Is it mathematically wrong to say “infinite number”?
I often hear the phrase "an infinite number of..." in mathematics. Is this phrase mathematically ungrammatical, since infinity is not a "number"?
I'm sure some people will say that whether this phrase is right or wrong is a matter of opinion or taste, or that if people are saying it and know what it means, then who's to say it's wrong. Fine. I'm just curious about how most mathematicians feel about it. Could it lead to conceptual pitfalls? Or should the most modern and broad notion of "number" include "infinity" as a member?
terminology
asked 1 hour ago
WillG
45438
add a comment |
I often hear the phrase "an infinite number of..." in mathematics. Is this phrase mathematically ungrammatical, since infinity is not a "number"?
I'm sure some people will say that whether this phrase is right or wrong is a matter of opinion or taste, or that if people are saying it and know what it means, then who's to say it's wrong. Fine. I'm just curious about how most mathematicians feel about it. Could it lead to conceptual pitfalls? Or should the most modern and broad notion of "number" include "infinity" as a member?
terminology
asked 1 hour ago
WillG
45438
6
"Infinitely many..." is more idiomatic.
– Lord Shark the Unknown
1 hour ago
3
When people say that infinite is not a number, it simply means that there is no way to reconcile the naive notion of 'positive/negative infinity' and the algebraic rules over the real numbers. In other words, for other purposes one can actually come up with rigorous theories dealing with infinities. Infinite cardinals are quintessential examples.
– Sangchul Lee
1 hour ago
add a comment |
I often hear the phrase "an infinite number of..." in mathematics. Is this phrase mathematically ungrammatical, since infinity is not a "number"?
I'm sure some people will say that whether this phrase is right or wrong is a matter of opinion or taste, or that if people are saying it and know what it means, then who's to say it's wrong. Fine. I'm just curious about how most mathematicians feel about it. Could it lead to conceptual pitfalls? Or should the most modern and broad notion of "number" include "infinity" as a member?
terminology
asked 1 hour ago
WillG
45438
I often hear the phrase "an infinite number of..." in mathematics. Is this phrase mathematically ungrammatical, since infinity is not a "number"?
I'm sure some people will say that whether this phrase is right or wrong is a matter of opinion or taste, or that if people are saying it and know what it means, then who's to say it's wrong. Fine. I'm just curious about how most mathematicians feel about it. Could it lead to conceptual pitfalls? Or should the most modern and broad notion of "number" include "infinity" as a member?
terminology
terminology
asked 1 hour ago
WillG
45438
asked 1 hour ago
WillG
45438
asked 1 hour ago
WillG
45438
asked 1 hour ago
WillG
45438
asked 1 hour ago
WillG
45438
45438
6
"Infinitely many..." is more idiomatic.
– Lord Shark the Unknown
1 hour ago
3
When people say that infinite is not a number, it simply means that there is no way to reconcile the naive notion of 'positive/negative infinity' and the algebraic rules over the real numbers. In other words, for other purposes one can actually come up with rigorous theories dealing with infinities. Infinite cardinals are quintessential examples.
– Sangchul Lee
1 hour ago
add a comment |
6
"Infinitely many..." is more idiomatic.
– Lord Shark the Unknown
1 hour ago
3
When people say that infinite is not a number, it simply means that there is no way to reconcile the naive notion of 'positive/negative infinity' and the algebraic rules over the real numbers. In other words, for other purposes one can actually come up with rigorous theories dealing with infinities. Infinite cardinals are quintessential examples.
– Sangchul Lee
1 hour ago
6
6
"Infinitely many..." is more idiomatic.
– Lord Shark the Unknown
1 hour ago
"Infinitely many..." is more idiomatic.
– Lord Shark the Unknown
1 hour ago
3
3
When people say that infinite is not a number, it simply means that there is no way to reconcile the naive notion of 'positive/negative infinity' and the algebraic rules over the real numbers. In other words, for other purposes one can actually come up with rigorous theories dealing with infinities. Infinite cardinals are quintessential examples.
– Sangchul Lee
1 hour ago
When people say that infinite is not a number, it simply means that there is no way to reconcile the naive notion of 'positive/negative infinity' and the algebraic rules over the real numbers. In other words, for other purposes one can actually come up with rigorous theories dealing with infinities. Infinite cardinals are quintessential examples.
– Sangchul Lee
1 hour ago
add a comment |
2 Answers
2
active
oldest
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Infinity is not a number, as you said. However, it is a cardinality. For example, the cardinality of the naturals is $aleph_0$ and the cardinality of the reals is $aleph_1$ , basically, different sizes of infinity. Thus, although infinity cannot be considered a number, it can be used as a measure of size. Hence, it is mathematically and grammatically correct to use phrases such as 'infinitely many', or 'an infinite number of '.
answered 1 hour ago
Haran
775318
2
The cardinality of the reals is $aleph_1$ assuming the continuum hypothesis, but it could be much bigger in $mathsf{ZFC}$!
