Inexplicable Cardinals

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Because they are on his blog post, it should already be known that these large cardinals were invented by the writer, Zetapology, and should not be considered absolute or necessarily true information; I implore the readers to look into these themselves.

Inexplicable Cardinals

Inexplicable cardinals are a type of large cardinal which have no known place in the large cardinal hierarchy. Every Reinhardt cardinal is inexplicable. The reason that they have no known place in the large cardinal hierarchy is because their existence is inconsistent with $V=\mathrm{HOD}$, which itself is commonly thought to be consistent with almost all other known large cardinal axioms. However, they do not appear to be "large" cardinals; in fact, they are quite small. If an inexplicable cardinal exists, there is an inexplicable cardinal which is not even weakly inaccessible. This is because it has cofinality $\omega$.


The ordinals $\theta_\alpha(n)$

Let $\theta_0(n)$ be the smallest ordinal such that there is no formula $\phi$ in $n+1$-th order logic such that: $$\forall x(\phi(x)\Leftrightarrow x=\theta_0(n))$$

Let $\theta_\alpha(n)$ for $\alpha>0$ be the smallest ordinal such that there is no formula $\phi$ in $n+1$-th order logic such that: $$\forall A\in{}^{<\omega}\alpha(\phi(x,\theta_{A_0},\theta_{A_1}...)\Leftrightarrow x=\theta_\alpha(n))$$

Already $\theta_\alpha(n)$ is inconsistent with $V=\mathrm{HOD}$ for any $\alpha$ or $n$, because every model of $\mathrm{ZFC}+V=\mathrm{HOD}$ is pointwise definable, and they satisfy that every ordinal can be constructed in first-order logic with no parameters.

$\theta_\alpha(n)$, if assumed to exist, are surprisingly bounded in cardinality. Specifically, $\alpha\leq\theta_\alpha(n)<|\alpha|^+$ for any $\alpha$ and $n$. However, this "well-bounded" property is still not very secure. In fact, if $\theta_0(0)$ exists, it is larger than, the supremum of the eventually writable ordinals. It is assumed to also be larger than $\Sigma$, but that bound is not yet known.

$\theta_0(n)$ has another way of being defined. Specifically, it is the supremum of all suprema of $n+1$-order definable ordinal notations, which includes Klev's $\mathcal{O}^{++}$.

$n$-Inexplicable cardinals

Let an initial ordinal (i.e. a cardinal assuming AC) $\kappa$ be $n$-Inexplicable when $\theta_\kappa(n)=\kappa$. This is equivalent to $\exists\alpha(\theta_\alpha(n)=\kappa)$. Each of these are $\aleph$-fixed points, and fixed points of the enumerations of those, and so on. However, it is easy to construct such a cardinal, assuming it's existence, which has cofinality $\omega$ and is therefore not even weakly inaccessible this construction is due to Noah Schweber from MathOverflow. To start, we let $M_n(\alpha)$ be the smallest ordinal not definable using $n+1$-th order logic and parameters $<\alpha$. Note that Then we let:

$$\beta_0=0$$ $$\beta_1=|M_n(\theta_{\beta_0}(n))|^+=|M_n(\theta_0(n))|^+=\aleph_1$$ $$\beta_2=|M_n(\theta_{\beta_1}(n))|^+=|M_n(\theta_\Omega(n))|^+$$ $$...$$ $$\beta_{i+1}=|M_n(\theta_{\beta_i}(n))|^+$$

The supremum $\kappa$ of $\beta_i$ for finite $i$ is an initial ordinal because it is the supremum of initial ordinals. Since $M_n(\alpha)<\kappa$ for each $\alpha<\kappa$, (by definition), it is a $\theta$-fixed point.

An Inexplicable cardinal is one which is 0-Inexplicable.