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I Understanding cofinality

  1. Mar 8, 2016 #1
    if [itex] x= \aleph_{a} [/itex] where a is a limit ordinal. then cf(x)=cf(a)
    why is the cf(x) not eqaul to [itex] \aleph_{a} [/itex]
    is it constructing an order type from the previous cardinals, and using the previous cardinals to construct a sequence
     
  2. jcsd
  3. Mar 8, 2016 #2

    stevendaryl

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    You have to go back to the definition of cofinality of an ordinal. It's slightly confusing, but here's what I think the definition amounts to:
    (I'm going to assume the axiom of choice, because everything gets messier without it)

    Let [itex]A[/itex] be an ordinal (which we can represent as the set of all smaller ordinals). Let [itex]B[/itex] be a proper subset of [itex]A[/itex] (that is, [itex]B[/itex] is a set of ordinals, all of which are smaller than [itex]A[/itex]). Then [itex]B[/itex] is cofinal in [itex]A[/itex] if for every [itex]\alpha < A[/itex], there is a [itex]\beta \epsilon B[/itex] such that [itex]\alpha \leq \beta[/itex]. In other words, [itex]B[/itex] contains arbitrarily large elements of [itex]A[/itex]. So the definition of the cofinality of [itex]A[/itex]: It's the smallest cardinal [itex]\alpha[/itex] such that there is a set [itex]B[/itex] of size [itex]\alpha[/itex] that is cofinal in [itex]A[/itex].

    So a couple of examples: If [itex]n[/itex] is finite ordinal greater than zero, then the cofinality of [itex]n[/itex] is 1. That's because we can let [itex]B[/itex] just be the one-element set [itex]B = \{ n-1 \}[/itex]: If [itex]n' < n[/itex], then [itex]n' \leq n-1[/itex].

    Another example is [itex]\omega[/itex]: the cofinality of [itex]\omega[/itex] is [itex]\omega[/itex]. To see that, let [itex]B[/itex] be any finite set of natural numbers. Then it has a largest element, [itex]max(B)[/itex]. Clearly, this number can't be greater than or equal to every element of [itex]\omega[/itex]. So [itex]B[/itex] is not cofinal in [itex]\omega[/itex]. Turning that around, if [itex]B[/itex] IS cofinal in [itex]\omega[/itex], then [itex]B[/itex] must be infinite, so its cardinality is [itex]\omega[/itex].

    So now, let's look at the case of [itex]\aleph_a[/itex]. If [itex]a[/itex] is a limit ordinal, then we can let [itex]B = \{ \aleph_{a'} | a' < a \}[/itex]. Then [itex]B[/itex] will be cofinal in [itex]\aleph_a[/itex]. So the cofinality of [itex]\aleph_a[/itex] would be less than or equal to the cardinality of [itex]B[/itex], which is just [itex]a[/itex].
     
  4. Mar 15, 2016 #3
    And notice that stevendaryl's argument that ##cf(\aleph_a) \leq |a|## can be slightly modified to show ##cf(\aleph_a) \leq cf(a)##. Indeed, letting ##C## be some cofinal subset of ##a##, one can verify that ##\{\aleph_c: \ c\in C\}## is cofinal in ##\aleph_a##.
     
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