– Alessandro Codenotti
36 mins ago
@AlessandroCodenotti (+1) for that, but the argument remains unaffected.
– Haran
43 secs ago
add a comment |
I think "an infinite number of _____" is a fine use but here's a maybe far fetched potential conceptual pitfall.
There is an infinite number of ways to answer the posed question. In fact (in some sense, because I am limited to using symbols from a finite alphabet), there is $aleph_0$. That is, there is countably many different answers one could type up.
And if you sequence the rational numbers $q_1,q_2,dots $ then this sequence has an infinite number of subsequential limit points. In fact, there are continuum such limit points.
A potential pitfall would be to think that these are same. Just because there is an infinite number of ways of doing $X$ and infinite number of ways of doing $Y$ doesn't mean there is the same number of ways of doing $X$ as there are ways of doing $Y$.
Anyway: I think this pitfall is somewhat far fetched but it's definitely worth keeping in mind that the rules that obvious to us when dealing with number do not apply to infinities.
"If there $z$ ways of doing $X$ and there $z$ ways of doing $Y$ then there are the same number of ways of doing $X$ as there are ways of doing $Y$. Is a perfectly valid conclusion when we are working with $z=17$... but not when $z$ is infinite.
answered 24 mins ago
Mason
1,9181530
add a comment |
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2 Answers
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2 Answers
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active
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Infinity is not a number, as you said. However, it is a cardinality. For example, the cardinality of the naturals is $aleph_0$ and the cardinality of the reals is $aleph_1$ , basically, different sizes of infinity. Thus, although infinity cannot be considered a number, it can be used as a measure of size. Hence, it is mathematically and grammatically correct to use phrases such as 'infinitely many', or 'an infinite number of '.
answered 1 hour ago
Haran
775318
2
The cardinality of the reals is $aleph_1$ assuming the continuum hypothesis, but it could be much bigger in $mathsf{ZFC}$!
– Alessandro Codenotti
36 mins ago
@AlessandroCodenotti (+1) for that, but the argument remains unaffected.
– Haran
43 secs ago
add a comment |
Infinity is not a number, as you said. However, it is a cardinality. For example, the cardinality of the naturals is $aleph_0$ and the cardinality of the reals is $aleph_1$ , basically, different sizes of infinity. Thus, although infinity cannot be considered a number, it can be used as a measure of size. Hence, it is mathematically and grammatically correct to use phrases such as 'infinitely many', or 'an infinite number of '.
answered 1 hour ago
Haran
775318
2
The cardinality of the reals is $aleph_1$ assuming the continuum hypothesis, but it could be much bigger in $mathsf{ZFC}$!
– Alessandro Codenotti
36 mins ago
@AlessandroCodenotti (+1) for that, but the argument remains unaffected.
– Haran
43 secs ago
add a comment |
Infinity is not a number, as you said. However, it is a cardinality. For example, the cardinality of the naturals is $aleph_0$ and the cardinality of the reals is $aleph_1$ , basically, different sizes of infinity. Thus, although infinity cannot be considered a number, it can be used as a measure of size. Hence, it is mathematically and grammatically correct to use phrases such as 'infinitely many', or 'an infinite number of '.
answered 1 hour ago
Haran
775318
Infinity is not a number, as you said. However, it is a cardinality. For example, the cardinality of the naturals is $aleph_0$ and the cardinality of the reals is $aleph_1$ , basically, different sizes of infinity. Thus, although infinity cannot be considered a number, it can be used as a measure of size. Hence, it is mathematically and grammatically correct to use phrases such as 'infinitely many', or 'an infinite number of '.
answered 1 hour ago
Haran
775318
answered 1 hour ago
Haran
775318
answered 1 hour ago
Haran
775318
answered 1 hour ago
Haran
775318
775318
2
The cardinality of the reals is $aleph_1$ assuming the continuum hypothesis, but it could be much bigger in $mathsf{ZFC}$!
– Alessandro Codenotti
36 mins ago
@AlessandroCodenotti (+1) for that, but the argument remains unaffected.
– Haran
43 secs ago
add a comment |
2
The cardinality of the reals is $aleph_1$ assuming the continuum hypothesis, but it could be much bigger in $mathsf{ZFC}$!
– Alessandro Codenotti
36 mins ago
@AlessandroCodenotti (+1) for that, but the argument remains unaffected.
– Haran
43 secs ago
2
2
The cardinality of the reals is $aleph_1$ assuming the continuum hypothesis, but it could be much bigger in $mathsf{ZFC}$!
– Alessandro Codenotti
36 mins ago
The cardinality of the reals is $aleph_1$ assuming the continuum hypothesis, but it could be much bigger in $mathsf{ZFC}$!
– Alessandro Codenotti
36 mins ago
@AlessandroCodenotti (+1) for that, but the argument remains unaffected.
– Haran
43 secs ago
@AlessandroCodenotti (+1) for that, but the argument remains unaffected.
– Haran
43 secs ago
add a comment |
I think "an infinite number of _____" is a fine use but here's a maybe far fetched potential conceptual pitfall.
