Mixing
definition of countable infinite character without the use of N
$begingroup$
Let's say a family of sets, $F$ has finite character iff $forall X: [X in F leftrightarrow forall E subseteq X: E , finite rightarrow E in F)$
or more precise: let's say a family $F$ has character k iff $forall X: X in F leftrightarrow (forall E subseteq X: |E| leq k rightarrow E in F)$.
Now I'm wondering, what would be a sensible definition for a family to be of (at most) countable infinite character?)
And can you think of an example for a family of countable infinite character, but not finite character?
EDIT: Would something like $forall X: X in F leftrightarrow (forall E subseteq X: |E| leq |omega| rightarrow E in F)$ be legitimate? (whereby $omega$ is the smallest inductive set, i.e. the set-theory-description of $mathbb{N_0}$)
EDIT: Would the power set of $mathbb{N}$ be such an example and if so, how could we write this without using $mathbb{N}$? And would there be a maximal element in it?
elementary-set-theory definition
asked Jan 20 at 22:47
StudentuStudentu
1279
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show 2 more comments
$begingroup$
Let's say a family of sets, $F$ has finite character iff $forall X: [X in F leftrightarrow forall E subseteq X: E , finite rightarrow E in F)$
or more precise: let's say a family $F$ has character k iff $forall X: X in F leftrightarrow (forall E subseteq X: |E| leq k rightarrow E in F)$.
Now I'm wondering, what would be a sensible definition for a family to be of (at most) countable infinite character?)
And can you think of an example for a family of countable infinite character, but not finite character?
EDIT: Would something like $forall X: X in F leftrightarrow (forall E subseteq X: |E| leq |omega| rightarrow E in F)$ be legitimate? (whereby $omega$ is the smallest inductive set, i.e. the set-theory-description of $mathbb{N_0}$)
EDIT: Would the power set of $mathbb{N}$ be such an example and if so, how could we write this without using $mathbb{N}$? And would there be a maximal element in it?
elementary-set-theory definition
asked Jan 20 at 22:47
StudentuStudentu
1279
$endgroup$
2
$begingroup$
You ask about "sensible," but that's going to be relative to a possible application, so it might help if you can comment on what that might be. Given the definition of "at most countable character" in your edit, $mathcal{P}(mathbb{N})$ is such a set, though I'm not sure how or why you would define the power set of $mathbb{N}$ without reference to $mathbb{N}$. And obviously power sets have a maximal element.
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– Malice Vidrine
Jan 20 at 23:04
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No, $omega$ is wrong. You want either $aleph_0$ or |$omega_0$|.
$endgroup$
– William Elliot
Jan 21 at 2:56
$begingroup$
In the first line of symbolic logic you have unbalanced [ ) and all such lines have ambiguous quantifer scopes.
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– William Elliot
Jan 21 at 3:09
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@WilliamElliot Thanks for commenting! You are right, $|omega|$ is what I meant (in my definition it contains the empty set). But what do you mean by your second comment?
$endgroup$
– Studentu
Jan 21 at 5:20
$begingroup$
Maybe instead of interjections, either edit the information into the question, or make one concentrated edit at the end?
$endgroup$
– Asaf Karagila♦
Jan 21 at 8:07
|
show 2 more comments
$begingroup$
Let's say a family of sets, $F$ has finite character iff $forall X: [X in F leftrightarrow forall E subseteq X: E , finite rightarrow E in F)$
or more precise: let's say a family $F$ has character k iff $forall X: X in F leftrightarrow (forall E subseteq X: |E| leq k rightarrow E in F)$.
Now I'm wondering, what would be a sensible definition for a family to be of (at most) countable infinite character?)
And can you think of an example for a family of countable infinite character, but not finite character?
