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In his book [1], Paul Larson remarks (Remark 1.1.22) that in L there is a function $h:\omega_1\rightarrow\omega_1$ such that for any countable elementary submodel $X$ of $V_\gamma$ (where $\gamma$ is the first strong limit cardinal), we have the order-type of $X\cap Ord$ is strictly less than $h(X\cap\omega_1)$.

What is known about the relationship between the existence of such a function and large cardinal phenomena?

Larson remarks that the existence of such a function is consistent with many large cardinals, and later in the book sketches a result of Velickovic showing that no such function exists in the presence of a precipitous ideal on $\omega_1$. What more is known? Happy to also learn results concerning other related functions, or be pointed to associated references.

[1] Larson, Paul B., The stationary tower. Notes on a course by W. Hugh Woodin, University Lecture Series 32. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3604-8/pbk). x, 132 p. (2004). ZBL1072.03031.

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    $\begingroup$ I don't know if Boban's argument in Larson's book is the same as that in his paper below, but I think it is worth taking a look at least at the introduction of his paper since in his forcing construction he uses some approximations of height functions, which I think resemble things you are looking for. www.logique.jussieu.fr/~boban/pdf/PFA_and_NS.pdf $\endgroup$ – Rahman. M Feb 9 at 13:44
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    $\begingroup$ Seems related to bounding by canonical functions (the example you pointed out in $L$ is a function that is, in a very strong way, NOT bounded by a canonical function). Some relevant sources might be: Deiser-Donder "Canonical functions..."; Schimmerling-Velickovic "Collapsing functions"; Jech-Shelah "A note on countable functions". $\endgroup$ – Sean Cox Feb 10 at 0:57
  • $\begingroup$ Yes, I think the canonical function angle is exactly the sort of thing I was looking for! $\endgroup$ – Todd Eisworth Feb 10 at 1:11
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In the paper "The consistency strength of the perfect set property for universally Baire sets of reals", Ralf Schindler and I show that the negation of a very similar statement is equiconsistent with the existence of what we call a virtually Shelah cardinal.

We say that a cardinal $\kappa$ is virtually Shelah if for every function $f:\kappa \to \kappa$ there is an ordinal $\lambda > \kappa$, a transitive set $M$ with $V_\lambda \subset M$, and a generic elementary embedding $j : V_\lambda \to M$ with critical point $\kappa$ and $j(f)(\kappa) \le \lambda$.

(By "generic elementary embedding" we mean an elementary embedding between structures in $V$ that exists in some generic extension of $V$. By absoluteness it suffices to consider any generic extension in which the domain structure is countable.)

This is a fairly weak large cardinal hypothesis. If $0^\sharp$ exists, then every Silver indiscernible is a virtually Shelah cardinal in $L$. On the other hand, every virtually Shelah cardinal is ineffable and a limit of ineffable cardinals.

By Theorem 1.2 of our paper, the following statements are equiconsistent modulo ZFC:

  1. There is a virtually Shelah cardinal.
  2. Every universally Baire set of reals has the perfect set property.
  3. For every function $f : \omega_1 \to \omega_1$ there is an ordinal $\lambda > \omega_1$ such that for a stationary set of $\sigma \in \mathcal{P}_{\omega_1}(\lambda)$ we have $\sigma \cap \omega_1 \in \omega_1$ and the order type of $\sigma$ is at least $f(\sigma \cap \omega_1)$.

More specifically, we showed:

  • Statement 1 implies that after forcing with $\text{Col}(\omega,\mathord{<}\kappa)$ where $\kappa$ is a virtually Shelah cardinal, statement 2 holds.
  • Statement 2 implies statement 3.
  • Statement 3 implies that $\omega_1^V$ is a virtually Shelah cardinal in $L$.

I would also be interested to hear what else is known about statement 3 or the related statement in your question. In the paper, we only considered statement 3 because it came up naturally as a convenient intermediary between statements 1 and 2.

