Mixing
Order of the natural numbers
Clash Royale CLAN TAG#URR8PPP
up vote
3
down vote
favorite
The set of natural numbers as given from the Peano axioms $(N,S)$ has an order.
I saw in wikipedia that $(N,+)$ is a commutative monoid, but since the naturals have an order structure by construction shouldn't $(N,+)$ be an ordered commutative monoid ?
Thanks
abstract-algebra terminology definition peano-axioms
edited Jul 17 at 15:21
rschwieb
100k1193227
asked Jul 17 at 15:11
kot
497
add a comment |Â
up vote
3
down vote
favorite
The set of natural numbers as given from the Peano axioms $(N,S)$ has an order.
I saw in wikipedia that $(N,+)$ is a commutative monoid, but since the naturals have an order structure by construction shouldn't $(N,+)$ be an ordered commutative monoid ?
Thanks
abstract-algebra terminology definition peano-axioms
edited Jul 17 at 15:21
rschwieb
100k1193227
asked Jul 17 at 15:11
kot
497
4
Well, it is indeed.
– Bernard
Jul 17 at 15:13
5
Are you asking this because you think it is mandatory that every set be described by all appellations it has for every structure it has? Then it is also a commutative, ordered semiring. And probably 57 other things. Saying it is one thing does not mean it isn't anything else.
– rschwieb
Jul 17 at 15:16
4
Did you run across a source that said it wasn't an ordered commutative monoid?
– fleablood
Jul 17 at 15:19
1
A dolphin is a carnivorous marine mammal. Is it wrong to say it is a marine mammal? Not completely saying something is what it is is not wrong. Saying something is what it is not is wrong.
– fleablood
Jul 17 at 15:22
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
The set of natural numbers as given from the Peano axioms $(N,S)$ has an order.
I saw in wikipedia that $(N,+)$ is a commutative monoid, but since the naturals have an order structure by construction shouldn't $(N,+)$ be an ordered commutative monoid ?
Thanks
abstract-algebra terminology definition peano-axioms
edited Jul 17 at 15:21
rschwieb
100k1193227
asked Jul 17 at 15:11
kot
497
The set of natural numbers as given from the Peano axioms $(N,S)$ has an order.
I saw in wikipedia that $(N,+)$ is a commutative monoid, but since the naturals have an order structure by construction shouldn't $(N,+)$ be an ordered commutative monoid ?
Thanks
abstract-algebra terminology definition peano-axioms
edited Jul 17 at 15:21
rschwieb
100k1193227
asked Jul 17 at 15:11
kot
497
edited Jul 17 at 15:21
rschwieb
100k1193227
edited Jul 17 at 15:21
rschwieb
100k1193227
edited Jul 17 at 15:21
rschwieb
100k1193227
100k1193227
asked Jul 17 at 15:11
kot
497
asked Jul 17 at 15:11
kot
497
asked Jul 17 at 15:11
kot
497
497
4
Well, it is indeed.
– Bernard
Jul 17 at 15:13
5
Are you asking this because you think it is mandatory that every set be described by all appellations it has for every structure it has? Then it is also a commutative, ordered semiring. And probably 57 other things. Saying it is one thing does not mean it isn't anything else.
– rschwieb
Jul 17 at 15:16
4
Did you run across a source that said it wasn't an ordered commutative monoid?
– fleablood
Jul 17 at 15:19
1
A dolphin is a carnivorous marine mammal. Is it wrong to say it is a marine mammal? Not completely saying something is what it is is not wrong. Saying something is what it is not is wrong.
– fleablood
Jul 17 at 15:22
add a comment |Â
4
Well, it is indeed.
– Bernard
Jul 17 at 15:13
5
Are you asking this because you think it is mandatory that every set be described by all appellations it has for every structure it has? Then it is also a commutative, ordered semiring. And probably 57 other things. Saying it is one thing does not mean it isn't anything else.
– rschwieb
Jul 17 at 15:16
4
Did you run across a source that said it wasn't an ordered commutative monoid?
– fleablood
Jul 17 at 15:19
1
A dolphin is a carnivorous marine mammal. Is it wrong to say it is a marine mammal? Not completely saying something is what it is is not wrong. Saying something is what it is not is wrong.
– fleablood
Jul 17 at 15:22
4
4
Well, it is indeed.