EDIT: Would something like $forall X: X in F leftrightarrow (forall E subseteq X: |E| leq |omega| rightarrow E in F)$ be legitimate? (whereby $omega$ is the smallest inductive set, i.e. the set-theory-description of $mathbb{N_0}$)
EDIT: Would the power set of $mathbb{N}$ be such an example and if so, how could we write this without using $mathbb{N}$? And would there be a maximal element in it?
elementary-set-theory definition
asked Jan 20 at 22:47
StudentuStudentu
1279
$endgroup$
Let's say a family of sets, $F$ has finite character iff $forall X: [X in F leftrightarrow forall E subseteq X: E , finite rightarrow E in F)$
or more precise: let's say a family $F$ has character k iff $forall X: X in F leftrightarrow (forall E subseteq X: |E| leq k rightarrow E in F)$.
Now I'm wondering, what would be a sensible definition for a family to be of (at most) countable infinite character?)
And can you think of an example for a family of countable infinite character, but not finite character?
EDIT: Would something like $forall X: X in F leftrightarrow (forall E subseteq X: |E| leq |omega| rightarrow E in F)$ be legitimate? (whereby $omega$ is the smallest inductive set, i.e. the set-theory-description of $mathbb{N_0}$)
EDIT: Would the power set of $mathbb{N}$ be such an example and if so, how could we write this without using $mathbb{N}$? And would there be a maximal element in it?
elementary-set-theory definition
elementary-set-theory definition
asked Jan 20 at 22:47
StudentuStudentu
1279
asked Jan 20 at 22:47
StudentuStudentu
1279
edited Jan 21 at 16:40
Studentu
asked Jan 20 at 22:47
StudentuStudentu
1279
asked Jan 20 at 22:47
StudentuStudentu
1279
asked Jan 20 at 22:47
StudentuStudentu
1279
1279
2
$begingroup$
You ask about "sensible," but that's going to be relative to a possible application, so it might help if you can comment on what that might be. Given the definition of "at most countable character" in your edit, $mathcal{P}(mathbb{N})$ is such a set, though I'm not sure how or why you would define the power set of $mathbb{N}$ without reference to $mathbb{N}$. And obviously power sets have a maximal element.
$endgroup$
– Malice Vidrine
Jan 20 at 23:04
$begingroup$
No, $omega$ is wrong. You want either $aleph_0$ or |$omega_0$|.
$endgroup$
– William Elliot
Jan 21 at 2:56
$begingroup$
In the first line of symbolic logic you have unbalanced [ ) and all such lines have ambiguous quantifer scopes.
$endgroup$
– William Elliot
Jan 21 at 3:09
$begingroup$
@WilliamElliot Thanks for commenting! You are right, $|omega|$ is what I meant (in my definition it contains the empty set). But what do you mean by your second comment?
$endgroup$
– Studentu
Jan 21 at 5:20
$begingroup$
Maybe instead of interjections, either edit the information into the question, or make one concentrated edit at the end?
$endgroup$
– Asaf Karagila♦
Jan 21 at 8:07
|
show 2 more comments
2
$begingroup$
You ask about "sensible," but that's going to be relative to a possible application, so it might help if you can comment on what that might be. Given the definition of "at most countable character" in your edit, $mathcal{P}(mathbb{N})$ is such a set, though I'm not sure how or why you would define the power set of $mathbb{N}$ without reference to $mathbb{N}$. And obviously power sets have a maximal element.
$endgroup$
– Malice Vidrine
Jan 20 at 23:04
$begingroup$
No, $omega$ is wrong. You want either $aleph_0$ or |$omega_0$|.
$endgroup$
– William Elliot
Jan 21 at 2:56
$begingroup$
In the first line of symbolic logic you have unbalanced [ ) and all such lines have ambiguous quantifer scopes.
$endgroup$
– William Elliot
Jan 21 at 3:09
$begingroup$
@WilliamElliot Thanks for commenting! You are right, $|omega|$ is what I meant (in my definition it contains the empty set). But what do you mean by your second comment?