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    $\begingroup$ In the terminology of the Schimmerling-Velickovic "Collapsing functions" paper mentioned in Sean's comment, statement 3 above becomes "there is no collapsing function for Ord". $\endgroup$ – Trevor Wilson Feb 10 at 2:41
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    $\begingroup$ I’d love to accept both answers, but have to choose one! $\endgroup$ – Todd Eisworth Feb 18 at 15:31
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Not sure if this is what you're interested in, but let me take up the question of what large cardinal axioms are known to be consistent with the existence of $h$. The answer is all large cardinal axioms known to have a canonical inner model. Beyond that, the consistency question remains open: indeed, it remains open whether large cardinal axioms beyond the reach of inner model theory could outright imply the existence of a precipitous ideal on $\omega_1$, and thereby refute the existence of $h$.

It seems plausible, however, that $h$ in fact exists in the canonical inner models yet to be discovered. This is because one can construct a function that is almost exactly like $h$ (see the fourth-to-last paragraph below) using a general condensation principle due to Woodin called Strong Condensation, which likely holds in all canonical inner models. If $\kappa$ is a cardinal, Strong Condensation at $\kappa$ states that there is a surjective function $f : \kappa\to H(\kappa)$ of such that for all $M\prec (H(\kappa),f)$, letting $(H_M,f_M)$ be the transitive collapse of $(M,f)$, $f_M = {f}\restriction H_M$.

All the known canonical inner models satisfy Strong Condensation at their least strong limit cardinal. (Strong condensation cannot hold at any cardinals past the first Ramsey cardinal.) Moreover by a theorem of Woodin, under $\text{AD}^+ + V = L(P(\mathbb R))$, for a Turing cone of reals $x$, $\text{HOD}_x$ satisfies Strong Condensation at its least strong limit cardinal. This heuristically argues that Strong Condensation should hold at the least strong limit cardinal in canonical inner models satisfying arbitrarily strong large cardinal axioms, assuming that such models exist. The reason is that the pattern observed in inner model theory to date suggests that these models should locally resemble the $\text{HOD}$s of determinacy models.

Here is the approximation to the existence of $h$ one gets assuming Strong Condensation at the least strong limit cardinal $\gamma$: There is a set $a\subseteq \omega_1$ and a function $g:\omega_1\to\omega_1$ such that for any $N\prec V_\gamma$ with $a\in N$, $g(N\cap \omega_1) > \text{ot}(N\cap \gamma)$. The rest of this answer consists of a proof of this fact.

Fix $f : \gamma\to H(\gamma)$ witnessing Strong Condensation at $\gamma$. The first step of the proof is cosmetic. One uses a theorem of Woodin which states that $f$ is definable over $H(\gamma)$ from the parameter $f \restriction \omega_1$. Let $a\subseteq \omega_1$ code $f\restriction \omega_1$. (It is an easy exercise to show that $f\restriction \omega_1\in H(\omega_2)$.) Then every $N\prec V_\gamma$ with $a\in N$ has the property that $N\cap H(\gamma)\prec (H(\gamma),f)$.

For $\alpha < \gamma$, let $P_\alpha = f[\alpha]$. Note that the $P_\alpha$ are increasing with union $H(\gamma)$, and if $M\prec (H(\gamma),f)$ and $\text{ot}(M\cap \gamma) = \alpha$, then the transitive collapse of $M$ is equal to $P_\alpha$. The structures $P_\alpha$ will play the role of the $L_\alpha$ hierarchy in Larson's proof.

For every $\xi < \omega_1$, let $g(\xi)$ be the least ordinal $\alpha$ such that there is a surjection from $\omega$ to $\xi$ in $P_\alpha$. Suppose $N\prec V_\gamma$ and $a\in N$. We will show that $g(N\cap \omega_1) > \text{ot}(N\cap \gamma)$. Let $M = N\cap H(\gamma)$, so $M\prec (H(\gamma),<)$. Clearly it suffices to show that $g(M\cap \omega_1) > \text{ot}(M\cap \gamma)$. Let $H_M$ be the transitive collapse of $M$. Then $M\cap \omega_1 = \omega_1^{H_M}$ and letting $\beta = \text{ot}(M\cap \gamma)$, $H_M = P_\beta$. Assume towards a contradiction that $\beta \geq g(\omega_1^{H_M})$. By the definition of $g$, there is a surjection from $\omega$ to $\omega_1^{H_M}$ in $P_\beta$. But since $P_\beta = H_M$, this contradicts that $\omega_1^{H_M}$ is uncountable in $H_M$.

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    $\begingroup$ Thank you for this contribution! $\endgroup$ – Todd Eisworth Feb 10 at 17:45

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