## On anti-foundation and coding the hereditarily finite sets

August 27, 2016

I would like to highlight a cute question in a recent paper,

MR3400774
Giovanna D’Agostino, Alberto Policriti, Eugenio G. Omodeo, and Alexandru I. Tomescu.
Mapping sets and hypersets into numbers.
Fund. Inform. 140 (2015), no. 3-4, 307–328.

Recall that W. Ackermann verified what in modern terms we call the bi-interpretability of $\mathsf{ZFfin}$ and $\mathsf{PA}$, where the latter is (first-order) Peano arithmetic, and the former is finite set theory, the result of replacing in $\mathsf{ZF}$ the axiom of infinity with its negation (and with foundation formulated as the schema of $\in$-induction). The reference is

MR1513141
Wilhelm Ackermann.
Die Widerspruchsfreiheit der allgemeinen Mengenlehre.
Math. Ann. 114 (1937), no. 1, 305–315.

I have written about this before. Briefly, one exhibits (definable) translations between the collection $\mathsf{HF}$ of hereditarily finite sets and $\mathbb{N},$ and verifies that the translation extends to a definable translation of the relations, functions and constants of the language of each structure in a way that $\mathsf{PA}$ verifies that $\mathsf{ZFfin}$ holds in the translation of $(\mathsf{HF},\in),$ and $\mathsf{ZFfin}$ verifies that $\mathsf{PA}$ holds in the translation of ${\mathbb N}=(\omega,+,\times,<,0,1)$. Recall that $\mathsf{HF}$ consists of those sets $a$ whose transitive closure is finite, that is, $a$ is finite, and all its elements are finite, and all the elements of its elements are finite, and so on. Using foundation, one easily verifies that $\mathsf{HF}=V_\omega=\bigcup_{n\in\omega}V_n$, that is, it is the collection of sets resulting from iterating the power-set operation (any finite number of times) starting from the empty set.

In the direction relevant here, one defines a map $h:\mathsf{HF}\to\mathbb{N}$ by

$h(a)=\sum_{b\in a}2^{h(b)}.$

One easily verifies, using induction on the set-theoretic rank of the sets involved, that this recursive definition makes sense and is injective (and, indeed, bijective).

Of course this argument uses foundation. In the D’Agostino-Policriti-Omodeo-Tomescu paper they consider instead the theory resulting from replacing foundation with the  anti-foundation axiom, and proceed to describe a suitable replacement for $h$ that injects (codes) $\mathsf{HF}$ into the real numbers. They do quite a bit more in the paper but, for the coding itself, I highly recommend the nice review by Randall Holmes in MathSciNet, linked to above.

The anti-foundation axiom $\mathsf{AFA}$ became known thanks to the work of Peter Aczel, and it is his formulation that I recall below, although it was originally introduced in work of Forti and Honsell from 1983, where they call it $X_1$. Aczel’s presentation appears in the excellent book

MR0940014 (89j:03039)
Peter Aczel.
Non-well-founded sets. With a foreword by Jon Barwise.
CSLI Lecture Notes, 14. Stanford University, Center for the Study of Language and Information, Stanford, CA, 1988. xx+137 pp.
ISBN: 0-937073-22-9.

The original paper is

MR0739920 (85f:03054)
Marco Forti, Furio Honsell.
Set theory with free construction principles.
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), no. 3, 493–522.

Given a binary relation $R$, its field $\mathrm{fld}(R)$ is the union of its domain and codomain. A decoration of $R$ is a function $d:\mathrm{fld}(R)\to V$ satisfying

$d(x)=\{d(y)\mid y\mathrel{R}x\}$

for all $x,y\in\mathrm{fld}(R)$. When $R$ is $\in$ and the sets in question are well-founded, the only decoration is the identity. Similarly, any well-founded relation $R$ admits a unique decoration. Define $\mathsf{AFA}$ as the statement that any binary $R$ (whether well-founded or not) admits a unique decoration.

