# What are Moschovakis cardinals?

The question is exactly that of the title: what are Moschovakis cardinals?

Background. In a recent answer to the question, "Are there examples of statements that have been proven whose consistency proofs came before their proofs?," user14111 posted (Are there examples of statements that have been proven whose consistency proofs came before their proofs?) an answer involving "Moschovakis cardinals," a large cardinal notion which was shown to be inconsistent at some point in time. Now, googling for Moschovakis cardinals reveals nothing besides that answer and this (http://mathforum.org/kb/thread.jspa?forumID=13&threadID=22263&messageID=59655#59655) Math Forum Discussion post, which seems(?) to be responding to a post which was then deleted.

According to user14111, the notion of a Moschovakis cardinal arose in an unpublished manuscript circulated around the late 60s; given the timing, my current guess is that "Moschovakis cardinal" is just a synonym for "Reinhardt cardinal," but I'll admit there is no real basis for my guess.

Why I'm interested. (Assuming these aren't just Reinhardts in disguise) I'm always interested in large cardinal axioms inconsistent with ZFC; in particular, can Moschovakis cardinals survive in ZF? Also, on a purely historical level, it would be interesting to know about.

Even if Moschovakis=Reinhardt, I'm still intrigued: why would that name be used? I've heard Reinhardt cardinals called Kunen cardinals before, since Kunen proved their inconsistency; but Moschovakis seems to have no relation to the subject that I'm aware of.

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@Asaf, what do you mean? –  Noah Schweber Jul 20 '13 at 9:22
That was a bad joke. Don't worry about it. –  Asaf Karagila Jul 20 '13 at 9:23
Noah, why not ask Yiannis directly? –  Andres Caicedo Jul 20 '13 at 14:40

The following quotations are from "A survey of recent results in set theory" by A. R. D. Mathias, a preprint dated July 1968, and stated to be "a draft of a survey to be published in the Proceedings from the UCLA Set Theory Institute." (I found this today while doing some housecleaning.)

The first quotation begins on p. 28, column 2, and continues onto p. 29, column 1.

D 2001 (Moschovakis) Let $\ \kappa,\lambda,\nu$ be cardinals, $\mu$ an ordinal.
$\kappa\underset{\lambda}\to(\mu)_{\nu}^{\lt\omega}$ iff for every $f$ with domain the set of sequences of
length $\lambda$ of finite subsets of $\kappa$ and range a subset of $\nu$ (in
symbols, $f:([\kappa]^{\lt\omega})^{\lambda}\to\nu$), there is a sequence $\langle x_{\eta}\ |\eta\lt\lambda\rangle$ of
subsets of $\kappa$, each of order type $\mu$, (i.e. $\langle x_{\eta}, \in\restriction x_{\eta}\rangle\cong\mu$),
with the property that for every $s$ and $t$ in the domain of $f$,
if for all $\eta\lt\lambda,\ s_{\eta}\subseteq x_{\eta},t_{\eta}\subseteq x_{\eta}$, and $\overline{\overline s}_{\eta}=\overline{\overline t}_{\eta}$, then
$f(s)=f(t)$.

The case $\lambda=1$ is the Erdös-Rado property $\kappa\to(\mu)_{\nu}^{\lt\omega}$. The case
$\nu=2$ will be written $\kappa\underset{\lambda}\to(\mu)^{\lt\omega}$.

$\kappa$ is a $\underline{\text{Ramsey}}$ cardinal iff $\kappa\to(\kappa)^{\lt\omega}$. $\quad\kappa$ is a $\underline{\text{Moschovakis}}$ cardinal
iff $\kappa\underset{\omega_1}\to(\omega_1)^{\lt\omega}$

From p. 30, column 2:

Little is known about the size of Moschovakis cardinals relative to
other large numbers; Silver has made the following simple observation:

T 2011 $\text{ZF}+\text{AC}\vdash$ If there are both Moschovakis and strongly compact
cardinals, then the first Moschovakis cardinal is smaller than
the first strongly compact.

From p. 51, what must have been the motivation for considering such awful things:

T 3025 (Moschovakis; Prikry) $\text{ZF}+\text{AC}\vdash$ If there is a Moschovakis
cardinal, then every $\underset{\sim}\Sigma^1_2$ game is determined.

This is all pretty much Greek to me but I suppose it will mean something to you, if I didn't botch the typing too badly.

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Nice catch. Shameful, really, I have the published paper, and didn't know the answer was there. The printed version is full of typos, actually. The definition is in page 127. The reference is MR0539225 (81b:03057), Mathias, A. R. D. Surrealist landscape with figures (a survey of recent results in set theory). Period. Math. Hungar. 10 (1979), no. 2-3, 109–175. –  Andres Caicedo Dec 7 '13 at 6:10
By the way, T 3025 should be that every $\mathbf\Sigma^1_2$ game is determined. (All this is pre-Woodin cardinals, of course.) –  Andres Caicedo Dec 7 '13 at 6:12
OK. The paper closes with some "Notes on subsequent developments". On page 161 we read: "D 2001. There are no Moschovakis cardinals (MARTIN; KUNEN; NEIL WILLIAMS)." Finally, on page 166, a further update: "T 3025. Now vacuous, as there are no Moschovakis cardinals." –  Andres Caicedo Dec 7 '13 at 6:16
Wow, thanks! I really thought this wasn't an actual term. –  Noah Schweber Dec 7 '13 at 7:27

Not sure if this is an answer or a comment, but when I asked Moschovakis himself, he dismissed the question immediately and he urged me to focus on more serious questions.

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All work and no play? –  Monroe Eskew Aug 13 '13 at 16:00
Combined with the lack of mention in the available literature, this has a very "last nail in the coffin" sort of sound. I think this answers my question. –  Noah Schweber Aug 15 '13 at 18:38