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The following sentence is from my discrete mathematics textbook:

Each rational number has as infinitely many representations as a ratio.

Is this correct? Are there various degrees of infinitely many, or am I misunderstanding?

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It is incorrectly phrased. What would be correct is:

Each rational number has infinitely many distinct representations as a ratio of integers.

At least in American English, I do not think it is as clearly expressed as possible. The meaning is

Each rational number can be represented by any of an infinite number of fractions with integers in the numerator and denominator.

A mathematician might say that

A rational number may have multiple representations, but it can be expressed uniquely in lowest terms as p/q, where q is a positive integer, p is an integer, and p and q share no prime factors.

The idea is that 1/3, 18/54, -12/(-4) are three of the infinitely many representations of the same number that can be expressed most simply as 1/3.

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"Each rational number has as infinitely many representations as a ratio does." There are, indeed, degrees of 'infinitely many', as Georg Cantor (1845-1918), the 'father of set theory' showed. A set of numbers has a cardinality, that is, a number which is the count of its elements (its members). This applies to sets of numbers with 'infinite' members, even if we cannot actually count them. The cardinality of the set of integers (of which there are an infinite number) is the same as that of the set of rational numbers, which, in Cantor's set theory, is called ℵ0 (aleph zero or aleph null). Cantor showed that the set of real numbers, which also has an 'infinite' number of members, has a higher cardinality (there are more of them), (I shall not show how he did that here), which is called ℵ1 (aleph one). This character ℵ is Aleph, the first letter of the Hebrew alphabet.

Cardinality (infinite sets)

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    Your math history is correct (except perhaps for the continuum hypothesis), but the statement asked about is neither good English nor good math. The ratio pi/e is a ratio but not a rational number.
    – Jeff Morrow
    Aug 23, 2020 at 20:01
  • Doesn't the continuum theory say that there is no infinite set whose cardinality is strictly between that of the integers and the real numbers? I thought Cantorian infinite sets were pretty well established; I am convinced by his uncountability proof, and diagonal argument, although I freely admit that I am neither a mathematician nor a historian of science.
    – Michael Harvey
    Aug 23, 2020 at 20:11
  • That is, that there is no cardinal between ℵ0 and ℵ1 by definition, not by CH.
    – Michael Harvey
    Aug 23, 2020 at 20:18
  • Yes, you understand correctly what the continuum hypothesis says. But it cannot be proven. I said "perhaps" because I was not sure whether you intended to imply anything about that hypothesis. See en.wikipedia.org/wiki/Continuum_hypothesis. But you are correct that, except for finitists, who deny the legitimacy of any type of infinity, Cantor's diagonal proof is completely accepted by mathematicians. It is after all a very simple and elegant proof.
    – Jeff Morrow
    Aug 23, 2020 at 20:22
  • "I said "perhaps" because I was not sure whether you intended to imply anything about that hypothesis." - I would be venturing on what would be, for me, very shaky ground if I dared to do that, and not just because it would be a bit off-topic here.
    – Michael Harvey
    Aug 23, 2020 at 20:36

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