In $\mathsf{ZF}$ with foundation replaced with $\mathsf{AFA}$ one can prove the existence of many non-well-founded sets. One of the appealing aspects of $\mathsf{AFA}$ is that the resulting univere is actually quite structured: Other anti-foundation axioms allow the existence of infinitely many Quine atoms, sets $x$ such that $x=\{x\}$, for instance. Under $\mathsf{AFA}$, there is exactly one such $x$, usually called $\Omega$. The axiom is sometimes described as saying that it provides solutions to many “equations” among sets. For instance, consider the system of equations $x=\{y\}$ and $y=\{x\}$. Under $\mathsf{AFA}$ the system has $x=y=\Omega$ as its unique solution. Note that assuming $\mathsf{AFA}$, $\Omega$ is in $\mathsf{HF}$, as are many other non-well-founded sets.

Here is the open question from the D’Agostino-Policriti-Omodeo-Tomescu paper: Work in set theory with $\mathsf{AFA}$ instead of foundation. Is there a unique, injective, function $h:\mathsf{HF}\to \mathbb{R}$  satisfying

$h(x)=\sum_{y\in x}2^{-h(y)}$

for all $x,y\in\mathsf{HF}$?

Note that there is a unique such $h$ on the well-founded hereditarily finite sets, and it is in fact injective. In general, existence, uniqueness and injectivity of $h$ appear to be open. The claim that there is such a function $h$ is a statement about solutions of certain equations on the reals, and the claim that $h$ is unique requires moreover uniqueness of such solutions. The expectation is that $h(x)$ is transcendental for all non-well-founded hereditarily finite $x$ but, even assuming this, the injectivity of $h$ seems to require additional work.

For example, consider $x=\Omega$. The function $h$ must satisfy

$h(\Omega)=2^{-h(\Omega)}$

and, indeed $h(\Omega)=0.6411857\dots$ is the unique solution $x$ of the equation $x=2^{-x}$

I would be curious to hear of any progress regarding this problem.

## Monochromatic colorings

August 20, 2016

Caïus Wojcik and Luca Zamboni recently posted a paper on the arXiv solving an interesting problem in combinatorics on words.

http://arxiv.org/abs/1608.03519
Monochromatic factorisations of words and periodicity.
Caïus Wojcik, Luca Q. Zamboni.

I had recently learned of the problem through another paper by Zamboni and a collaborator,

MR3425965
Aldo de Luca, Luca Q. Zamboni
On prefixal factorizations of words.
European J. Combin. 52 (2016), part A, 59–73.

It is a nice result and I think it may be enjoyable to work through the argument here. Everything that follows is either straightforward, standard, or comes from these papers.

1. The problem

To make the post reasonably self-contained, I begin by recalling some conventions, not all of which we need here.

By an alphabet we simply mean a set $A$, whose elements we refer to as letters. A word $w$ is a sequence $w:N\to A$ of letters from $A$ where $N$ is a (not necessarily non-empty, not necessarily proper) initial segment of $\mathbb N$. If we denote $w_i=w(i)$ for all $i\in N$, it is customary to write the word simply as

$w_0w_1\dots$

and we will follow the convention. The empty word is typically denoted by $\Lambda$ or $\varepsilon$. By $A^*$ we denote the collection of all finite words from $A$, and $A^+=A^*\setminus \{\varepsilon\}.$ By $|x|$ we denote the length of the word $x$ (that is, the size of the domain of the corresponding function).

We define concatenation of words in the obvious way, and denote by $x_0x_1$ the word resulting from concatenating the words $x_0$ and $x_1$, where $x_0\in A^*$. This operation is associative, and we extend it as well to infinite concatenations.

If a word $w$ can be written as the concatenation of words $x_0,x_1,\dots,$

$w=x_0x_1\dots,$

we refer to the right-hand side as a factorization of $w$. If $w=xy$ and $x$ is non-empty, we say that $x$ is a prefix of $w$. Similarly, if $y$ is non-empty, it is a suffix of $w$. By $x^n$ for $n\in\mathbb N$ we denote the word resulting form concatenating $n$ copies of $x$. Similarly, $x^{\mathbb N}$ is the result of concatenating infinitely many copies.

By a coloring we mean here a function $c:A^+\to C$ where $C$ is a finite set of “colors”.

Apparently the problem I want to discuss was first considered by T.C. Brown around 2006 and, independently, by Zamboni around 2010. It is a question about monochromatic factorizations of infinite words. To motivate it, let me begin with a cute observation.

