Is it mathematically wrong to say “infinite number”?












3














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?










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  • 5




    "Infinitely many..." is more idiomatic.
    – Lord Shark the Unknown
    42 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
    26 mins ago


















3














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?










share|cite|improve this question


















  • 5




    "Infinitely many..." is more idiomatic.
    – Lord Shark the Unknown
    42 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
    26 mins ago
















3












3








3







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?










share|cite|improve this question













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?







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asked 43 mins ago









WillG

44938




44938








  • 5




    "Infinitely many..." is more idiomatic.
    – Lord Shark the Unknown
    42 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
    26 mins ago
















  • 5




    "Infinitely many..." is more idiomatic.
    – Lord Shark the Unknown
    42 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
    26 mins ago










5




5




"Infinitely many..." is more idiomatic.
– Lord Shark the Unknown
42 mins ago






"Infinitely many..." is more idiomatic.
– Lord Shark the Unknown
42 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
26 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
26 mins ago












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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 '.






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    1 Answer
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    7














    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 '.






    share|cite|improve this answer


























      7














      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 '.






      share|cite|improve this answer
























        7












        7








        7






        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 '.






        share|cite|improve this answer












        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 '.







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        answered 36 mins ago









        Haran

        757317




        757317






























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