Prove that there exist $2^{aleph_{0}}$ well-orderings of the set of all natural numbers.
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From Chapter 6 Hrbacek and Jech (Ordinal Numbers) Problem 1.3. I don't understand how to begin. Below is the only solution I could find online, but even then I still don't understand how they arrived with those figures.
elementary-set-theory
asked Nov 19 at 4:40
Barycentric_Bash
31528
add a comment |
up vote
1
down vote
favorite
From Chapter 6 Hrbacek and Jech (Ordinal Numbers) Problem 1.3. I don't understand how to begin. Below is the only solution I could find online, but even then I still don't understand how they arrived with those figures.
elementary-set-theory
asked Nov 19 at 4:40
Barycentric_Bash
31528
2
How would a chapter and exercise number be helpful without the name of the reference?
– spaceisdarkgreen
Nov 19 at 5:34
1
Of course there are only $aleph_1$ distinct types of well-orderings on $mathbb{N}$.
– Henno Brandsma
Nov 19 at 22:18
Any bijection of $mathbb{N}$ can be declared an order isomorphism with another well-order (of type $omega$), e.g.
– Henno Brandsma
Nov 19 at 22:20
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
From Chapter 6 Hrbacek and Jech (Ordinal Numbers) Problem 1.3. I don't understand how to begin. Below is the only solution I could find online, but even then I still don't understand how they arrived with those figures.
elementary-set-theory
asked Nov 19 at 4:40
Barycentric_Bash
31528
From Chapter 6 Hrbacek and Jech (Ordinal Numbers) Problem 1.3. I don't understand how to begin. Below is the only solution I could find online, but even then I still don't understand how they arrived with those figures.
elementary-set-theory
elementary-set-theory
asked Nov 19 at 4:40
Barycentric_Bash
31528
asked Nov 19 at 4:40
Barycentric_Bash
31528
edited Nov 20 at 4:22
asked Nov 19 at 4:40
Barycentric_Bash
31528
asked Nov 19 at 4:40
Barycentric_Bash
31528
asked Nov 19 at 4:40
Barycentric_Bash
31528
31528
2
How would a chapter and exercise number be helpful without the name of the reference?
– spaceisdarkgreen
Nov 19 at 5:34
1
Of course there are only $aleph_1$ distinct types of well-orderings on $mathbb{N}$.
– Henno Brandsma
Nov 19 at 22:18
Any bijection of $mathbb{N}$ can be declared an order isomorphism with another well-order (of type $omega$), e.g.
– Henno Brandsma
Nov 19 at 22:20
add a comment |
2
How would a chapter and exercise number be helpful without the name of the reference?
– spaceisdarkgreen
Nov 19 at 5:34
1
Of course there are only $aleph_1$ distinct types of well-orderings on $mathbb{N}$.
– Henno Brandsma
Nov 19 at 22:18
Any bijection of $mathbb{N}$ can be declared an order isomorphism with another well-order (of type $omega$), e.g.
– Henno Brandsma
Nov 19 at 22:20
2
2
How would a chapter and exercise number be helpful without the name of the reference?
– spaceisdarkgreen
Nov 19 at 5:34
How would a chapter and exercise number be helpful without the name of the reference?
– spaceisdarkgreen
Nov 19 at 5:34
1
1
Of course there are only $aleph_1$ distinct types of well-orderings on $mathbb{N}$.
– Henno Brandsma
Nov 19 at 22:18
Of course there are only $aleph_1$ distinct types of well-orderings on $mathbb{N}$.
– Henno Brandsma
Nov 19 at 22:18
Any bijection of $mathbb{N}$ can be declared an order isomorphism with another well-order (of type $omega$), e.g.
– Henno Brandsma
Nov 19 at 22:20
Any bijection of $mathbb{N}$ can be declared an order isomorphism with another well-order (of type $omega$), e.g.