Fact. Suppose $w=w_0w_1\dots$ is an infinite word, and $c$ is a coloring. There is then a factorization

$w=px_0x_1\dots$

where all the $x_i\in A^+$ have the same color.

Proof. The proof is a straightforward application of Ramsey’s theorem: Assign to $c$ the coloring of the set $[\mathbb N]^2$ of $2$-sized subsets of $\mathbb N$ given by $d(\{i,j\})=c(w_iw_{i+1}\dots w_{j-1})$ whenever $i. Ramsey’s theorem ensures that there is an infinite set $I=\{n_0 such that all $w_{n_i}w_{n_i+1}\dots w_{n_j-1}$ with $i have the same color. We can then take $p=w_0\dots w_{n_0-1}$ and $x_i=w_{n_i}\dots w_{n_{i+1}-1}$ for all $i$. $\Box$

In the fact above, the word $w$ was arbitrary, and we obtained a monochromatic factorization of a suffix of $w$. However, without additional assumptions, it is not possible to improve this to a monochromatic factorization of $w$ itself. For example, consider the word $w=01^{\mathbb N}$ and the coloring

$c(x)=\left\{\begin{array}{cl}0&\mbox{if }0\mbox{ appears in }x,\\ 1&\mbox{otherwise.}\end{array}\right.$

If nothing else, it follows that if $w$ is an infinite word that admits a monochromatic factorization for any coloring, then the first letter of $w$ must appear infinitely often. The same idea shows that each letter in $w$ must appear infinitely often.

Actually, significantly more should be true. For example, consider the word

$w=010110111\dots 01^n0 1^{n+1}\dots,$

and the coloring

$c(x)=\left\{\begin{array}{cl}0&\mbox{if }x\mbox{ is a prefix of }w,\\1&\mbox{otherwise.}\end{array}\right.$

This example shows that in fact any such $w$ must admit a prefixal factorization, a factorization

$w=x_0x_1\dots$

where each $x_i$ is a prefix of $w$.

Problem. Characterize those infinite words $w$ with the property P that given any coloring, there is a monochromatic factorization of $w$.

The above shows that any word with property P admits a prefixal factorization. But it is easy to see that this is not enough. For a simple example, consider

$w=010^210^31\dots0^n10^{n+1}1\dots$

Consider the coloring $c$ where $c(x)=0$ if $x$ is not a prefix of $w$, $c(0)=$1, and $c(x)=2$ otherwise. If

$w=x_0x_1\dots$

is a monochromatic factorization of $w$, then $x_0=01\dots$ so $c(x_0)=2$ and each $x_i$ must be a prefix of $w$ of length at least $2$. But it is easy to see that $w$ admits no such factorization: For any $n>2$, consider the first appearance in $w$ of $0^{n+1}$ and note that none of the first $n$ zeros can be the beginning of an $x_i$, so for some $j$ we must have $x_j=01\dots 10^n$ and since $n>2$, in fact $x_j=01\dots 10^n10^n$, but this string only appears once in $w$, so actually $j=0$. Since $n$ was arbitrary, we are done.

Here is a more interesting example: The Thue-Morse word

$t=0110100110010110\dots$

was defined by Axel Thue in 1906 and became known through the work of Marston Morse in the 1920s. It is defined as the limit (in the natural sense) of the sequence $x_0,x_1,\dots$ of finite words given by $x_0=0$ and $x_{n+1}=x_n\bar{x_n}$ where, for $x\in\{0,1\}^*$, $\bar x$ is the result of replacing each letter $i$ in $x$ with $1-i$.

This word admits a prefixal factorization, namely

$t=(011)(01)0(011)0(01)(011)(01)0(01)(011)0(011)(01)0\dots$

To see this, note that the sequence of letters of $t$ can be defined recursively by $t_0=0$, $t_{2n}=t_n$ and $t_{2n+1}=1-t_n$. To see this, note in turn that the sequence given by this recursive definition actually satisfies that $t_n$ is the parity of the number of $1$s in the binary expansion of $n,$ from which the recursive description above as the limit of the $x_n$ should be clear. The relevance of this observation is that no three consecutive letters in $t$ can be the same (since $t_{2n+1}=1-t_{2n}$ for all $n$), and from this it is clear that $t$ can be factored using only the words $0$, $01$, and $011$.

