Worldly cardinals are not always downwards absolute

I recently came to realize that worldly cardinals are not necessarily downward absolute to transitive inner models. That is, it can happen that a cardinal $\kappa$ is worldly in the full set-theoretic universe $V$, but not in some transitive inner model $W$, even when $W$ is itself a model of ZFC. The observation came out of some conversations I had with Alexander Block from Hamburg during his recent research visit to New York. Let me explain the argument.

A cardinal $\kappa$ is inaccessible, if it is an uncountable regular strong limit cardinal. The structure $V_\kappa$, consisting of the rank-initial segment of the set-theoretic universe up to $\kappa$, which can be generated from the empty set by applying the power set operation $\kappa$ many times, has many nice features. In particular, it is transitive model of $\newcommand\ZFC{\text{ZFC}}\ZFC$. The models $V_\kappa$ for $\kappa$ inaccessible are precisely the uncountable Grothendieck universes used in category theory.

Although the inaccessible cardinals are often viewed as the entryway to the large cardinal hierarchy, there is a useful large cardinal concept weaker than inaccessibility. Namely, a cardinal $\kappa$ is worldly, if $V_\kappa$ is a model of $\ZFC$. Every inaccessible cardinal is worldly, and in fact a limit of worldly cardinals, because if $\kappa$ is inaccessible, then there is an elementary chain of cardinals $\lambda<\kappa$ with $V_\lambda\prec V_\kappa$, and all such $\lambda$ are worldly. The regular worldly cardinals are precisely the inaccessible cardinals, but the least worldly cardinal is always singular of cofinality $\omega$.

The worldly cardinals can be seen as a kind of poor-man’s inaccessible cardinal, in that worldliness often suffices in place of inaccessibility in many arguments, and this sometimes allows one to weaken a large cardinal hypothesis. But meanwhile, they do have some significant strengths. For example, if $\kappa$ is worldly, then $V_\kappa$ satisfies the principle that every set is an element of a transitive model of $\ZFC$.

It is easy to see that inaccessibility is downward absolute, in the sense that if $\kappa$ is inaccessible in the full set-theoretic universe $V$ and $W\newcommand\of{\subseteq}\of V$ is a transitive inner model of $\ZFC$, then $\kappa$ is also inaccessible in $W$. The reason is that $\kappa$ cannot be singular in $W$, since any short cofinal sequence in $W$ would still exist in $V$; and it cannot fail to be a strong limit there, since if some $\delta<\kappa$ had $\kappa$-many distinct subsets in $W$, then this injection would still exist in $V$. So inaccessibility is downward absolute.

The various degrees of hyper-inaccessibility are also downwards absolute to inner models, so that if $\kappa$ is an inaccessible limit of inaccessible limits of inaccessible cardinals, for example, then this is also true in any inner model. This downward absoluteness extends all the way through the hyperinaccessibility hierarchy and up to the Mahlo cardinals and beyond. A cardinal $\kappa$ is Mahlo, if it is a strong limit and the regular cardinals below $\kappa$ form a stationary set. We have observed that being regular is downward absolute, and it is easy to see that every stationary set $S$ is stationary in every inner model, since otherwise there would be a club set $C$ disjoint from $S$ in the inner model, and this club would still be a club in $V$. Similarly, the various levels of hyper-Mahloness are also downward absolute.

So these smallish large cardinals are generally downward absolute. How about the worldly cardinals? Well, we can prove first off that worldliness is downward absolute to the constructible universe $L$.

Observation. If $\kappa$ is worldly, then it is worldly in $L$.

Proof. If $\kappa$ is worldly, then $V_\kappa\models\ZFC$. This implies that $\kappa$ is a beth-fixed point. The $L$ of $V_\kappa$, which is a model of $\ZFC$, is precisely $L_\kappa$, which is also the $V_\kappa$ of $L$, since $\kappa$ must also be a beth-fixed point in $L$. So $\kappa$ is worldly in $L$. QED

But meanwhile, in the general case, worldliness is not downward absolute.

Theorem. Worldliness is not necessarily downward absolute to all inner models. It is relatively consistent with $\ZFC$ that there is a worldly cardinal $\kappa$ and an inner model $W\of V$, such that $\kappa$ is not worldly in $W$.