There is an infinite number of ways to answer the posed question. In fact (in some sense, because I am limited to using symbols from a finite alphabet), there is $aleph_0$. That is, there is countably many different answers one could type up.
And if you sequence the rational numbers $q_1,q_2,dots $ then this sequence has an infinite number of subsequential limit points. In fact, there are continuum such limit points.
A potential pitfall would be to think that these are same. Just because there is an infinite number of ways of doing $X$ and infinite number of ways of doing $Y$ doesn't mean there is the same number of ways of doing $X$ as there are ways of doing $Y$.
Anyway: I think this pitfall is somewhat far fetched but it's definitely worth keeping in mind that the rules that obvious to us when dealing with number do not apply to infinities.
"If there $z$ ways of doing $X$ and there $z$ ways of doing $Y$ then there are the same number of ways of doing $X$ as there are ways of doing $Y$. Is a perfectly valid conclusion when we are working with $z=17$... but not when $z$ is infinite.
answered 24 mins ago
Mason
1,9181530
add a comment |
I think "an infinite number of _____" is a fine use but here's a maybe far fetched potential conceptual pitfall.
There is an infinite number of ways to answer the posed question. In fact (in some sense, because I am limited to using symbols from a finite alphabet), there is $aleph_0$. That is, there is countably many different answers one could type up.
And if you sequence the rational numbers $q_1,q_2,dots $ then this sequence has an infinite number of subsequential limit points. In fact, there are continuum such limit points.
A potential pitfall would be to think that these are same. Just because there is an infinite number of ways of doing $X$ and infinite number of ways of doing $Y$ doesn't mean there is the same number of ways of doing $X$ as there are ways of doing $Y$.
Anyway: I think this pitfall is somewhat far fetched but it's definitely worth keeping in mind that the rules that obvious to us when dealing with number do not apply to infinities.
"If there $z$ ways of doing $X$ and there $z$ ways of doing $Y$ then there are the same number of ways of doing $X$ as there are ways of doing $Y$. Is a perfectly valid conclusion when we are working with $z=17$... but not when $z$ is infinite.
answered 24 mins ago
Mason
1,9181530
add a comment |
I think "an infinite number of _____" is a fine use but here's a maybe far fetched potential conceptual pitfall.
There is an infinite number of ways to answer the posed question. In fact (in some sense, because I am limited to using symbols from a finite alphabet), there is $aleph_0$. That is, there is countably many different answers one could type up.
And if you sequence the rational numbers $q_1,q_2,dots $ then this sequence has an infinite number of subsequential limit points. In fact, there are continuum such limit points.
A potential pitfall would be to think that these are same. Just because there is an infinite number of ways of doing $X$ and infinite number of ways of doing $Y$ doesn't mean there is the same number of ways of doing $X$ as there are ways of doing $Y$.
Anyway: I think this pitfall is somewhat far fetched but it's definitely worth keeping in mind that the rules that obvious to us when dealing with number do not apply to infinities.
"If there $z$ ways of doing $X$ and there $z$ ways of doing $Y$ then there are the same number of ways of doing $X$ as there are ways of doing $Y$. Is a perfectly valid conclusion when we are working with $z=17$... but not when $z$ is infinite.
answered 24 mins ago
Mason
1,9181530
I think "an infinite number of _____" is a fine use but here's a maybe far fetched potential conceptual pitfall.
There is an infinite number of ways to answer the posed question. In fact (in some sense, because I am limited to using symbols from a finite alphabet), there is $aleph_0$. That is, there is countably many different answers one could type up.
And if you sequence the rational numbers $q_1,q_2,dots $ then this sequence has an infinite number of subsequential limit points. In fact, there are continuum such limit points.
A potential pitfall would be to think that these are same. Just because there is an infinite number of ways of doing $X$ and infinite number of ways of doing $Y$ doesn't mean there is the same number of ways of doing $X$ as there are ways of doing $Y$.
Anyway: I think this pitfall is somewhat far fetched but it's definitely worth keeping in mind that the rules that obvious to us when dealing with number do not apply to infinities.
"If there $z$ ways of doing $X$ and there $z$ ways of doing $Y$ then there are the same number of ways of doing $X$ as there are ways of doing $Y$. Is a perfectly valid conclusion when we are working with $z=17$... but not when $z$ is infinite.
answered 24 mins ago
Mason
1,9181530
answered 24 mins ago
Mason
1,9181530
answered 24 mins ago
Mason
1,9181530
answered 24 mins ago
Mason
1,9181530
1,9181530
add a comment |
add a comment |
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6
"Infinitely many..." is more idiomatic.
– Lord Shark the Unknown
1 hour ago
3
When people say that infinite is not a number, it simply means that there is no way to reconcile the naive notion of 'positive/negative infinity' and the algebraic rules over the real numbers. In other words, for other purposes one can actually come up with rigorous theories dealing with infinities. Infinite cardinals are quintessential examples.
– Sangchul Lee
1 hour ago