– Bernard
Jul 17 at 15:13
Well, it is indeed.
– Bernard
Jul 17 at 15:13
5
5
Are you asking this because you think it is mandatory that every set be described by all appellations it has for every structure it has? Then it is also a commutative, ordered semiring. And probably 57 other things. Saying it is one thing does not mean it isn't anything else.
– rschwieb
Jul 17 at 15:16
Are you asking this because you think it is mandatory that every set be described by all appellations it has for every structure it has? Then it is also a commutative, ordered semiring. And probably 57 other things. Saying it is one thing does not mean it isn't anything else.
– rschwieb
Jul 17 at 15:16
4
4
Did you run across a source that said it wasn't an ordered commutative monoid?
– fleablood
Jul 17 at 15:19
Did you run across a source that said it wasn't an ordered commutative monoid?
– fleablood
Jul 17 at 15:19
1
1
A dolphin is a carnivorous marine mammal. Is it wrong to say it is a marine mammal? Not completely saying something is what it is is not wrong. Saying something is what it is not is wrong.
– fleablood
Jul 17 at 15:22
A dolphin is a carnivorous marine mammal. Is it wrong to say it is a marine mammal? Not completely saying something is what it is is not wrong. Saying something is what it is not is wrong.
– fleablood
Jul 17 at 15:22
add a comment |Â
3 Answers
3
active
oldest
votes
up vote
11
down vote
accepted
You can have a square and call it a rectangle. There is no inherent problem with that, as long as all you really need from your quadrilateral is that all angles are right.
In the exact same way, you can take an ordered, commutative monoid and call it a commutative monoid.
answered Jul 17 at 15:20
Arthur
98.9k793175
Thanks Arthur, your example made it very clear!!
– kot
Jul 17 at 15:22
Why, as a Math major Arthur, and computer programmer of twenty years (who knows what Haskell and LISP are) is this language of category theory (I think), so opaque to me as to its function? Set theory and Axiom of Choice are pretty much where I draw the line lol..
– Dean Radcliffe
Jul 17 at 18:51
So can i say: 1. $N$ is a set (True) 2. A set has no structure (True). Therefore $N$ has no structure. But $N$ has structure!
– kot
Jul 18 at 1:03
@kot 2. A set does not necessarily have structure. It may have structure arising from some other source. Euclidean space has a huge amount of structure (as a metrizable vector space). But we can also just think of it as a set. In fact, a number of definitions can be phrased: "[term] is a set that also satisfies the following properties ..."
– Teepeemm
Jul 18 at 1:18
@Teepeem So one can think of Euclidean space only as a set therefore without structure or one should not think of Euclidean space only as a set ?
– kot
Jul 18 at 4:53
 |Â
show 1 more comment
up vote
6
down vote
Yes, but you can discuss addition without discussing order.
The same way that you can say that $(mathbb R,+)$ is a group. It is also an abelian group. It is a ring. It is a field. It is an ordered field. It is a module over $mathbb Z$. Etc., etc.
answered Jul 17 at 15:17
Martin Argerami
116k1071164
On a somewhat related note : if there are exactly two cows on a picture, is it mathematically correct to say that there is one cow on the picture?
– Eric Duminil
Jul 17 at 19:18
1
Of course. Otherwise, the sentence "the set of natural numbers contains a natural number" would be false :)
– Martin Argerami
Jul 17 at 19:50
Related: is it true that there are finitely many prime numbers? Yes. (But it is not true that there are only finitely many prime numbers.)
– David
Jul 17 at 23:54
add a comment |Â
up vote
4
down vote
$(N,+)$ is both a commutative monoid and an ordered commutative monoid.
answered Jul 17 at 15:15
user145640
1665
add a comment |Â
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
11
down vote
accepted
You can have a square and call it a rectangle. There is no inherent problem with that, as long as all you really need from your quadrilateral is that all angles are right.
In the exact same way, you can take an ordered, commutative monoid and call it a commutative monoid.
answered Jul 17 at 15:20
Arthur
98.9k793175
Thanks Arthur, your example made it very clear!!
– kot
Jul 17 at 15:22
Why, as a Math major Arthur, and computer programmer of twenty years (who knows what Haskell and LISP are) is this language of category theory (I think), so opaque to me as to its function? Set theory and Axiom of Choice are pretty much where I draw the line lol..