But it is not so straightforward as in the previous example to check whether $t$ admits a factorization into prefixes of length larger than $1$.

Instead, I recall a basic property of $t$ and use it to exhibit an explicit coloring for which $t$ admits no monochromatic factorization.

## Ramsey theory of very small countable ordinals

November 6, 2014

I was an undergraduate student at Los Andes, from 1992 to 1996. This year, their mathematics program is turning 50. There was a conference in September to celebrate the event, and I had the honor to give one of the talks (see here for the English version of the slides).

The Faculty of Science publishes a magazine, Hipótesis, and a special edition will be devoted to the conference. I have submitted an expository paper, based on my talk.

The topic is the partition calculus of very small countable ordinals (mainly ordinals below $\omega^2$, actually). The paper reviews Ramsey’s theorem and a few finite examples, before discussing the two main results.

1.

One is an old theorem by Haddad and Sabbagh, unfortunately not well known. In 1969, they computed the Ramsey numbers $r(\omega+n,m)$ for $n,m$ finite.

Given nonzero ordinals $\alpha,\beta$, recall that $r(\alpha,\beta)$ is the least $\gamma$ such that any red-blue coloring of the edges of $K_\gamma$ either admits a red copy of $K_{\alpha}$ or a blue copy of $K_\beta$. Clearly $r(\alpha,1)=1$, $r(\alpha,\beta)\ge r(\alpha,2)=\alpha$ if $\beta\ge2$, and $r(\alpha,\beta)=r(\beta,\alpha)$, so we may as well assume that $\alpha\ge\beta>2$, and we adopt this convention in what follows.

Ramsey proved two theorems about this function in a famous 1928 paper that introduced the topic. In the notation we have just set up, his first result asserts that $r(n,m)$ is finite whenever $n,m$ are finite, and his second result states that $r(\omega,\omega)=\omega$. The computation of the numbers $r(n,m)$ is an extremely difficult, most likely unfeasible, problem, though $r$ is obviously a recursive function. We are concerned here with the values of the function when at least one of the arguments is infinite.

It turns out that $r(\omega+1,\omega)$ is already $\omega_1$. Hence, if we are interested in studying the countable values of the function $r(\alpha,\beta)$, then we must assume that either $\omega=\alpha$, in which case $r(\alpha,\beta)=\omega$ and there is nothing more to say, or else (that is, if $\alpha$ is countable and strictly larger than $\omega$) we must assume that $\beta$ is finite.

The function has been intensively studied when $\alpha$ is a limit ordinal, particularly a power of $\omega$. Here we look at the much humbler setting where $\omega<\alpha<\omega2$. Recalling that each ordinal equals the set of its predecessors, and using interval notation to describe sets of ordinals, the Haddad-Sabbagh result is as follows:

Lemma. For all positive integers $n, m$ there exists a positive integer $k\ge n, m$ such that for any red-blue coloring of the edges of $K_{[0,k)}$, and such that $K_{[0,m)}$ is blue, there is a subset $H$ of ${}[0, k)$ with $K_H$ monochromatic, and either $K_H$ is blue and $|H| = m + 1$, or else $K_H$ is red, $|H|=n+1$, and $H\cap[0,m)\ne\emptyset$.

Denote by $r_{HS}(n+1,m+1)$ the smallest number $k$ witnessing the lemma.

Theorem. If $n,m$ are positive integers, then $r(\omega +n,m)=\omega(m-1)+t$, where $r_{HS}(n+1,m)=(m-1)+t$.

The theorem was announced in 1969, but the proof never appeared. I have written a survey on the topic, including what should be the first proof in print of this result.

## Square principles in Pmax extensions

May 21, 2012

As mentioned previously, I am part of a SQuaRE (Structured Quartet Research Ensemble), “Descriptive aspects of inner model theory”. The other members of the group are Paul Larson, Grigor Sargsyan, Ralf Schindler, John Steel, and Martin Zeman. We have just submitted our paper Square principles in ${\mathbb P}_{\rm max}$ extensions to the Israel Journal of Mathematics. The paper can be downloaded form my papers page, or from the arXiv.

