Can someone please explain what the term "faithful enumeration" of an infinite set means?
Meaning of the term faithful enumeration
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2In what context did you encounter the term? – 2012-03-23
1 Answers
In my experience it most often means simply a bijective enumeration, one without repetitions. A faithful enumeration of $\Bbb Q$, for instance, would be one of the form $\Bbb Q=\{q_n:n\in\omega\}$, where $q_n\ne q_m$ whenever $n\ne m$, while a faithful enumeration of $\Bbb R$ would be one of the form $\Bbb R=\{x_\xi:\xi<2^\omega\}$, where $x_\xi\ne x_\zeta$ whenever $\xi\ne\zeta$. The domain of the enumeration is typically the cardinality of the set; here that’s $\omega$ in the case of the countably infinite set $\Bbb Q$ and $2^\omega$ in the case of $\Bbb R$.
If the set $S$ already bears a well-ordering $\prec$, the term might possibly be used for a bijective enumeration $S=\{s_\xi:\xi<\alpha\}$, where $\alpha$ is the order type of $\langle S,\prec\rangle$, and $s_\xi\prec s_\zeta$ whenever $\xi<\zeta<\alpha$.
But you’d have to provide a context in order for us to be sure what was intended in the particular case that you’ve encountered.
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0@user6495: If $X$ is an infinite set, then there is an injection from $f:\Bbb N\to X$. Technically this requires (some part of) the axiom of choice, but it’s a routine assumption in most of mathematics. The intuitive idea is simple. Let $f(0)$ be any element of $X$, $f(1)$ be any element of $X\setminus\{f(0)\}$, and so on; since $X$ is infinite, you’re never unable to find a new element to pick. To return to the original question, in this context *faithful enumeration* simply means *injective*. – 2012-03-26