– Dean Radcliffe
Jul 17 at 18:51
So can i say: 1. $N$ is a set (True) 2. A set has no structure (True). Therefore $N$ has no structure. But $N$ has structure!
– kot
Jul 18 at 1:03
@kot 2. A set does not necessarily have structure. It may have structure arising from some other source. Euclidean space has a huge amount of structure (as a metrizable vector space). But we can also just think of it as a set. In fact, a number of definitions can be phrased: "[term] is a set that also satisfies the following properties ..."
– Teepeemm
Jul 18 at 1:18
@Teepeem So one can think of Euclidean space only as a set therefore without structure or one should not think of Euclidean space only as a set ?
– kot
Jul 18 at 4:53
 |Â
show 1 more comment
up vote
11
down vote
accepted
You can have a square and call it a rectangle. There is no inherent problem with that, as long as all you really need from your quadrilateral is that all angles are right.
In the exact same way, you can take an ordered, commutative monoid and call it a commutative monoid.
answered Jul 17 at 15:20
Arthur
98.9k793175
Thanks Arthur, your example made it very clear!!
– kot
Jul 17 at 15:22
Why, as a Math major Arthur, and computer programmer of twenty years (who knows what Haskell and LISP are) is this language of category theory (I think), so opaque to me as to its function? Set theory and Axiom of Choice are pretty much where I draw the line lol..
– Dean Radcliffe
Jul 17 at 18:51
So can i say: 1. $N$ is a set (True) 2. A set has no structure (True). Therefore $N$ has no structure. But $N$ has structure!
– kot
Jul 18 at 1:03
@kot 2. A set does not necessarily have structure. It may have structure arising from some other source. Euclidean space has a huge amount of structure (as a metrizable vector space). But we can also just think of it as a set. In fact, a number of definitions can be phrased: "[term] is a set that also satisfies the following properties ..."
– Teepeemm
Jul 18 at 1:18
@Teepeem So one can think of Euclidean space only as a set therefore without structure or one should not think of Euclidean space only as a set ?
– kot
Jul 18 at 4:53
 |Â
show 1 more comment
up vote
11
down vote
accepted
up vote
11
down vote
accepted
You can have a square and call it a rectangle. There is no inherent problem with that, as long as all you really need from your quadrilateral is that all angles are right.
In the exact same way, you can take an ordered, commutative monoid and call it a commutative monoid.
answered Jul 17 at 15:20
Arthur
98.9k793175
You can have a square and call it a rectangle. There is no inherent problem with that, as long as all you really need from your quadrilateral is that all angles are right.
In the exact same way, you can take an ordered, commutative monoid and call it a commutative monoid.
answered Jul 17 at 15:20
Arthur
98.9k793175
answered Jul 17 at 15:20
Arthur
98.9k793175
answered Jul 17 at 15:20
Arthur
98.9k793175
answered Jul 17 at 15:20
Arthur
98.9k793175
98.9k793175
Thanks Arthur, your example made it very clear!!
– kot
Jul 17 at 15:22
Why, as a Math major Arthur, and computer programmer of twenty years (who knows what Haskell and LISP are) is this language of category theory (I think), so opaque to me as to its function? Set theory and Axiom of Choice are pretty much where I draw the line lol..
– Dean Radcliffe
Jul 17 at 18:51
So can i say: 1. $N$ is a set (True) 2. A set has no structure (True). Therefore $N$ has no structure. But $N$ has structure!
– kot
Jul 18 at 1:03
@kot 2. A set does not necessarily have structure. It may have structure arising from some other source. Euclidean space has a huge amount of structure (as a metrizable vector space). But we can also just think of it as a set. In fact, a number of definitions can be phrased: "[term] is a set that also satisfies the following properties ..."
– Teepeemm
Jul 18 at 1:18
@Teepeem So one can think of Euclidean space only as a set therefore without structure or one should not think of Euclidean space only as a set ?
– kot
Jul 18 at 4:53
 |Â
show 1 more comment
Thanks Arthur, your example made it very clear!!
– kot
Jul 17 at 15:22
Why, as a Math major Arthur, and computer programmer of twenty years (who knows what Haskell and LISP are) is this language of category theory (I think), so opaque to me as to its function? Set theory and Axiom of Choice are pretty much where I draw the line lol..
