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Must a sequence be well-founded?
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up vote
1
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Must a sequence be well-founded?
Is $Bbb Z=(ldots-1,0,1,2,ldots)$ a sequence?
Conventionally we think of $Bbb N=(0,1,2,3,ldots)$ as a sequence, but what about if it has no starting value?
Obviously we can reorder any countable set into a sequence e.g. by $f:Bbb Zto Bbb N$ but what about in its raw form, do we call $Bbb Z$ a sequence?
sequences-and-series set-theory terminology order-theory integers
asked 28 mins ago
Robert Frost
3,9561036
add a comment |Â
up vote
1
down vote
favorite
Must a sequence be well-founded?
Is $Bbb Z=(ldots-1,0,1,2,ldots)$ a sequence?
Conventionally we think of $Bbb N=(0,1,2,3,ldots)$ as a sequence, but what about if it has no starting value?
Obviously we can reorder any countable set into a sequence e.g. by $f:Bbb Zto Bbb N$ but what about in its raw form, do we call $Bbb Z$ a sequence?
sequences-and-series set-theory terminology order-theory integers
asked 28 mins ago
Robert Frost
3,9561036
1
What is your definition of a sequence?
– Nosrati
25 mins ago
How about $BbbQ$ ? Even worse....
– dmtri
12 mins ago
@dmtri there is of course $f:Bbb QtoBbb N$ too but I was inquiring only about loss of the well-foundedness property, aside from which, $Bbb Z$ is isomorphic to $Bbb N$. Whereas $Bbb Q$ plus well-foundedness remains far from isomorphic to $Bbb N$.
– Robert Frost
9 mins ago
@Nosrati That is in flux at the moment and I'm inquiring into standard terminology so as to solidify my definition in a way which will facilitate seamless communication with others.
– Robert Frost
8 mins ago
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Must a sequence be well-founded?
Is $Bbb Z=(ldots-1,0,1,2,ldots)$ a sequence?
Conventionally we think of $Bbb N=(0,1,2,3,ldots)$ as a sequence, but what about if it has no starting value?
Obviously we can reorder any countable set into a sequence e.g. by $f:Bbb Zto Bbb N$ but what about in its raw form, do we call $Bbb Z$ a sequence?
sequences-and-series set-theory terminology order-theory integers
asked 28 mins ago
Robert Frost
3,9561036
Must a sequence be well-founded?
Is $Bbb Z=(ldots-1,0,1,2,ldots)$ a sequence?
Conventionally we think of $Bbb N=(0,1,2,3,ldots)$ as a sequence, but what about if it has no starting value?
Obviously we can reorder any countable set into a sequence e.g. by $f:Bbb Zto Bbb N$ but what about in its raw form, do we call $Bbb Z$ a sequence?
sequences-and-series set-theory terminology order-theory integers
sequences-and-series set-theory terminology order-theory integers
asked 28 mins ago
Robert Frost
3,9561036
asked 28 mins ago
Robert Frost
3,9561036
edited 6 mins ago
asked 28 mins ago
Robert Frost
3,9561036
asked 28 mins ago
Robert Frost
3,9561036
asked 28 mins ago
Robert Frost
3,9561036
3,9561036
1
What is your definition of a sequence?
– Nosrati
25 mins ago
How about $BbbQ$ ? Even worse....
– dmtri
12 mins ago
@dmtri there is of course $f:Bbb QtoBbb N$ too but I was inquiring only about loss of the well-foundedness property, aside from which, $Bbb Z$ is isomorphic to $Bbb N$. Whereas $Bbb Q$ plus well-foundedness remains far from isomorphic to $Bbb N$.
– Robert Frost
9 mins ago
@Nosrati That is in flux at the moment and I'm inquiring into standard terminology so as to solidify my definition in a way which will facilitate seamless communication with others.
– Robert Frost
8 mins ago
add a comment |Â
1
What is your definition of a sequence?
