Re: Cardinality of an Infinite set, which is smaller than Alef 0?
From: Dennis May (dennis_at_NOkernelSPAMthread.e7even.com)
Date: 11/28/04
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Date: Sun, 28 Nov 2004 18:55:25 +0000
Kaligula wrote:
> Without the Axiom of Choice, can anyone prove that Alef 0, is the
> smallest cardinality of an infinite set?
>
It depends what you mean.
On the one hand if a set A has cardinality less than or equal to
aleph_0, then by definition there is an injection f:A -> omega. This
means A is wellorderable. If it is also infinite it must have
cardinality aleph_0.
On the other hand without AC there may be sets which are not comparable
with aleph_0 in cardinality. So aleph_0 is minimal among infinite
cardinals but not necessarily the minimum without AC. It is the minimum
among the cardinals of infinite wellorderable sets.
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