The forcing ${\mathbb P}_{\rm max}$ was developed by Hugh Woodin in his book The axiom of determinacy, forcing axioms, and the nonstationary ideal${\mathbb P}_{\rm max}$ belongs to $L({\mathbb R})$ and, if determinacy holds, the theory that it forces is combinatorially rich, and we do not currently know how to replicate it with traditional forcing methods.

In particular, Woodin showed that, starting with a model of ${\sf AD}_{\mathbb R}+$$\Theta$ is regular”, a strong form of determinacy, the ${\mathbb P}_{\rm max}$ extension satisfies ${\sf MM}({\mathfrak c})$, the restriction of Martin’s maximum to posets of size at most ${}|{\mathbb R}|$. It is natural to wonder to what extent this can be extended. In this paper, we study the effect of ${\mathbb P}_{\rm max}$ on square principles, centering on those that would be decided by ${\sf MM}({\mathfrak c}^+)$.

These square principles are combinatorial statements stating that a specific version of compactness fails in the universe, namely, there is a certain tree without branches. They were introduced by Ronald Jensen in his paper on The fine structure of the constructible universe. The most well known is the principle $\square_\kappa$:

Definition. Given a cardinal $\kappa$, the principle $\square_{\kappa}$ holds iff there exists a sequence $\langle C_{\alpha} \mid \alpha < \kappa^{+} \rangle$ such that for each $\alpha < \kappa^{+}$,

1. Each $C_{\alpha}$ is club in $\alpha$;
2. For each limit point $\beta$ of $C_{\alpha}$, $C_{\beta} = C_{\alpha} \cap \beta$; and
3. The order type of each $C_\alpha$ is at most $\kappa$.

For $\kappa=\omega$ this is true, but uninteresting. The principle holds in Gödel’s $L$, for all uncountable $\kappa$. It is consistent, relative to a supercompact cardinal, that it fails for all uncountable $\kappa$. For example, this is a consequence of Martin’s maximum.

Recently, the principle $\square(\kappa)$ has been receiving some attention.

Definition. Given an ordinal $\gamma$, the principle $\square(\gamma)$ holds iff there exists a sequence $\langle C_{\alpha} \mid \alpha <\gamma \rangle$ such that

1. For each $\alpha < \gamma$, each $C_{\alpha}$ is club in $\alpha$;
2. For each $\alpha<\gamma$, and each limit point $\beta$ of $C_{\alpha}$, $C_{\beta} = C_{\alpha} \cap \beta$; and
3. There is no thread through the sequence, i.e., there is no club $E\subseteq \gamma$ such that $C_{\alpha} = E \cap \alpha$ for each limit point $\alpha$ of $E$.

Using the Core Model Induction technique developed by Woodin, work of Ernest Schimmerling, extended by Steel, has shown that the statement

$2^{\aleph_1}=\aleph_2+\lnot\square(\omega_2)+\lnot\square_{\omega_2}$

implies that determinacy holds in $L({\mathbb R})$, and the known upper bounds in consistency strength are much higher.

Here are some of our results: First, if one wants to obtain $\lnot\square_{\omega_2}$ in a ${\mathbb P}_{\rm max}$ extension, one needs to start from a reasonably strong determinacy assumption:

Theorem. Assume ${\sf AD}_{\mathbb R}+$$\Theta$ is regular”, and that there is no $\Gamma \subseteq \mathcal{P}(\mathbb{R})$ such that $L(\Gamma, \mathbb{R}) \models$$\Theta$ is Mahlo in ${\sf HOD}$“. Then $\square_{\omega_{2}}$ holds in the ${\mathbb P}_{\rm max}$ extension.

This results uses a blend of fine structure theory with the techniques developed by Sargsyan on his work on hybrid mice. The assumption cannot be improved since we also have the following, the hypothesis of which are a consequence of  ${\sf AD}_{\mathbb R}+V=L({\mathcal P}({\mathbb R}))+$$\Theta$ is Mahlo in ${\sf HOD}$”.

Theorem. Assume that ${\sf AD}^+$ holds and that $\theta$ is a limit on the Solovay sequence such that that there are cofinally many $\kappa<\theta$ that are limits of the Solovay sequence and are regular in ${\sf HOD}$. Then $\square_{\omega_2}$ fails in the ${\mathbb P}_{\rm max}$ extension of ${\sf HOD}_{\mathcal{P}_{\theta}(\mathbb{R})}$.