– Nosrati
25 mins ago
How about $BbbQ$ ? Even worse....
– dmtri
12 mins ago
@dmtri there is of course $f:Bbb QtoBbb N$ too but I was inquiring only about loss of the well-foundedness property, aside from which, $Bbb Z$ is isomorphic to $Bbb N$. Whereas $Bbb Q$ plus well-foundedness remains far from isomorphic to $Bbb N$.
– Robert Frost
9 mins ago
@Nosrati That is in flux at the moment and I'm inquiring into standard terminology so as to solidify my definition in a way which will facilitate seamless communication with others.
– Robert Frost
8 mins ago
1
1
What is your definition of a sequence?
– Nosrati
25 mins ago
What is your definition of a sequence?
– Nosrati
25 mins ago
How about $BbbQ$ ? Even worse....
– dmtri
12 mins ago
How about $BbbQ$ ? Even worse....
– dmtri
12 mins ago
@dmtri there is of course $f:Bbb QtoBbb N$ too but I was inquiring only about loss of the well-foundedness property, aside from which, $Bbb Z$ is isomorphic to $Bbb N$. Whereas $Bbb Q$ plus well-foundedness remains far from isomorphic to $Bbb N$.
– Robert Frost
9 mins ago
@dmtri there is of course $f:Bbb QtoBbb N$ too but I was inquiring only about loss of the well-foundedness property, aside from which, $Bbb Z$ is isomorphic to $Bbb N$. Whereas $Bbb Q$ plus well-foundedness remains far from isomorphic to $Bbb N$.
– Robert Frost
9 mins ago
@Nosrati That is in flux at the moment and I'm inquiring into standard terminology so as to solidify my definition in a way which will facilitate seamless communication with others.
– Robert Frost
8 mins ago
@Nosrati That is in flux at the moment and I'm inquiring into standard terminology so as to solidify my definition in a way which will facilitate seamless communication with others.
– Robert Frost
8 mins ago
add a comment |Â
4 Answers
4
active
oldest
votes
up vote
4
down vote
Given any set $I$, it makes sense to talk about "$I$-indexed sequences".
We don't need to make any extra assumptions on $I$ for this to make sense. However, some features we might equip $I$ can induce additional structure on $I$-indexed sequences.
For example, choosing a total ordering on $I$ lets us talk about whether one place in the sequence comes before or after another place. Well-orderings are common to consider, because you can do transfinite induction over a well-ordered index set.
answered 23 mins ago
Hurkyl
110k9114256
So e.g. I would call $Bbb Z$ a "$Bbb Z$-indexed sequence", the same as @Kavi's answer, so as to avoid any ambiguity?
– Robert Frost
13 mins ago
add a comment |Â
up vote
3
down vote
Calling it a sequence can result in conflict with statements of some theorems. It is better to call it a sequence indexed by integers.
answered 19 mins ago
Kavi Rama Murthy
26.4k31438
add a comment |Â
up vote
2
down vote
The terminology of "sequence" is not completely nailed down.
In the strictest sense, "a sequence in $X$" is a function $mathbb N to X$. In a looser sense, $mathbb N$ might be replaced by another upwards unbounded countable linear order, by any countable linear order, or even by any linear order. In the loosest sense it can refer to any function $I to X$ from some index set $I$.
If you say sequence without further context, I expect the strictest definition; but if you talk about an $I$-sequence, I accept that no matter what kind of index set $I$ is.
answered 15 mins ago
Mees de Vries
14.4k12348
This would really depend on the context, though. Sometimes when you say a sequence, I expect it to just be a function. Just like when you say $pi$, sometimes you expect to be the ratio of a circle's circumference and diameter, and sometimes you expect it to be some function or whatnot.
– Asaf Karagila♦
47 secs ago
add a comment |Â
up vote
0
down vote
You can also list $mathbbZ$ as $0,1,-1,2,-2,3,-3,...$. Then you have the same case as for $mathbbN$ and it has a starting value.
answered 18 mins ago
YukiJ
1,6912624
I intended to exclude this answer by my comment about taking it as given that $f:Bbb ZtoBbb N$ turns it into a well-founded sequence.