First, I shall prove that $\kappa$ is worldly in the forcing extension $V[G]$. Since every set of rank less than $\kappa$ is added by some stage less than $\kappa$, it follows that $V_\kappa^{V[G]}$ is precisely $\bigcup_{\gamma<\kappa} V_\kappa[G_\gamma]$. Most of the $\ZFC$ axioms hold easily in $V_\kappa^{V[G]}$; the only difficult case is the collection axiom. And for this, by considering the ranks of witnesses, it suffices to show for every $\gamma<\kappa$ that every function $f:\gamma\to\kappa$ that is definable from parameters in $V_\kappa^{V[G]}$ is bounded. Suppose we have such a function, defined by $f(\alpha)=\beta$ just in case $\varphi(\alpha,\beta,p)$ holds in $V_\kappa^{V[G]}$. Let $\delta<\kappa$ be larger than the rank of $p$. Now consider $V_\kappa[G_\delta]$, which is a set-forcing extension of $V_\kappa$ and therefore a model of $\ZFC$. The fail forcing, from stage $\delta$ up to $\kappa$, is homogeneous in this model. And therefore we know that $f(\alpha)=\beta$ just in case $1$ forces $\varphi(\check\alpha,\check\beta,\check p)$, since these arguments are all in the ground model $V_\kappa[G_\delta]$. So the function is already definable in $V_\kappa[G_\delta]$. Because this is a model of $\ZFC$, the function $f$ is bounded below $\kappa$. So we get the collection axiom in $V_\kappa^{V[G]}$ and hence all of $\ZFC$ there, and so $\kappa$ is worldly in $V[G]$.

For any $A\of\kappa$, let $\P_A$ be the restriction of the Easton product forcing to include only the stages in $A$, and let $G_A$ be the corresponding generic filter. The full forcing $\P$ factors as $\P_A\times\P_{\kappa\setminus A}$, and so $V[G_A]\of V[G]$ is a transitive inner model of $\ZFC$.

But if we pick $A\of\kappa$ to be a short cofinal set in $\kappa$, which is possible because $\kappa$ is singular, then $\kappa$ will not be worldly in the inner model $V[G_A]$, since in $V_\kappa[G_A]$ we will be able to identify that sequence as the places where the GCH fails. So $\kappa$ is not worldly in $V[G_A]$.

In summary, $\kappa$ was worldly in $V[G]$, but not in the transitive inner model $W=V[G_A]$, and so worldliness is not downward absolute. QED

Same structure, different truths, Stanford University CSLI, May 2016

This will be a talk for the Workshop on Logic, Rationality, and Intelligent Interaction at the CSLI, Stanford University, May 27-28, 2016.

Abstract. To what extent does a structure determine its theory of truth? I shall discuss several surprising mathematical results illustrating senses in which it does not, for the satisfaction relation of first-order logic is less absolute than one might have expected. Two models of set theory, for example, can have exactly the same natural numbers and the same arithmetic structure $\langle\mathbb{N},+,\cdot,0,1,<\rangle$, yet disagree on what is true in this structure; they have the same arithmetic, but different theories of arithmetic truth; two models of set theory can have the same natural numbers and a computable linear order in common, yet disagree on whether it is a well-order; two models of set theory can have the same natural numbers and the same reals, yet disagree on projective truth; two models of set theory can have a rank initial segment of the universe $\langle V_\delta,{\in}\rangle$ in common, yet disagree about whether it is a model of ZFC. These theorems and others can be proved with elementary classical model-theoretic methods, which I shall explain. On the basis of these observations, Ruizhi Yang (Fudan University, Shanghai) and I argue that the definiteness of the theory of truth for a structure, even in the case of arithmetic, cannot be seen as arising solely from the definiteness of the structure itself in which that truth resides, but rather is a higher-order ontological commitment.

Slides | Main article: Satisfaction is not absolute | CLSI 2016 | Abstract at CLSI

The absolute truth about non-absolute truth, JAF – Weak Arithmetics Days, New York, July 2015

This will be a talk for the Journées sur les Arithmétiques Faibles – Weak Arithmetics Days conference, held in New York at the CUNY Graduate Center, July 7 – 9, 2015.

Abstract. I will discuss several fun theorems and folklore results illustrating that the satisfaction relation of first-order logic is less absolute than one might have expected. Two models of set theory, for example, can have the same natural numbers $\langle\mathbb{N},+,\cdot,0,1,<\rangle$, yet disagree on their theories of arithmetic truth; two models of set theory can have the same natural numbers and a computable linear order in common, yet disagree on whether it is a well-order and hence disagree about $\omega_1^{CK}$; two models of set theory can have the same natural numbers and the same reals, yet disagree on projective truth; two models of set theory can have a rank initial segment of the universe $\langle V_\delta,{\in}\rangle$ in common, yet disagree about whether it is a model of ZFC. These theorems and others can be proved with elementary classical model-theoretic methods. On the basis of these observations, Ruizhi Yang (Fudan University, Shanghai) and I have argued that the definiteness of the theory of truth for a structure, even in the case of arithmetic, cannot be seen as arising solely from the definiteness of the structure itself in which that truth resides, but rather is a higher-order ontological commitment.