$endgroup$
– Studentu
Jan 21 at 5:20
$begingroup$
Maybe instead of interjections, either edit the information into the question, or make one concentrated edit at the end?
$endgroup$
– Asaf Karagila♦
Jan 21 at 8:07
2
2
$begingroup$
You ask about "sensible," but that's going to be relative to a possible application, so it might help if you can comment on what that might be. Given the definition of "at most countable character" in your edit, $mathcal{P}(mathbb{N})$ is such a set, though I'm not sure how or why you would define the power set of $mathbb{N}$ without reference to $mathbb{N}$. And obviously power sets have a maximal element.
$endgroup$
– Malice Vidrine
Jan 20 at 23:04
$begingroup$
You ask about "sensible," but that's going to be relative to a possible application, so it might help if you can comment on what that might be. Given the definition of "at most countable character" in your edit, $mathcal{P}(mathbb{N})$ is such a set, though I'm not sure how or why you would define the power set of $mathbb{N}$ without reference to $mathbb{N}$. And obviously power sets have a maximal element.
$endgroup$
– Malice Vidrine
Jan 20 at 23:04
$begingroup$
No, $omega$ is wrong. You want either $aleph_0$ or |$omega_0$|.
$endgroup$
– William Elliot
Jan 21 at 2:56
$begingroup$
No, $omega$ is wrong. You want either $aleph_0$ or |$omega_0$|.
$endgroup$
– William Elliot
Jan 21 at 2:56
$begingroup$
In the first line of symbolic logic you have unbalanced [ ) and all such lines have ambiguous quantifer scopes.
$endgroup$
– William Elliot
Jan 21 at 3:09
$begingroup$
In the first line of symbolic logic you have unbalanced [ ) and all such lines have ambiguous quantifer scopes.
$endgroup$
– William Elliot
Jan 21 at 3:09
$begingroup$
@WilliamElliot Thanks for commenting! You are right, $|omega|$ is what I meant (in my definition it contains the empty set). But what do you mean by your second comment?
$endgroup$
– Studentu
Jan 21 at 5:20
$begingroup$
@WilliamElliot Thanks for commenting! You are right, $|omega|$ is what I meant (in my definition it contains the empty set). But what do you mean by your second comment?
$endgroup$
– Studentu
Jan 21 at 5:20
$begingroup$
Maybe instead of interjections, either edit the information into the question, or make one concentrated edit at the end?
$endgroup$
– Asaf Karagila♦
Jan 21 at 8:07
$begingroup$
Maybe instead of interjections, either edit the information into the question, or make one concentrated edit at the end?
$endgroup$
– Asaf Karagila♦
Jan 21 at 8:07
|
show 2 more comments
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$begingroup$
You ask about "sensible," but that's going to be relative to a possible application, so it might help if you can comment on what that might be. Given the definition of "at most countable character" in your edit, $mathcal{P}(mathbb{N})$ is such a set, though I'm not sure how or why you would define the power set of $mathbb{N}$ without reference to $mathbb{N}$. And obviously power sets have a maximal element.
$endgroup$
– Malice Vidrine
Jan 20 at 23:04
$begingroup$
No, $omega$ is wrong. You want either $aleph_0$ or |$omega_0$|.
$endgroup$
– William Elliot
Jan 21 at 2:56
$begingroup$
In the first line of symbolic logic you have unbalanced [ ) and all such lines have ambiguous quantifer scopes.
$endgroup$
– William Elliot
Jan 21 at 3:09
$begingroup$
@WilliamElliot Thanks for commenting! You are right, $|omega|$ is what I meant (in my definition it contains the empty set). But what do you mean by your second comment?
$endgroup$
– Studentu
Jan 21 at 5:20
$begingroup$
Maybe instead of interjections, either edit the information into the question, or make one concentrated edit at the end?
$endgroup$
– Asaf Karagila♦
Jan 21 at 8:07