– Robert Frost
16 mins ago
add a comment |Â
4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
Given any set $I$, it makes sense to talk about "$I$-indexed sequences".
We don't need to make any extra assumptions on $I$ for this to make sense. However, some features we might equip $I$ can induce additional structure on $I$-indexed sequences.
For example, choosing a total ordering on $I$ lets us talk about whether one place in the sequence comes before or after another place. Well-orderings are common to consider, because you can do transfinite induction over a well-ordered index set.
answered 23 mins ago
Hurkyl
110k9114256
So e.g. I would call $Bbb Z$ a "$Bbb Z$-indexed sequence", the same as @Kavi's answer, so as to avoid any ambiguity?
– Robert Frost
13 mins ago
add a comment |Â
up vote
4
down vote
Given any set $I$, it makes sense to talk about "$I$-indexed sequences".
We don't need to make any extra assumptions on $I$ for this to make sense. However, some features we might equip $I$ can induce additional structure on $I$-indexed sequences.
For example, choosing a total ordering on $I$ lets us talk about whether one place in the sequence comes before or after another place. Well-orderings are common to consider, because you can do transfinite induction over a well-ordered index set.
answered 23 mins ago
Hurkyl
110k9114256
So e.g. I would call $Bbb Z$ a "$Bbb Z$-indexed sequence", the same as @Kavi's answer, so as to avoid any ambiguity?
– Robert Frost
13 mins ago
add a comment |Â
up vote
4
down vote
up vote
4
down vote
Given any set $I$, it makes sense to talk about "$I$-indexed sequences".
We don't need to make any extra assumptions on $I$ for this to make sense. However, some features we might equip $I$ can induce additional structure on $I$-indexed sequences.
For example, choosing a total ordering on $I$ lets us talk about whether one place in the sequence comes before or after another place. Well-orderings are common to consider, because you can do transfinite induction over a well-ordered index set.
answered 23 mins ago
Hurkyl
110k9114256
Given any set $I$, it makes sense to talk about "$I$-indexed sequences".
We don't need to make any extra assumptions on $I$ for this to make sense. However, some features we might equip $I$ can induce additional structure on $I$-indexed sequences.
For example, choosing a total ordering on $I$ lets us talk about whether one place in the sequence comes before or after another place. Well-orderings are common to consider, because you can do transfinite induction over a well-ordered index set.
answered 23 mins ago
Hurkyl
110k9114256
edited 14 mins ago
answered 23 mins ago
Hurkyl
110k9114256
answered 23 mins ago
Hurkyl
110k9114256
answered 23 mins ago
Hurkyl
110k9114256
110k9114256
So e.g. I would call $Bbb Z$ a "$Bbb Z$-indexed sequence", the same as @Kavi's answer, so as to avoid any ambiguity?
– Robert Frost
13 mins ago
add a comment |Â
So e.g. I would call $Bbb Z$ a "$Bbb Z$-indexed sequence", the same as @Kavi's answer, so as to avoid any ambiguity?
– Robert Frost
13 mins ago
So e.g. I would call $Bbb Z$ a "$Bbb Z$-indexed sequence", the same as @Kavi's answer, so as to avoid any ambiguity?
– Robert Frost
13 mins ago
So e.g. I would call $Bbb Z$ a "$Bbb Z$-indexed sequence", the same as @Kavi's answer, so as to avoid any ambiguity?