Here, ${\mathcal P}_\alpha({\mathbb R})$ denotes the collection of subsets of ${\mathbb R}$ of Wadge rank less than $\alpha$. The Solovay sequence, introduced by Robert Solovay in The independence of ${\sf DC}$ from ${\sf AD}$, is a refinement of the definition of $\Theta$, the least ordinal $\alpha$ for which there is no surjection $f:{\mathbb R}\to\alpha$:

Definition. The Solovay sequence if the sequence of ordinals $\langle \theta_{\alpha} \mid \alpha \leq \gamma \rangle$ such that

1. $\theta_{0}$ is the least ordinal that is not the surjective image of the reals by an ordinal definable function;
2. For each $\alpha < \gamma$, $\theta_{\alpha + 1}$ is the least ordinal that is not the surjective image of the reals by a function definable from an ordinal and a set of reals of Wadge rank $\theta_{\alpha}$;
3. For each limit ordinal $\beta \leq \gamma$, $\theta_{\beta} = \sup\{\theta_{\alpha} \mid \alpha < \beta\}$; and
4. $\theta_{\gamma} = \Theta$.

The problem with the result just stated is that choice fails in the resulting model. To remedy this, we need to start with stronger assumptions. Still, these assumptions greatly improve the previous upper bounds for the consistency (with ${\sf ZFC}$) of $2^{\aleph_1}=\aleph_2+\lnot\square(\omega_2)+\lnot\square_{\omega_2}$. In particular, we now know that this theory is strictly weaker than a Woodin limit of Woodin cardinals.

Theorem. Assume that ${\sf AD}_{\mathbb R}$ holds, that $V = L(\mathcal{P}(\mathbb{R}))$, and that stationarily many elements $\theta$ of cofinality $\omega_{1}$ in the Solovay sequence are regular in ${\sf HOD}$. Then in the ${\mathbb P}_{\rm max} * {\rm Add}(\omega_{3},1)$-extension, $\square_{\omega_2}$ fails.

(The forcing ${\rm Add}(\omega_3,1)$ adds a Cohen subset of $\omega_3$. This suffices to well-order ${\mathcal P}({\mathbb R})$, and therefore to force choice. That in the resulting model we also have $2^{\aleph_1}=\aleph_2+\lnot\square(\omega_2)$ follows from prior work of Woodin.)

Finally, I think I should mention a bit of notation. In the paper, we say that $\kappa^+$ is square inaccessible iff $\square_\kappa$ fails. We also say that $\gamma$ is threadable iff $\square(\gamma)$ fails. This serves to put the emphasis on the negations of the square principles, which we feel is where the interest resides. It also solves the slight notational inconvenience of calling $\square_\kappa$ a principle that is actually a statement about $\kappa^+$.

## Determinacy and Jónsson cardinals

April 9, 2012

It is a well known result of Kleinberg that the axiom of determinacy implies that the $\aleph_n$ are Jónsson cardinals for all $n\le\omega$. This follows from a careful analysis of partition properties and the computation of ultrapowers of associated measures. See also here for extensions of this result, references, and some additional context.

Using his theory of descriptions of measures, Steve Jackson proved (in unpublished work) that, assuming determinacy, every cardinal below $\aleph_{\omega_1}$ is in fact Rowbottom. See for example these slides: 12. Woodin mentioned after attending a talk on this result that the HOD analysis shows that every cardinal is Jónsson below $\Theta$.

During the Second Conference on the Core Model Induction and HOD Mice at Münster, Jackson, Ketchersid, and Schlutzenberg reconstructed what we believe is Woodin’s argument.

I have posted a short note with the proof on my papers page. The note is hastily written, and I would appreciate any

you ﬁnd appropriate. I posted a version of the note in August last year, and Grigor Sargsyan soon emailed me some comments, that I have incorporated in the current version.

[I mentioned this on Twitter last year, but apparently not here. These are the relevant links (while discussing the Münster meeting): 1, 2, 3, 4, 5.]

## Downward transference of mice and universality of local core models

April 4, 2012

Martin Zeman and I have just submitted our paper Downward transference of mice and universality of local core models to the Journal of Symbolic Logic, downloadable from my papers page. We have also uploaded it on the ArXiv. (I should have been doing this for years; this is the first time I post there.)

