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Inspiration for the following question comes from an exercise in Spivak's Calculus, there too are considered finite sets of real numbers in interval $[0,1]$ but in completely different setting. I will state formulation of the question and my attempt to solve it. I should note that all my knowledge of set theory mostly comes from reading wikipedia, math.SE, and some abstract algebra textbook introductions, so I have no "real" knowledge of it.


Consider collection of sets $A_i$ for every natural number $i$ such that every set in that collection contains a finite amount of real numbers in $[0,1]$. Then the question is: what is the cardinality of set $C = \bigcup_{i=1}^{\infty} A_i$


My intuition says that it is equal to cardinality of integers. My first thoughts of bijection were bijecting all the elements in sets with particular subsets of rational numbers, but then I thought of way that almost seems too easy: Biject first $|A_1|$ natural numbers with elements of $A_1$, biject elements of $A_2$ with next $|A_2|$ numbers, in general, biject elements of $A_n$ to natural numbers from $k = \sum\limits_{i=1}^{n-1}|A_i| + 1$ to $k + |A_n|$. Does this resolve the issue?

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    @joriki: Well, I used the word 'real' in quotation marks in lieu of a better word, but what I was trying to convey was that my knowledge isn't complete even at elementary level. If it was, I guess I would have had more confidence in my result.2011-05-29

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The fact that the elements of $A_i$ are real numbers in $[0,1]$ is irrelevant. You only need to know that each $A_i$ is finite.
Also, repetitions are easy to handle: if an element of $A_i$ has already occurred in $A_j$ for some $j < i$, just use the value that you already gave it.
Finally, the union of any countable collection of countable (not just finite) sets is also countable. But this requires the axiom of (countable) choice.

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    You need choice in general to show that a countable union of finite sets is countable. This is not necessary here, though, since the ordering of $\mathbb{R}$ gives explicit well-orderings on each finite set.2011-05-30