– Dean Radcliffe
Jul 17 at 18:51
So can i say: 1. $N$ is a set (True) 2. A set has no structure (True). Therefore $N$ has no structure. But $N$ has structure!
– kot
Jul 18 at 1:03
@kot 2. A set does not necessarily have structure. It may have structure arising from some other source. Euclidean space has a huge amount of structure (as a metrizable vector space). But we can also just think of it as a set. In fact, a number of definitions can be phrased: "[term] is a set that also satisfies the following properties ..."
– Teepeemm
Jul 18 at 1:18
@Teepeem So one can think of Euclidean space only as a set therefore without structure or one should not think of Euclidean space only as a set ?
– kot
Jul 18 at 4:53
Thanks Arthur, your example made it very clear!!
– kot
Jul 17 at 15:22
Thanks Arthur, your example made it very clear!!
– kot
Jul 17 at 15:22
Why, as a Math major Arthur, and computer programmer of twenty years (who knows what Haskell and LISP are) is this language of category theory (I think), so opaque to me as to its function? Set theory and Axiom of Choice are pretty much where I draw the line lol..
– Dean Radcliffe
Jul 17 at 18:51
Why, as a Math major Arthur, and computer programmer of twenty years (who knows what Haskell and LISP are) is this language of category theory (I think), so opaque to me as to its function? Set theory and Axiom of Choice are pretty much where I draw the line lol..
– Dean Radcliffe
Jul 17 at 18:51
So can i say: 1. $N$ is a set (True) 2. A set has no structure (True). Therefore $N$ has no structure. But $N$ has structure!
– kot
Jul 18 at 1:03
So can i say: 1. $N$ is a set (True) 2. A set has no structure (True). Therefore $N$ has no structure. But $N$ has structure!
– kot
Jul 18 at 1:03
@kot 2. A set does not necessarily have structure. It may have structure arising from some other source. Euclidean space has a huge amount of structure (as a metrizable vector space). But we can also just think of it as a set. In fact, a number of definitions can be phrased: "[term] is a set that also satisfies the following properties ..."
– Teepeemm
Jul 18 at 1:18
@kot 2. A set does not necessarily have structure. It may have structure arising from some other source. Euclidean space has a huge amount of structure (as a metrizable vector space). But we can also just think of it as a set. In fact, a number of definitions can be phrased: "[term] is a set that also satisfies the following properties ..."
– Teepeemm
Jul 18 at 1:18
@Teepeem So one can think of Euclidean space only as a set therefore without structure or one should not think of Euclidean space only as a set ?
– kot
Jul 18 at 4:53
@Teepeem So one can think of Euclidean space only as a set therefore without structure or one should not think of Euclidean space only as a set ?
– kot
Jul 18 at 4:53
 |Â
show 1 more comment
up vote
6
down vote
Yes, but you can discuss addition without discussing order.
The same way that you can say that $(mathbb R,+)$ is a group. It is also an abelian group. It is a ring. It is a field. It is an ordered field. It is a module over $mathbb Z$. Etc., etc.
answered Jul 17 at 15:17
Martin Argerami
116k1071164
On a somewhat related note : if there are exactly two cows on a picture, is it mathematically correct to say that there is one cow on the picture?
– Eric Duminil
Jul 17 at 19:18
1
Of course. Otherwise, the sentence "the set of natural numbers contains a natural number" would be false :)
– Martin Argerami
Jul 17 at 19:50
Related: is it true that there are finitely many prime numbers? Yes. (But it is not true that there are only finitely many prime numbers.)
– David
Jul 17 at 23:54
add a comment |Â
up vote
6
down vote
Yes, but you can discuss addition without discussing order.
The same way that you can say that $(mathbb R,+)$ is a group. It is also an abelian group. It is a ring. It is a field. It is an ordered field. It is a module over $mathbb Z$. Etc., etc.
answered Jul 17 at 15:17
Martin Argerami
116k1071164
On a somewhat related note : if there are exactly two cows on a picture, is it mathematically correct to say that there is one cow on the picture?
– Eric Duminil
Jul 17 at 19:18
1
Of course. Otherwise, the sentence "the set of natural numbers contains a natural number" would be false :)
– Martin Argerami
Jul 17 at 19:50
Related: is it true that there are finitely many prime numbers? Yes. (But it is not true that there are only finitely many prime numbers.)