Slides |  Main article: Satisfaction is not absolute

Does definiteness-of-truth follow from definiteness-of-objects? NY Philosophical Logic Group, NYU, November 2014

This will be a talk for the New York Philosophical Logic Group, November 10, 2014, 5-7pm, at the NYU Philosophy Department, 5 Washington Place, Room 302.

Abstract. This talk — a mix of mathematics and philosophy — concerns the extent to which we may infer definiteness of truth in a mathematical context from definiteness of the underlying objects and structure of that context. The philosophical analysis is based in part on the mathematical observation that the satisfaction relation for model-theoretic truth is less absolute than often supposed.  Specifically, two models of set theory can have the same natural numbers and the same structure of arithmetic in common, yet disagree about whether a particular arithmetic sentence is true in that structure. In other words, two models can have the same arithmetic objects and the same formulas and sentences in the language of arithmetic, yet disagree on their corresponding theories of truth for those objects. Similarly, two models of set theory can have the same natural numbers, the same arithmetic structure, and the same arithmetic truth, yet disagree on their truths-about-truth, and so on at any desired level of the iterated truth-predicate hierarchy.  These mathematical observations, for which I shall strive to give a very gentle proof in the talk (using only elementary classical methods), suggest that a philosophical commitment to the determinate nature of the theory of truth for a structure cannot be seen as a consequence solely of the determinateness of the structure in which that truth resides. The determinate nature of arithmetic truth, for example, is not a consequence of the determinate nature of the arithmetic structure N = {0,1,2,…} itself, but rather seems to be an additional higher-order commitment requiring its own analysis and justification.

This work is based on my recent paper, Satisfaction is not absolute, joint with Ruizhi Yang (Fudan University, Shanghai).

Local properties in set theory

Set-theoretic arguments often make use of the fact that a particular property $\varphi$ is local, in the sense that instances of the property can be verified by checking certain facts in only a bounded part of the set-theoretic universe, such as inside some rank-initial segment $V_\theta$ or inside the collection $H_\kappa$ of all sets of hereditary size less than $\kappa$. It turns out that this concept is exactly equivalent to the property being $\Sigma_2$ expressible in the language of set theory.

Theorem. For any assertion $\varphi$ in the language of set theory, the following are equivalent:

1. $\varphi$ is ZFC-provably equivalent to a $\Sigma_2$ assertion.
2. $\varphi$ is ZFC-provably equivalent to an assertion of the form “$\exists \theta\, V_\theta\models\psi$,” where $\psi$ is a statement of any complexity.
3. $\varphi$ is ZFC-provably equivalent to an assertion of the form “$\exists \kappa\, H_\kappa\models\psi$,” where $\psi$ is a statement of any complexity.

Just to clarify, the $\Sigma_2$ assertions in set theory are those of the form $\exists x\,\forall y\,\varphi_0(x,y)$, where $\varphi_0$ has only bounded quantifiers. The set $V_\theta$ refers to the rank-initial segment of the set-theoretic universe, consisting of all sets of von Neumann rank less than $\theta$. The set $H_\kappa$ consists of all sets of hereditary size less than $\kappa$, that is, whose transitive closure has size less than $\kappa$.

Proof. ($3\to 2$) Since $H_\kappa$ is correctly computed inside $V_\theta$ for any $\theta>\kappa$, it follows that to assert that some $H_\kappa$ satisfies $\psi$ is the same as to assert that some $V_\theta$ thinks that there is some cardinal $\kappa$ such that $H_\kappa$ satisfies $\psi$.

($2\to 1$) The statement $\exists \theta\, V_\theta\models\psi$ is equivalent to the assertion $\exists\theta\,\exists x\,(x=V_\theta\wedge x\models\psi)$. The claim that $x\models\psi$ involves only bounded quantifiers, since the quantifiers of $\psi$ become bounded by $x$. The claim that $x=V_\theta$ is $\Pi_1$ in $x$ and $\theta$, since it is equivalent to saying that $x$ is transitive and the ordinals of $x$ are precisely $\theta$ and $x$ thinks every $V_\alpha$ exists, plus a certain minimal set theory (so far this is just $\Delta_0$, since all quantifiers are bounded), plus, finally, the assertion that $x$ contains every subset of each of its elements. So altogether, the assertion that some $V_\theta$ satisfies $\psi$ has complexity $\Sigma_2$ in the language of set theory.

