# Does the set of all ordinals strictly dominated by a given set exist in ZF?

How do I prove that

$$\{\alpha\in\mathsf{On}\,|\,\alpha\prec A\}\in V,$$

assuming $A\in V$? I know that if AC is assumed, this set is equal to $\mbox{card}(A):=\mu_\alpha(\alpha\approx A)$, and hence it exists. But in the absence of the AC, we allow the possibility of sets such that $\forall\alpha\in\mathsf{On}\ \,\alpha\not\approx A$, but it seems that if $\forall\alpha\in\mathsf{On}\ \,\alpha\prec A$, there should be some argument to prove that $A$ is "too big" to be a set. Side note: would it be possible to prove in this case that $\mathsf{On}\approx A$?

Notation notes: $A\approx B$ means that there is a bijection from $A$ to $B$, and $A\preccurlyeq B$ means there is an injection from $A$ to $B$. $\mathsf{On}$ is the proper class of ordinal numbers, and $\mu_\alpha(P)$ is the smallest ordinal $\alpha$ satisfying $P$, or $\emptyset$ if none exist.

The set $\{x\in\mathsf{On}\,|\,x\prec A\}$ (which is an ordinal) is actually a very interesting set in ZF, because it shares many properties with $\mbox{card}(A)$, but for classes which are not equinumerous to any ordinal, $\mbox{card}(A)=\emptyset$, but it can be proved by finite induction that $\omega\subseteq\{x\in\mathsf{On}\,|\,x\prec A\}$ for any such class, so in some sense it doesn't "give up" as easily as the conventional definition.

Edit: For clarification, I want to prove that $\{\alpha\in\mathsf{On}\,|\,\alpha\prec A\}\ne\mathsf{On}$, which is equivalent to $\{\alpha\in\mathsf{On}\,|\,\alpha\prec A\}\in V$, as well as to $\exists\alpha\in\mathsf{On}\ \,\alpha\not\prec A$.

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The existence of the set that you have presented is an immediate consequence of the result that a Hartogs Number exists for any set $A$.

Theorem (ZF) Let $\alpha$ be the class of all ordinals that inject into $A$. Then $\alpha$ is an ordinal that does not inject into $A$.

Proof:

• By the Power-Set Axiom, the classes $\mathcal{P}(A)$, $A \times A$ and $\mathcal{P}(A \times A)$ are sets.

• Let $\mathcal{B}$ be the class of subsets of $A$ that can be well-ordered. As $\mathcal{B}$ is a definable subclass of $\mathcal{P}(A)$, it is a set by the Axiom Schema of Separation.

• For each $B \in \mathcal{B}$, let $W(B)$ denote the class of all well-orderings on $B$. As $W(B)$ is a definable subclass of $\mathcal{P}(A \times A)$, it is a set by the Axiom Schema of Separation.

• For each $B \in \mathcal{B}$ and each $R \in W(B)$, the well-ordered set $(B,R)$ is order-isomorphic to a unique ordinal $\beta$. Clearly, $\beta$ must inject into $A$, so $\beta \in \alpha$.

• By the definition of $\alpha$, any $\beta \in \alpha$ is equipollent to some $B \in \mathcal{P}(A)$. The well-ordering on $\beta$ then induces a well-ordering $R$ on $B$.

• By the Power-Set Axiom, the class $\mathcal{P}(A) \times \mathcal{P}(A \times A)$ is a set. As $W := \{ (B,R) ~|~ B \in \mathcal{P}(A) \land R \in W(B) \}$ is a definable subclass of $\mathcal{P}(A) \times \mathcal{P}(A \times A)$, it follows from the Axiom Schema of Separation that $W$ is a set.

• By the foregoing argument, there exists a surjective class-function $f$ from $W$ to $\alpha$.

• As $W$ is a set and $\alpha = \text{Range}(f)$, it follows from the Axiom Schema of Replacement that $\alpha$ is a set.

• We shall now prove that $\alpha$ is an ordinal.

• As $\alpha$ is a set of ordinals, it is automatically well-ordered.

• Next, suppose that $\gamma \in \beta \in \alpha$. Then as $\beta$ is an ordinal, its transitivity yields $\gamma \subseteq \beta$.

• As $\beta \in \alpha$, we see that $\beta$ injects into $A$. Let $\phi: \beta \to A$ be an injection.

• The restriction of $\phi$ to $\gamma$ is an injective class-function from $\gamma$ to $A$. This restriction is a definable subclass of $\phi$, so it is an injective set-function by the Axiom Schema of Separation.

• Hence, $\gamma \in \alpha$.

• Therefore, $\alpha$ is transitive, which implies that it is an ordinal.