– David
Jul 17 at 23:54
add a comment |Â
up vote
6
down vote
up vote
6
down vote
Yes, but you can discuss addition without discussing order.
The same way that you can say that $(mathbb R,+)$ is a group. It is also an abelian group. It is a ring. It is a field. It is an ordered field. It is a module over $mathbb Z$. Etc., etc.
answered Jul 17 at 15:17
Martin Argerami
116k1071164
Yes, but you can discuss addition without discussing order.
The same way that you can say that $(mathbb R,+)$ is a group. It is also an abelian group. It is a ring. It is a field. It is an ordered field. It is a module over $mathbb Z$. Etc., etc.
answered Jul 17 at 15:17
Martin Argerami
116k1071164
answered Jul 17 at 15:17
Martin Argerami
116k1071164
answered Jul 17 at 15:17
Martin Argerami
116k1071164
answered Jul 17 at 15:17
Martin Argerami
116k1071164
116k1071164
On a somewhat related note : if there are exactly two cows on a picture, is it mathematically correct to say that there is one cow on the picture?
– Eric Duminil
Jul 17 at 19:18
1
Of course. Otherwise, the sentence "the set of natural numbers contains a natural number" would be false :)
– Martin Argerami
Jul 17 at 19:50
Related: is it true that there are finitely many prime numbers? Yes. (But it is not true that there are only finitely many prime numbers.)
– David
Jul 17 at 23:54
add a comment |Â
On a somewhat related note : if there are exactly two cows on a picture, is it mathematically correct to say that there is one cow on the picture?
– Eric Duminil
Jul 17 at 19:18
1
Of course. Otherwise, the sentence "the set of natural numbers contains a natural number" would be false :)
– Martin Argerami
Jul 17 at 19:50
Related: is it true that there are finitely many prime numbers? Yes. (But it is not true that there are only finitely many prime numbers.)
– David
Jul 17 at 23:54
On a somewhat related note : if there are exactly two cows on a picture, is it mathematically correct to say that there is one cow on the picture?
– Eric Duminil
Jul 17 at 19:18
On a somewhat related note : if there are exactly two cows on a picture, is it mathematically correct to say that there is one cow on the picture?
– Eric Duminil
Jul 17 at 19:18
1
1
Of course. Otherwise, the sentence "the set of natural numbers contains a natural number" would be false :)
– Martin Argerami
Jul 17 at 19:50
Of course. Otherwise, the sentence "the set of natural numbers contains a natural number" would be false :)
– Martin Argerami
Jul 17 at 19:50
Related: is it true that there are finitely many prime numbers? Yes. (But it is not true that there are only finitely many prime numbers.)
– David
Jul 17 at 23:54
Related: is it true that there are finitely many prime numbers? Yes. (But it is not true that there are only finitely many prime numbers.)
– David
Jul 17 at 23:54
add a comment |Â
up vote
4
down vote
$(N,+)$ is both a commutative monoid and an ordered commutative monoid.
answered Jul 17 at 15:15
user145640
1665
add a comment |Â
up vote
4
down vote
$(N,+)$ is both a commutative monoid and an ordered commutative monoid.
answered Jul 17 at 15:15
user145640
1665
add a comment |Â
up vote
4
down vote
up vote
4
down vote
$(N,+)$ is both a commutative monoid and an ordered commutative monoid.
answered Jul 17 at 15:15
user145640
1665
$(N,+)$ is both a commutative monoid and an ordered commutative monoid.
answered Jul 17 at 15:15
user145640
1665
answered Jul 17 at 15:15
user145640
1665
answered Jul 17 at 15:15
user145640
1665
answered Jul 17 at 15:15
user145640
1665
1665
add a comment |Â
add a comment |Â
Â
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4
Well, it is indeed.
– Bernard
Jul 17 at 15:13
5
Are you asking this because you think it is mandatory that every set be described by all appellations it has for every structure it has? Then it is also a commutative, ordered semiring. And probably 57 other things. Saying it is one thing does not mean it isn't anything else.
– rschwieb
Jul 17 at 15:16
4
Did you run across a source that said it wasn't an ordered commutative monoid?
– fleablood
Jul 17 at 15:19
1
A dolphin is a carnivorous marine mammal. Is it wrong to say it is a marine mammal? Not completely saying something is what it is is not wrong. Saying something is what it is not is wrong.
– fleablood
Jul 17 at 15:22