# Massive Cardinals

## Consistency Strength

Massive cardinals are a very strong kind of large cardinal which is so far indeterminate in strength. It is known that $0$-Massive cardinals lie beneath I3, and each of these lie beneath Reinhardt cardinals.

## The Ordinals $\theta_\alpha$

We let $\theta_0$ be the smallest ordinal for which there is no first-order formula $\phi$ such that $\forall x(\phi(x)\Leftrightarrow x=\theta_0)$. It is known that $\theta_0$ is infinite. It is also known that $\theta_0>$ $\omega_1^{ck}$. Generally, we let $\theta_\alpha$ be defined as the smallest ordinal such that there is no first-order formula $\phi$ such that: $$\exists n\in\omega\exists \beta_0,\beta_1...\in\alpha\forall x(\phi(x,\theta_{\beta_0},\theta_{\beta_1}...\theta_{\beta_n})\Leftrightarrow x=\theta_\alpha)$$ More simply put, $\theta_\alpha$ is the smallest ordinal such that finitely many uses $\theta_\beta$ cannot be used as parameters in a first-order formula to completely define it. These ordinals are very large, though there is no inconsistency known proving that they are uncountable. However, because $\theta_\beta < \theta_\alpha$ when $\beta<\alpha$, it is very easy to prove that $\theta_\alpha\geq\alpha$.

## $\alpha$-Massive Cardinals

If $\kappa$ is $\alpha$-Massive it is the critical point of some nontrivial elementary embedding $j:M\rightarrow M$ where $M$ is a transitive inner model of ZFC and: $$\forall\phi\forall v\in{}^{<\omega}\theta_\alpha(\phi\;\mathrm{is}\;\mathrm{first-order}\rightarrow(\{x:\phi(x,v)\}\in V\rightarrow\{x:\phi(x,v)\}\in M))$$ This means that every first-order definable set $X$ with finitely many parameters from ordinals $\beta<\theta_\alpha$ is in $M$. It is quite obvious that for $0$-Massive cardinals they are the critical point of some $j:M\rightarrow M$ where $\{\{x:\phi(x)\}:\phi\;\mathrm{is}\;\mathrm{first-order}\}\subseteq M$. This may seem like it is a proper class; yet the existence of a Reinhardt cardinal implies that $M$ can be a set with rank less than that of a Reinhardt cardinal.

## Massive Cardinals

A Massive cardinal $\kappa$ is $\alpha$-Massive for some $\alpha$ such that $\theta_\alpha\geq\kappa$.

## $\alpha$-Supermassive Cardinals

If $\kappa$ is $\alpha$-Supermassive it is the critical point of some nontrivial elementary embedding $j:M\rightarrow M$ where $M$ is a transitive inner model of ZFC and: $$\forall\phi\forall v\in {}^{<\omega}V_{\theta_\alpha}(\phi\;\mathrm{is}\;\mathrm{first-order}\rightarrow(\{x:\phi(x,v)\}\in V\rightarrow\{x:\phi(x,v)\}\in M))$$ This means that every first-order definable set $X$ with finitely many parameters from sets of rank $\beta<\theta_\alpha$ is in $M$. Because $M$ is transitive and (quite clearly) for every $X\in V_{\theta_\alpha}$, $\{x:x=X\}=\{X\}\in M$, thus $X\in M$. This means that $V_{\theta_\alpha}\subset M$. With this, one could prove that the existence of an $\alpha$-Supermassive cardinal implies the existence of an $I3(\kappa,\theta_\alpha)$ cardinal; which furthermore implies the existence of an I3 cardinal with critical point less than that of $\theta_\alpha$. This means that $\theta_\alpha$ must be very large in order for an $\alpha$-Supermassive cardinal to exist.

## Supermassive Cardinals

A Supermassive cardinal $\kappa$ is the critical point of some nontrivial elementary embedding $j:M\rightarrow M$ where $M$ is a transitive inner model of ZFC and for some proper class inner model $W$ such that $W\prec V$:

$$\forall\phi\forall v\in {}^{<\omega}W(\phi\;\mathrm{is}\;\mathrm{first-order}\rightarrow(\{x:\phi(x,v)\}\in V\rightarrow\{x:\phi(x,v)\}\in M))$$ This means that every first-order definable set $X$ with finitely many parameters of $W$ is in $M$. It is quite obvious that a Supermassive Cardinal is Massive and $\alpha$-Supermassive for every $V_{\theta_\alpha}\in W$ and thus is also $\alpha$-Massive for every $V_{\theta_\alpha}\in W$ and for every $\alpha\in W$. The size of a Supermassive cardinal is not quite clear. It is known that every Reinhardt cardinal is Supermassive (such is quite evident when $V=W$). It is also known that $j:Y\rightarrow Y$ must exist and have a critical point for each $Y$ a transitive inner model of ZFC which is a subclass of $W$.

## Why $\theta_\alpha$ is Used

The reason $\theta_\alpha$ is used to define $\alpha$-Massive and $\alpha$-Supermassive cardinals is simply because if instead of using $\theta_\alpha$ one used $\alpha$, then $0$-Massive cardinals would be precisely the $1$-Massive cardinals.

Generally with this naive definition, $\alpha$-Massive cardinals are precisely the $\beta$-Massive cardinals when $\theta_\gamma\leq\alpha\leq\beta<\theta_{\gamma+1}$. This is why $\theta_\alpha$ is used instead.