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I am learning set theory and one of my questions was this:

Is the set of fractions between 0 and 1 an infinite set?

I initially thought the answer was yes, but one of this possible answer confused me:

The statement is false; the set of all real numbers between 0 and 1, which includes both fractions and irrational numbers, is an infinite set, but the set of fractions alone is a finite set.

Could someone please help me rationalize whether this is true or not?

Asaf Karagila
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server_unknown
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    The statement in bold is false: the set ${1/1,1/2,1/3,1/4,\dots}$ is a subset of the set of fractions between $0$ and $1$, and it is clearly infinite. Therefore, the set of fractions between $0$ and $1$ is infinite. (The set of irrational numbers between $0$ and $1$ is also infinite.) – Joe Sep 06 '22 at 13:12
  • In general, one way to recognise a set $X$ is infinite, is to ask whether you can construct an injective map $\mathbb{N}\to X$, if so then it has at least the cardinality of the natural numbers – David Sheard Sep 06 '22 at 13:13
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    If you substitute "infinite set" with "uncountable set" and "finite set" with "countable set", then the statement makes sense. – Riccardo Sep 06 '22 at 13:20
  • @Morgan Rodgers ... I am taking Contemporary College Mathematics, and this specific question (and possible answer) was in my Set Theory homework. Like I said, I originally thought the answer was infinite, but this possible answer was so weird, I couldn't wrap my mind around it and it made me doubt everything. :) – server_unknown Sep 06 '22 at 13:55

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