It is a nice observation that goes back to Friedman that if $0^\sharp$ exists and ${\mathbf M}$ is an inner model that correctly computes $\omega_2$, then $0^\sharp\in {\mathbf M}$. Looking at a completely different problem, from the theory of forcing axioms, we were led to the question of how much this result can be generalized.

Our main result is that there is a significant transfer of structure going on, simply due to the agreement of cardinals. (The statement of the result and, of course, the argument, require familiarity with fine structure theory, as developed in Steel’s or Martin’s books.)

Theorem. Assume that ${\mathbf M}$ is a proper class inner model, and that $\delta$ is regular in ${\mathbf V}$.

1. If there are no inner models of ${\mathbf V}$ with Woodin cardinals, $\delta>\omega_1$, and

$\{x\in{\mathcal P}_\delta(\delta^+)\cap{\mathbf M}\mid{\rm cf}^{\mathbf M}(x\cap\delta)>\omega\}$

is stationary, then ${\mathbf K}^{\mathbf M}\|\delta$ is universal for all iterable 1-small premice in ${\mathbf V}$ of cardinality less than $\delta$.

2. If, in ${\mathbf M}$, $0^\P$ does not exist, and ${\mathcal P}_\delta(\delta^+)\cap{\mathbf M}$ is stationary, and $\delta>\omega_1$, then the same conclusion holds. If $\delta=\omega_1$, then ${\mathbf K}^{\mathbf M}\|\omega_2$ is universal for all countable iterable premice in ${\mathbf V}$.

Here, as usual, ${\mathcal P}_\kappa(\lambda)$ denotes the collection of subsets of $\lambda$ of size less than $\kappa$. It is easy to check that ${\mathcal P}_{\omega_1} (\omega_2)\cap{\mathbf M}$ is stationary if ${\mathbf M}$ computes $\omega_2$ correctly, so this result generalizes the statement about $0^\sharp$ mentioned above.

In fact, it follows that, if $\omega_2$ is computed correctly in ${\mathbf M}$, then any sound mouse in ${\mathbf V}$ projecting to $\omega$ and below $0^\P$, is in ${\mathbf M}$. Beyond $0^\P$, the argument becomes more complicated, and we need to assume a global anti-large cardinal assumption, namely, that there are no inner models in ${\mathbf V}$ with Woodin cardinals.

We expect that this restriction can be weakened, perhaps even dispensed with.

(This paper is the second in a series, aiming to explore the structure of inner models for which some agreement of cardinals holds. I briefly mentioned the first paper here.)

## On CH (After Hamkins)

April 2, 2012

This is my excuse to put this page to use. It all started, more or less, around here:

Moreno, Javier (bluelephant). “Es muy bueno este artículo de Hamkins sobre algo así como sociología de la teoría de conjuntos: http://arxiv.org/abs/1203.4026” 2 April 2012, 3:39 p.m. Tweet.

Villaveces, Andrés (gavbenyos). “@bluelephant lo siento bastante crudo – me parece que Joel simplemente pone en versión impresa cierto consenso, pero falta argumentar” 2 April 2012, 4:32 p.m. Tweet.

(Loosely translated: “This is a very good article by Hamkins on something akin to sociology of set theory” “I find it weak — I think that Joel is simply putting in writing a known consensus, but needs to argue for it more”) Long story short, I feel Andrés is right, but I thought I should elaborate my view, at least somewhat. Originally I considered writing a blog entry, but it quickly became apparent it would grow longer than I can afford time-wise. So, I used twitter instead.

(But there is a serious caveat, namely, it seems that the paper is intended for a general mathematical and philosophical audience, so the omission of technical issues is most likely intentional. Javier even remarked as much.)

What follows is the series of Tweets I posted. It is not a transcription; to ease reading, I have added a couple of links, reformatted the posts rather than continuing with the MLA suggested approach, very lightly edited the most obvious typos, and added a couple of phrases where I felt more clarity was needed. [Edit, April 22: I also removed a line on $MM^{++}$ vs Woodin’s $(*)$, as the proof of the underlying claim has been withdrawn.]

I started at 10:44 p.m., with a warning: “(Technical pseudo-philosophical thoughts for a few posts.)”