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
44938
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
44938
5
"Infinitely many..." is more idiomatic.
– Lord Shark the Unknown
58 mins ago
2
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
42 mins 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
44938
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
44938
asked 1 hour ago
WillG
44938
asked 1 hour ago
WillG
44938
asked 1 hour ago
WillG
44938
asked 1 hour ago
WillG
44938
44938
5
"Infinitely many..." is more idiomatic.
– Lord Shark the Unknown
58 mins ago
2
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
42 mins ago
add a comment |
5
"Infinitely many..." is more idiomatic.
– Lord Shark the Unknown
58 mins ago
2
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
42 mins ago
5
5
"Infinitely many..." is more idiomatic.
– Lord Shark the Unknown
58 mins ago
"Infinitely many..." is more idiomatic.
– Lord Shark the Unknown
58 mins ago
2
2
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
42 mins 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
42 mins ago
add a comment |
2 Answers
2
active
oldest
votes
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 52 mins ago
Haran
767317
The cardinality of the reals is $aleph_1$ assuming the continuum hypothesis, but it could be much bigger in $mathsf{ZFC}$!
– Alessandro Codenotti
15 mins 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 2 mins ago
Mason
1,9181530
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3053635%2fis-it-mathematically-wrong-to-say-infinite-number%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
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 52 mins ago
Haran
767317
The cardinality of the reals is $aleph_1$ assuming the continuum hypothesis, but it could be much bigger in $mathsf{ZFC}$!
– Alessandro Codenotti
15 mins 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 52 mins ago
Haran
767317
The cardinality of the reals is $aleph_1$ assuming the continuum hypothesis, but it could be much bigger in $mathsf{ZFC}$!
– Alessandro Codenotti
15 mins 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 52 mins ago
Haran
767317
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 52 mins ago
Haran
767317
answered 52 mins ago
Haran
767317
answered 52 mins ago
Haran
767317
answered 52 mins ago
Haran
767317
767317
The cardinality of the reals is $aleph_1$ assuming the continuum hypothesis, but it could be much bigger in $mathsf{ZFC}$!
– Alessandro Codenotti
15 mins ago
add a comment |
The cardinality of the reals is $aleph_1$ assuming the continuum hypothesis, but it could be much bigger in $mathsf{ZFC}$!
– Alessandro Codenotti
15 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
15 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
15 mins 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 2 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 2 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 2 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 2 mins ago
Mason
1,9181530
answered 2 mins ago
Mason
1,9181530
answered 2 mins ago
Mason
1,9181530
answered 2 mins ago
Mason
1,9181530
1,9181530
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3053635%2fis-it-mathematically-wrong-to-say-infinite-number%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
5
"Infinitely many..." is more idiomatic.
– Lord Shark the Unknown
58 mins ago
2
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
42 mins ago