• Finally, $\alpha$ cannot inject into $A$, otherwise $\alpha \in \alpha$, which is a contradiction. $\quad \spadesuit$

Conclusion: The class $\alpha := \{ \beta ~|~ \text{ON}(\beta) \land (\beta \prec A) \}$ is a set (in fact, an ordinal).

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And how would one go about proving that $\exists\beta\,\forall\alpha\,(\alpha\prec A\to\alpha<\beta)$? I agree that having proven this, the problem would be solved. – Mario Carneiro Jan 14 '13 at 3:57
By Hartogs’ result, there exists an ordinal $\beta$ such that $\beta$ does not inject into $A$. Now, for $\alpha \prec A$, we have (i) $\alpha < \beta$, (ii) $\alpha = \beta$ or (iii) $\alpha > \beta$. The last two cases are impossible because they imply that $\beta$ injects into $A$. Therefore, we are left with $\alpha < \beta$. – Haskell Curry Jan 14 '13 at 4:12
The whole point of the question is to prove Hartogs result. Note that if $\alpha$ injects into $A$, then there is a subset $B$ of $A$ and a binary relation $<$ on $B$ such that $(B,<)$ is isomorphic to $\alpha$. The collection of all binary relations $<$ defined on some subset of $A$ that define a well-ordering is a set (by separation), and by replacement, the collection of associated ordinals (the class you are asking about) is a set. This use of replacement is essential. – Andrés E. Caicedo Jan 14 '13 at 4:13
The Hartogs number is the keyword I was looking for. I'll accept this answer if I can complete the proof. (I am working in a strict formal system, so the proof will need some work.) us.metamath.org/mpeuni/mmset.html – Mario Carneiro Jan 14 '13 at 4:18
@MarioCarneiro: Sure! I’m glad that I could point you in the right direction. – Haskell Curry Jan 14 '13 at 4:19

First note that every bounded class of ordinals is a set. This is trivial using the subset schema applied to any upper bound of the class. Here is a nice proof as for why such classes classes of ordinals cannot be a proper class:

Assume by contradiction that $A$ is a set such that $\{\alpha\in\mathsf{Ord}\mid\alpha\prec A\}$ is not a set. Then for every initial ordinal $\omega_\alpha$ there is a subset of $A$ of cardinality $\omega_\alpha$. Using the power set axiom there is a set $\mathcal P(A)$. We define the following function: $$f(B)=\begin{cases}\omega_\alpha & \exists\omega_\alpha\sim B\\ 0 &\text{otherwise.}\end{cases}$$

This function is well-defined because every $B$ which can be well-ordered is equipotent with exactly one initial ordinal. Using the replacement schema we know that the range of $f$ (when applied to the set $\mathcal P(A)$) is a set. But this is a contradiction because this collection should contain all the initial ordinals which is a proper class.

There are other ways of proving this class is a set, for example by exhibiting an injection from it into $\mathcal{P(P(P(}A)))$. But I like this way because it's shorter, in some sense.

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That is a very clever setup, and looks to be easier to implement than the Hartogs proof. However, the necessary machinery for the actual proof may be more complicated than your second suggestion, of an injection into $\mathcal{P(P(P(}A)))$. What exactly were you thinking? (My head starts to spin past a single-level power set, and I have no intuition of the properties of the third iteration power set at all, except to note that it's damn big.) – Mario Carneiro Jan 14 '13 at 9:23
@Mario: This is an actual proof. If one wants to show an injection from the Hartogs number into some set, one needs to do so in three iterations of the power set. It is easier to write an injection into four iterations, actually. You just consider chains of subsets which are of order type $\alpha$, and since you can't really choose between them you map $\alpha$ into the collection of all chains in $\mathcal P(A)$ whose order type in $\subseteq$ is $\alpha$. This is three iterations. It was proved that you cannot do this in two iterations for every set (although sometimes you can). – Asaf Karagila Jan 14 '13 at 12:53
I I have no problem with the claim that your proof is a real proof in that it gives all necessary information for a mathematician to understand it (and I do understand it), but as I mentioned in another comment, I am working in a strict formal system, and no details can be omitted. – Mario Carneiro Jan 14 '13 at 13:53
@Mario: I'm not sure what details needs to be added, perhaps maybe explicitly write down the formula defining $f$. And besides, if you understand the proof you should write it in your formal system... – Asaf Karagila Jan 14 '13 at 14:22
I didn't mean to suggest that any more work is necessary on your part, and you have certainly done very well with this proof. It is merely full disclosure on my part that the criteria I am using for the "best proof" is the simplicity with which it can actually be written down in symbols and proven through formal manipulations. Of course I intend to do the proof myself, and I will post it here when it is done. I think I will accept your answer because it brings more original information to the question, though, and it is a fine proof. – Mario Carneiro Jan 14 '13 at 18:09