 13:59
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.
How about showing it's a proper class by showing it isn't a set?
– Lord Shark the Unknown
Nov 25 at 13:59
This is truly a diagonal argument kind of thing, though. Show that for any set, there is an element of the class not in that set.
– Asaf Karagila♦
Nov 25 at 14:01
add a comment |
How about showing it's a proper class by showing it isn't a set?
– Lord Shark the Unknown
Nov 25 at 13:59
This is truly a diagonal argument kind of thing, though. Show that for any set, there is an element of the class not in that set.
– Asaf Karagila♦
Nov 25 at 14:01
How about showing it's a proper class by showing it isn't a set?
– Lord Shark the Unknown
Nov 25 at 13:59
How about showing it's a proper class by showing it isn't a set?
– Lord Shark the Unknown
Nov 25 at 13:59
This is truly a diagonal argument kind of thing, though. Show that for any set, there is an element of the class not in that set.
– Asaf Karagila♦
Nov 25 at 14:01
This is truly a diagonal argument kind of thing, though. Show that for any set, there is an element of the class not in that set.
– Asaf Karagila♦
Nov 25 at 14:01
add a comment |
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How about showing it's a proper class by showing it isn't a set?
– Lord Shark the Unknown
Nov 25 at 13:59
This is truly a diagonal argument kind of thing, though. Show that for any set, there is an element of the class not in that set.
– Asaf Karagila♦
Nov 25 at 14:01