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  1. Every countable set has cardinality $\aleph_0$.
  2. The next larger cardinality is $\aleph_1$.
  3. Every uncountable set has cardinality $\geq 2^{\aleph_0}$

Now, an infinite set can only be countable or uncountable, so how does this concept not negate the possible existence of a set $S$ such that $\aleph_0<|S|<2^{\aleph_0}?$

I am guessing the issue is that claim $(3)$ is in fact not necessarily true. If so, I would be glad to hear more on why this is not the case. I've only started looking into this area of mathematics recently, so pardon me if my question is naive.

Asaf Karagila
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Lavender Lullaby
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2 Answers2

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The word "uncountable" should be seen as equivalent to "not countable." So any infinite set that can't be put into bijection with $\Bbb{N}$ is uncountable, regardless of anything involving CH.

  • Thank you, that was quick. So if I understand correctly, could one reformulate CH as "the smallest uncountable set has cardinality $2^{\aleph_0}?"$ – Lavender Lullaby Sep 16 '13 at 04:35
  • Yes, exactly.${}$ – TonyK Sep 16 '13 at 06:29
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    Or rather, not quite. "The smallest uncountable set" is not well-defined, because there are many such. So you have to say something like: "Every uncountable set has cardinality at least $2^{\aleph_0}$." – TonyK Sep 16 '13 at 17:19
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When we say that a set is finite if there is a bijection between the set and a proper initial segment of the natural numbers. We say that a set is infinite if it is not finite.

Similarly, we say that a set is countable if there is an injection from that set into the set of natural numbers. We say that it is uncountable if it is not countable. That is all.

The continuum hypothesis is a statement about a particular set and its cardinality. It has nothing to do with the definition of countable, uncountable or cardinals in general.

Asaf Karagila
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  • "Nothing to do with"?! – TonyK Sep 16 '13 at 17:21
  • Tony, you seem upset. – Asaf Karagila Sep 16 '13 at 17:29
  • Surprised, is all. – TonyK Sep 16 '13 at 18:21
  • I mean, sure it's about the integers and their power set. But it doesn't have much to do with the definition of countable and uncountable. That is all. – Asaf Karagila Sep 16 '13 at 18:36
  • "I mean, sure it's about the integers [a countable set] and their power set [an uncountable set]. But it doesn't have much to do with the definition of countable and uncountable." I see that you have changed your position from "nothing" to "not much"! But I still disagree $-$ I think these concepts are inescapably interrelated. – TonyK Sep 16 '13 at 20:02
  • Yes. That is what I meant. And I still don't see where the definition of countable and uncountable come in here. – Asaf Karagila Sep 16 '13 at 20:06