– Robert Frost
13 mins ago
add a comment |Â
up vote
3
down vote
Calling it a sequence can result in conflict with statements of some theorems. It is better to call it a sequence indexed by integers.
answered 19 mins ago
Kavi Rama Murthy
26.4k31438
add a comment |Â
up vote
3
down vote
Calling it a sequence can result in conflict with statements of some theorems. It is better to call it a sequence indexed by integers.
answered 19 mins ago
Kavi Rama Murthy
26.4k31438
add a comment |Â
up vote
3
down vote
up vote
3
down vote
Calling it a sequence can result in conflict with statements of some theorems. It is better to call it a sequence indexed by integers.
answered 19 mins ago
Kavi Rama Murthy
26.4k31438
Calling it a sequence can result in conflict with statements of some theorems. It is better to call it a sequence indexed by integers.
answered 19 mins ago
Kavi Rama Murthy
26.4k31438
answered 19 mins ago
Kavi Rama Murthy
26.4k31438
answered 19 mins ago
Kavi Rama Murthy
26.4k31438
answered 19 mins ago
Kavi Rama Murthy
26.4k31438
26.4k31438
add a comment |Â
add a comment |Â
up vote
2
down vote
The terminology of "sequence" is not completely nailed down.
In the strictest sense, "a sequence in $X$" is a function $mathbb N to X$. In a looser sense, $mathbb N$ might be replaced by another upwards unbounded countable linear order, by any countable linear order, or even by any linear order. In the loosest sense it can refer to any function $I to X$ from some index set $I$.
If you say sequence without further context, I expect the strictest definition; but if you talk about an $I$-sequence, I accept that no matter what kind of index set $I$ is.
answered 15 mins ago
Mees de Vries
14.4k12348
This would really depend on the context, though. Sometimes when you say a sequence, I expect it to just be a function. Just like when you say $pi$, sometimes you expect to be the ratio of a circle's circumference and diameter, and sometimes you expect it to be some function or whatnot.
– Asaf Karagila♦
47 secs ago
add a comment |Â
up vote
2
down vote
The terminology of "sequence" is not completely nailed down.
In the strictest sense, "a sequence in $X$" is a function $mathbb N to X$. In a looser sense, $mathbb N$ might be replaced by another upwards unbounded countable linear order, by any countable linear order, or even by any linear order. In the loosest sense it can refer to any function $I to X$ from some index set $I$.
If you say sequence without further context, I expect the strictest definition; but if you talk about an $I$-sequence, I accept that no matter what kind of index set $I$ is.
answered 15 mins ago
Mees de Vries
14.4k12348
This would really depend on the context, though. Sometimes when you say a sequence, I expect it to just be a function. Just like when you say $pi$, sometimes you expect to be the ratio of a circle's circumference and diameter, and sometimes you expect it to be some function or whatnot.
– Asaf Karagila♦
47 secs ago
add a comment |Â
up vote
2
down vote
up vote
2
down vote
The terminology of "sequence" is not completely nailed down.
In the strictest sense, "a sequence in $X$" is a function $mathbb N to X$. In a looser sense, $mathbb N$ might be replaced by another upwards unbounded countable linear order, by any countable linear order, or even by any linear order. In the loosest sense it can refer to any function $I to X$ from some index set $I$.
If you say sequence without further context, I expect the strictest definition; but if you talk about an $I$-sequence, I accept that no matter what kind of index set $I$ is.
answered 15 mins ago
Mees de Vries
14.4k12348
The terminology of "sequence" is not completely nailed down.
In the strictest sense, "a sequence in $X$" is a function $mathbb N to X$. In a looser sense, $mathbb N$ might be replaced by another upwards unbounded countable linear order, by any countable linear order, or even by any linear order. In the loosest sense it can refer to any function $I to X$ from some index set $I$.
If you say sequence without further context, I expect the strictest definition; but if you talk about an $I$-sequence, I accept that no matter what kind of index set $I$ is.
answered 15 mins ago
Mees de Vries
14.4k12348
answered 15 mins ago
Mees de Vries
14.4k12348
answered 15 mins ago
Mees de Vries
14.4k12348
answered 15 mins ago
Mees de Vries
14.4k12348
14.4k12348
This would really depend on the context, though. Sometimes when you say a sequence, I expect it to just be a function. Just like when you say $pi$, sometimes you expect to be the ratio of a circle's circumference and diameter, and sometimes you expect it to be some function or whatnot.
