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In this question, I will be working in ZF.

Let $h(\kappa)$ for a cardinal $\kappa$ (not necessarily an ordinal) be the smallest ordinal $\alpha$ such that there is no surjection from a set of size $\kappa$ onto $\alpha$.

$h(\mathfrak{c})$ is defined as $\Theta$.

Let an uncountable ordinal $\alpha$ be a high limit ordinal if and only if for any cardinal $\kappa$ with no surjection onto $\alpha$, there is no surjection of $2^\kappa$ onto $\alpha$.

Equivalently, for any cardinal $\kappa$ such that $h(\kappa)\leq\alpha$, $h(2^\kappa)\leq\alpha$.


Under AC, the high limit ordinals are precisely those which are strong limit cardinals.

Questions: Under AD, what properties does the smallest high limit ordinal have? Is it possible in ZF that there exists no high limit ordinal?

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    $\begingroup$ Using $h$ somehow hints at the name "Hartogs", and this is not the Hartogs number. Instead, you can use the standard notation $\aleph^*$. Also using $\kappa$ and $\alpha$ to denote arbitrary cardinals is again a bit misleading. I wouldn't want to be the one grading your calculus papers with this style, as my guess would be that they are full of "Let $\varepsilon$ be a large natural number, and let $n>0$ be an arbitrarily small real number..." $\endgroup$
    – Asaf Karagila
    Dec 11, 2017 at 6:36
  • $\begingroup$ Also, it might be worth getting familiar with choiceless terminology. I am sick and tired with every second person reinventing the wheel. (Including yours truly.) $\endgroup$
    – Asaf Karagila
    Dec 11, 2017 at 6:40
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    $\begingroup$ @AsafKaragila $\kappa$ is used for any cardinal, whilst $\alpha$ is only used for ordinals. $\endgroup$ Dec 11, 2017 at 7:25
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    $\begingroup$ And that very thing is a root for a mistake. $\endgroup$
    – Asaf Karagila
    Dec 11, 2017 at 7:45

2 Answers 2

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First, let me point out that the notion of "high limit cardinal" is misleading, and perhaps not quite what you want it to be.

The reason is that strong limit cardinals are defined, in the absence of choice, as $\aleph$ numbers $\alpha$, such that for all $\beta<\alpha$, there is no surjection from $V_\beta$ onto $\alpha$.

It is easy to see why a strong limit cardinal is a high limit cardinal. But the converse need not be true. Clearly if there is no map from $\omega$ onto $\alpha$, then there is no map from $V_\omega$ onto $\alpha$; and then by induction we get all the way to $V_{\omega+\omega}$. But there is no reasonable way without choosing surjections to prove that in that case there is no surjection from $V_{\omega+\omega}$ onto $\alpha$ either.


But regardless, $\Theta$ is not a high limit cardinal. Note that there is a surjection from $\mathcal P(\Bbb R)$ onto $\Theta$, simply by mapping $A$ to a pre-well-order it codes, or to $0$ if it doesn't code one.

Therefore it is certainly not a high limit ordinal.

And finally, as to the question about ZF, even without choice there are arbitrarily large strong limit cardinals, and they are all high limit cardinals. The proof is practically the same proof as the usual one. Start with $\alpha_0$, let $\alpha_{n+1}$ be what you denote as $h(V_{\alpha_n})$, and let $\alpha=\sup\alpha_n$. It is easy to see that $\alpha$ is a strong limit cardinal, and therefore a high limit cardinal.

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  • $\begingroup$ I'm getting statements contradicting yours from @NoahSchweber $\endgroup$ Dec 12, 2017 at 2:22
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    $\begingroup$ Yes. And if you look closely, I pointed out the mistake below Noah's answer. $\endgroup$
    – Asaf Karagila
    Dec 12, 2017 at 6:01
  • $\begingroup$ @KeithMillar As Asaf pointed out, my claim was wrong. You should unaccept my answer and accept his. $\endgroup$ Dec 12, 2017 at 13:55
  • $\begingroup$ @AsafKaragila The term I used was misleading, I guess. However, my definition was as I wanted it. What would be a better name for these ordinals? $\endgroup$ Dec 13, 2017 at 0:13
  • $\begingroup$ @Keith: I don't know. Do they really need a name? If pressed against the wall, I would go with a Lindenbaum cardinal. $\endgroup$
    – Asaf Karagila
    Dec 14, 2017 at 10:50
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EDIT: My claim that $\Theta$ is a high limit ordinal is of course nonsense, as Asaf pointed out.

Meanwhile, nothing interesting happens when you drop choice: ZF certainly does prove that high limit ordinals exist. Consider the sequence $(\lambda_\alpha)_{\alpha\in Ord}$ given by

  • $\lambda_0=\omega$.

  • $\lambda_\eta=\sup\{\lambda_\alpha: \alpha<\eta\}$ for $\eta$ a limit.

  • $\lambda_{\alpha+1}=\sup\{\beta\in Ord:$ there is a surjection from $2^{\lambda_\alpha}$ to $\beta\}$ (this is just $h(\lambda_\alpha)$, in your notation).

The existence of $\lambda_\alpha$ for each $\alpha\in Ord$ follows from Replacement, once we know that $\lambda_{\alpha+1}$ exists whenever $\lambda_\alpha$ does. This latter fact might look like it requires choice, but it is in fact provable in ZF (originally I believe by Lindenbaum) using Hartog's theorem: note that if a set $A$ surjects onto an ordinal $\delta$, then $\delta$ injects into the powerset $\mathcal{P}(\alpha)$.

Note that indeed this is exactly the proof that strong limit ordinals exist: the only place we use choice in that argument is in showing that $2^\alpha$ is an ordinal when $\alpha$ is, and that's not necessary here since your definition of $h$ already ensures that the output is an ordinal.

Then for any limit $\eta>0$, $\lambda_\eta$ is by definition a high limit ordinal. And it's easy to see that $\lambda_\omega$ is the least high limit ordinal.

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    $\begingroup$ Note that $\kappa$ denotes an arbitrary cardinal. Not an $\aleph$. So actually $\Theta$ is not a "high limit ordinal", since the continuum is a counterexample. And if not, then its power set is a counterexample. $\endgroup$
    – Asaf Karagila
    Dec 11, 2017 at 6:39
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    $\begingroup$ Note that our answers are in direct contradiction to each other. :) $\endgroup$
    – Asaf Karagila
    Dec 12, 2017 at 9:32
  • $\begingroup$ @AsafKaragila Aargh. $\endgroup$ Dec 12, 2017 at 13:53
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    $\begingroup$ Sorry buddy, but International Talk Like A Pirate day was almost three months ago! $\endgroup$
    – Asaf Karagila
    Dec 12, 2017 at 13:54

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