($1\to 3$) This implication is a consequence of the following absoluteness lemma.

Lemma. (Levy) If $\kappa$ is any uncountable cardinal, then $H_\kappa\prec_{\Sigma_1} V$.

Proof. Suppose that $x\in H_\kappa$ and $V\models\exists y\,\psi(x,y)$, where $\psi$ has only bounded quantifiers. Fix some such witness $y$, which exists inside some $H_\gamma$ for perhaps much larger $\gamma$. By the Löwenheim-Skolem theorem, there is $X\prec H_\gamma$ with $\text{TC}(\{x\})\subset X$, $y\in X$ and $X$ of size less than $\kappa$. Let $\pi:X\cong M$ be the Mostowski collapse of $X$, so that $M$ is transitive, and since it has size less than $\kappa$, it follows that $M\subset H_\kappa$. Since the transitive closure of $\{x\}$ was contained in $X$, it follows that $\pi(x)=x$. Thus, since $X\models\psi(x,y)$ we conclude that $M\models \psi(x,\pi(y))$ and so hence $\pi(y)$ is a witness to $\psi(x,\cdot)$ inside $H_\kappa$, as desired. QED

Using the lemma, we now prove the remaining part of the theorem. Consider any $\Sigma_2$ assertion $\exists x\,\forall y\, \varphi_0(x,y)$, where $\varphi_0$ has only bounded quantifiers. This assertion is equivalent to $\exists\kappa\, H_\kappa\models\exists x\,\forall y\,\varphi_0(x,y)$, simply because if there is such a $\kappa$ with $H_\kappa$ having such an $x$, then by the lemma this $x$ works for all $y\in V$ since $H_\kappa\prec_{\Sigma_1}V$; and conversely, if there is an $x$ such that $\forall y\, \varphi_0(x,y)$, then this will remain true inside any $H_\kappa$ with $x\in H_\kappa$. QED

In light of the theorem, it makes sense to refer to the $\Sigma_2$ properties as the locally verifiable properties, or perhaps as semi-local properties, since positive instances of $\Sigma_2$ assertions can be verified in some sufficiently large $V_\theta$, without need for unbounded search. A truly local property, therefore, would be one such that positive and negative instances can be verified this way, and these would be precisely the $\Delta_2$ properties, whose positive and negative instances are locally verifiable.

Tighter concepts of locality are obtained by insisting that the property is not merely verified in some $V_\theta$, perhaps very large, but rather is verified in a $V_\theta$ where $\theta$ has a certain closeness to the parameters or instance of the property. For example, a cardinal $\kappa$ is measurable just in case there is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$, and this is verified inside $V_{\kappa+2}$. Thus, the assertion “$\kappa$ is measurable,” has complexity $\Sigma^2_1$ over $V_\kappa$. One may similarly speak of $\Sigma^n_m$ or $\Sigma^\alpha_m$ properties, to refer to properties that can be verified with $\Sigma_m$ assertions in $V_{\kappa+\alpha}$. Alternatively, for any class function $f$ on the ordinals, one may speak of $f$-local properties, meaning a property that can be checked of $x\in V_\theta$ by checking a property inside $V_{f(\theta)}$.

This post was made in response to a question on MathOverflow.

Satisfaction is not absolute, Dartmouth Logic Seminar, January 2014

This will be a talk for the Dartmouth Logic Seminar on January 23rd, 2014.

Abstract. I will discuss a number of theorems showing that the satisfaction relation of first-order logic is less absolute than might have been supposed. Two models of set theory can have the same natural numbers, for example, and the same standard model of arithmetic $\langle\mathbb{N},{+},{\cdot},0,1,{\lt}\rangle$, yet disagree on their theories of arithmetic truth; two models of set theory can have the same natural numbers and a computable linear order in common, yet disagree on whether it is a well-order; two models of set theory can have the same natural numbers and the same reals, yet disagree on projective truth; two models of set theory can have a rank initial segment of the universe $\langle V_\delta,{\in}\rangle$ in common, yet disagree about whether it is a model of ZFC. The theorems are proved with elementary classical model-theoretic methods, and many of them can be considered folklore results in the subject of models of arithmetic.

On the basis of these mathematical results, Ruizhi Yang (Fudan University, Shanghai) and I have argued that the definiteness of truth in a structure, such as with arithmetic truth in the standard model of arithmetic, cannot arise solely from the definiteness of the structure itself in which that truth resides; rather, it must be seen as a separate, higher-order ontological commitment.

Main article: Satisfaction is not absolute