– Asaf Karagila♦
47 secs ago
add a comment |Â
This would really depend on the context, though. Sometimes when you say a sequence, I expect it to just be a function. Just like when you say $pi$, sometimes you expect to be the ratio of a circle's circumference and diameter, and sometimes you expect it to be some function or whatnot.
– Asaf Karagila♦
47 secs ago
This would really depend on the context, though. Sometimes when you say a sequence, I expect it to just be a function. Just like when you say $pi$, sometimes you expect to be the ratio of a circle's circumference and diameter, and sometimes you expect it to be some function or whatnot.
– Asaf Karagila♦
47 secs ago
This would really depend on the context, though. Sometimes when you say a sequence, I expect it to just be a function. Just like when you say $pi$, sometimes you expect to be the ratio of a circle's circumference and diameter, and sometimes you expect it to be some function or whatnot.
– Asaf Karagila♦
47 secs ago
add a comment |Â
up vote
0
down vote
You can also list $mathbbZ$ as $0,1,-1,2,-2,3,-3,...$. Then you have the same case as for $mathbbN$ and it has a starting value.
answered 18 mins ago
YukiJ
1,6912624
I intended to exclude this answer by my comment about taking it as given that $f:Bbb ZtoBbb N$ turns it into a well-founded sequence.
– Robert Frost
16 mins ago
add a comment |Â
up vote
0
down vote
You can also list $mathbbZ$ as $0,1,-1,2,-2,3,-3,...$. Then you have the same case as for $mathbbN$ and it has a starting value.
answered 18 mins ago
YukiJ
1,6912624
I intended to exclude this answer by my comment about taking it as given that $f:Bbb ZtoBbb N$ turns it into a well-founded sequence.
– Robert Frost
16 mins ago
add a comment |Â
up vote
0
down vote
up vote
0
down vote
You can also list $mathbbZ$ as $0,1,-1,2,-2,3,-3,...$. Then you have the same case as for $mathbbN$ and it has a starting value.
answered 18 mins ago
YukiJ
1,6912624
You can also list $mathbbZ$ as $0,1,-1,2,-2,3,-3,...$. Then you have the same case as for $mathbbN$ and it has a starting value.
answered 18 mins ago
YukiJ
1,6912624
answered 18 mins ago
YukiJ
1,6912624
answered 18 mins ago
YukiJ
1,6912624
answered 18 mins ago
YukiJ
1,6912624
1,6912624
I intended to exclude this answer by my comment about taking it as given that $f:Bbb ZtoBbb N$ turns it into a well-founded sequence.
– Robert Frost
16 mins ago
add a comment |Â
I intended to exclude this answer by my comment about taking it as given that $f:Bbb ZtoBbb N$ turns it into a well-founded sequence.
– Robert Frost
16 mins ago
I intended to exclude this answer by my comment about taking it as given that $f:Bbb ZtoBbb N$ turns it into a well-founded sequence.
– Robert Frost
16 mins ago
I intended to exclude this answer by my comment about taking it as given that $f:Bbb ZtoBbb N$ turns it into a well-founded sequence.
– Robert Frost
16 mins ago
add a comment |Â
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1
What is your definition of a sequence?
– Nosrati
25 mins ago
How about $BbbQ$ ? Even worse....
– dmtri
12 mins ago
@dmtri there is of course $f:Bbb QtoBbb N$ too but I was inquiring only about loss of the well-foundedness property, aside from which, $Bbb Z$ is isomorphic to $Bbb N$. Whereas $Bbb Q$ plus well-foundedness remains far from isomorphic to $Bbb N$.
– Robert Frost
9 mins ago
@Nosrati That is in flux at the moment and I'm inquiring into standard terminology so as to solidify my definition in a way which will facilitate seamless communication with others.
– Robert Frost
8 mins ago