– Henno Brandsma
Nov 19 at 22:20
add a comment |
1 Answer
1
active
oldest
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up vote
2
down vote
accepted
Given the usual order on $Bbb N$, you can show there are at least $2^{aleph_0}$ because you can break $Bbb N$ into $aleph_0$ pairs $(2n, 2n+1)$. You can then invert each pair or not, which is $aleph_0$ binary choices. There are at most $aleph_0^{aleph_0}$ because there are $aleph_0$ choices for the first element of the well order, and again for the next, and so on. If you know $2^{aleph_0}=aleph_0^{aleph_0}=mathfrak c$ you are done.
answered Nov 19 at 5:32
Ross Millikan
289k23195367
2
I think your upper bound argument is kind of vague (and might give the misleading impression that the order type is necessarily $omega$). A more simple way is just that the well orders are all subsets of $mathbb Ntimes mathbb N.$
– spaceisdarkgreen
Nov 19 at 5:46
@spaceisdarkgreen: good point. When I first read it I thought it claimed the well order was a set of pairs $(a,b)$ where $ain Bbb N$ and $bin Bbb N$ was the position of $a$ in the list, which again claims that the order type is $omega$, but the well order can just be all the pairs $(a,b)$ where $a$ comes before $b$.
– Ross Millikan
Nov 19 at 6:04
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Given the usual order on $Bbb N$, you can show there are at least $2^{aleph_0}$ because you can break $Bbb N$ into $aleph_0$ pairs $(2n, 2n+1)$. You can then invert each pair or not, which is $aleph_0$ binary choices. There are at most $aleph_0^{aleph_0}$ because there are $aleph_0$ choices for the first element of the well order, and again for the next, and so on. If you know $2^{aleph_0}=aleph_0^{aleph_0}=mathfrak c$ you are done.
answered Nov 19 at 5:32
Ross Millikan
289k23195367
2
I think your upper bound argument is kind of vague (and might give the misleading impression that the order type is necessarily $omega$). A more simple way is just that the well orders are all subsets of $mathbb Ntimes mathbb N.$
– spaceisdarkgreen
Nov 19 at 5:46
@spaceisdarkgreen: good point. When I first read it I thought it claimed the well order was a set of pairs $(a,b)$ where $ain Bbb N$ and $bin Bbb N$ was the position of $a$ in the list, which again claims that the order type is $omega$, but the well order can just be all the pairs $(a,b)$ where $a$ comes before $b$.
– Ross Millikan
Nov 19 at 6:04
add a comment |
up vote
2
down vote
accepted
Given the usual order on $Bbb N$, you can show there are at least $2^{aleph_0}$ because you can break $Bbb N$ into $aleph_0$ pairs $(2n, 2n+1)$. You can then invert each pair or not, which is $aleph_0$ binary choices. There are at most $aleph_0^{aleph_0}$ because there are $aleph_0$ choices for the first element of the well order, and again for the next, and so on. If you know $2^{aleph_0}=aleph_0^{aleph_0}=mathfrak c$ you are done.
answered Nov 19 at 5:32
Ross Millikan
289k23195367
2
I think your upper bound argument is kind of vague (and might give the misleading impression that the order type is necessarily $omega$). A more simple way is just that the well orders are all subsets of $mathbb Ntimes mathbb N.$
– spaceisdarkgreen
Nov 19 at 5:46
@spaceisdarkgreen: good point. When I first read it I thought it claimed the well order was a set of pairs $(a,b)$ where $ain Bbb N$ and $bin Bbb N$ was the position of $a$ in the list, which again claims that the order type is $omega$, but the well order can just be all the pairs $(a,b)$ where $a$ comes before $b$.
