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I'm having trouble understanding the statement:

If $\kappa$ is a cardinal, $\aleph$ is any $\aleph$-number, and if $\kappa\leq\aleph$ then $\kappa$ can be well ordered as well.

I understand the concept of well-ordering a set of cardinal numbers (any set of cardinal numbers is in fact well-ordered by the relation $\le$) but I'm not sure if I have a proper understanding of what exactly it means to well-order an individual cardinal number.

I know that to each cardinal number there is associated a unique ordinal number and an ordinal number is set consisting of ordinal numbers strictly less than it so it's well-ordered and so to say $\kappa$ is well-ordered, is all that we mean is that the ordinal number that this cardinal numbers is associated to is well-ordered?

If what I'm saying is correct, aren't all cardinal well-ordered? What do we need this condition for then: $\kappa \le \aleph$ (where $\aleph$ is a well-orderable cardinal)?

Sorry if I'm confusing you too.

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Are you possibly working without the assumption of the Axiom of Choice? –  Arthur Fischer Aug 24 '12 at 11:46
    
@Arthur: I don't really know. I'm getting this from here: math.stackexchange.com/questions/56466/… in the proof we're not assuming the AC holds so I guess working without it. –  Mark Aug 24 '12 at 11:54
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If you're asking about a detail in some previous question, you should say so in your question. –  Chris Eagle Aug 24 '12 at 12:00
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1 Answer 1

up vote 5 down vote accepted

Assuming the axiom of choice cardinals are exactly initial ordinals, that is an ordinal which is not bijectible with any of its members.

However without the axiom of choice there might be non well-orderable sets. For non well-orderable sets we define cardinality in a slightly different way. As it turns out cardinals are partially ordered in ZF, and the fact that they are totally ordered is equivalent to AC.

The partial order is defined by injections between sets, and it is not hard to see that if cardinality is invariant under bijections then the partial order over the sets translates well to a partial ordering of the cardinals.

When we say that $\kappa$ is a well-ordered cardinal we mean that $\kappa$ is an ordinal. When we say that $\kappa$ is a general cardinal we mean that it represents the cardinality of an arbitrary set, which may not be well-orderable.

To say that $\kappa\leq\aleph_\alpha$ for some $\alpha$, is to say that every set $K$ such that $|K|=\kappa$ has an injection into the $\alpha$-th initial ordinal. If a set $K$ has an injection into an ordinal it inherits the well-ordering of the ordinal, namely:

If $f\colon K\to\alpha$ is an injective function into an ordinal then $x\prec y\iff f(x)\in f(y)$ is a well-ordering of $K$.


To summarize, to say that $\kappa$ is an arbitrary cardinal and $\kappa\leq\aleph$ for some $\aleph$ number, is to say that any set of cardinality $\kappa$ can in fact be well-ordered, therefore by the definition of cardinals $\kappa$ itself is an ordinal.

To read more:

  1. Non-aleph infinite cardinals
  2. Defining cardinality in the absence of choice
  3. There's non-Aleph transfinite cardinals without the axiom of choice?
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Thank you very much for your help. –  Mark Aug 24 '12 at 13:48
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