– Ross Millikan
Nov 19 at 6:04
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Given the usual order on $Bbb N$, you can show there are at least $2^{aleph_0}$ because you can break $Bbb N$ into $aleph_0$ pairs $(2n, 2n+1)$. You can then invert each pair or not, which is $aleph_0$ binary choices. There are at most $aleph_0^{aleph_0}$ because there are $aleph_0$ choices for the first element of the well order, and again for the next, and so on. If you know $2^{aleph_0}=aleph_0^{aleph_0}=mathfrak c$ you are done.
answered Nov 19 at 5:32
Ross Millikan
289k23195367
Given the usual order on $Bbb N$, you can show there are at least $2^{aleph_0}$ because you can break $Bbb N$ into $aleph_0$ pairs $(2n, 2n+1)$. You can then invert each pair or not, which is $aleph_0$ binary choices. There are at most $aleph_0^{aleph_0}$ because there are $aleph_0$ choices for the first element of the well order, and again for the next, and so on. If you know $2^{aleph_0}=aleph_0^{aleph_0}=mathfrak c$ you are done.
answered Nov 19 at 5:32
Ross Millikan
289k23195367
answered Nov 19 at 5:32
Ross Millikan
289k23195367
answered Nov 19 at 5:32
Ross Millikan
289k23195367
answered Nov 19 at 5:32
Ross Millikan
289k23195367
289k23195367
2
I think your upper bound argument is kind of vague (and might give the misleading impression that the order type is necessarily $omega$). A more simple way is just that the well orders are all subsets of $mathbb Ntimes mathbb N.$
– spaceisdarkgreen
Nov 19 at 5:46
@spaceisdarkgreen: good point. When I first read it I thought it claimed the well order was a set of pairs $(a,b)$ where $ain Bbb N$ and $bin Bbb N$ was the position of $a$ in the list, which again claims that the order type is $omega$, but the well order can just be all the pairs $(a,b)$ where $a$ comes before $b$.
– Ross Millikan
Nov 19 at 6:04
add a comment |
2
I think your upper bound argument is kind of vague (and might give the misleading impression that the order type is necessarily $omega$). A more simple way is just that the well orders are all subsets of $mathbb Ntimes mathbb N.$
– spaceisdarkgreen
Nov 19 at 5:46
@spaceisdarkgreen: good point. When I first read it I thought it claimed the well order was a set of pairs $(a,b)$ where $ain Bbb N$ and $bin Bbb N$ was the position of $a$ in the list, which again claims that the order type is $omega$, but the well order can just be all the pairs $(a,b)$ where $a$ comes before $b$.
– Ross Millikan
Nov 19 at 6:04
2
2
I think your upper bound argument is kind of vague (and might give the misleading impression that the order type is necessarily $omega$). A more simple way is just that the well orders are all subsets of $mathbb Ntimes mathbb N.$
– spaceisdarkgreen
Nov 19 at 5:46
I think your upper bound argument is kind of vague (and might give the misleading impression that the order type is necessarily $omega$). A more simple way is just that the well orders are all subsets of $mathbb Ntimes mathbb N.$
– spaceisdarkgreen
Nov 19 at 5:46
@spaceisdarkgreen: good point. When I first read it I thought it claimed the well order was a set of pairs $(a,b)$ where $ain Bbb N$ and $bin Bbb N$ was the position of $a$ in the list, which again claims that the order type is $omega$, but the well order can just be all the pairs $(a,b)$ where $a$ comes before $b$.
– Ross Millikan
Nov 19 at 6:04
@spaceisdarkgreen: good point. When I first read it I thought it claimed the well order was a set of pairs $(a,b)$ where $ain Bbb N$ and $bin Bbb N$ was the position of $a$ in the list, which again claims that the order type is $omega$, but the well order can just be all the pairs $(a,b)$ where $a$ comes before $b$.
– Ross Millikan
Nov 19 at 6:04
add a comment |
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2
How would a chapter and exercise number be helpful without the name of the reference?
– spaceisdarkgreen
Nov 19 at 5:34
1
Of course there are only $aleph_1$ distinct types of well-orderings on $mathbb{N}$.
– Henno Brandsma
Nov 19 at 22:18
Any bijection of $mathbb{N}$ can be declared an order isomorphism with another well-order (of type $omega$), e.g.
– Henno Brandsma
Nov 19 at 22:20