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$\mathrm{K}(W)$ and $\mathcal{T}_\alpha(W)$


For a class $W$:

  • $\mathcal{T}_0(W)=W$
  • $\mathcal{T}_{\alpha+1}(W)=\{x\in V:x\subseteq\mathcal{T}_\alpha(W)\}\cup\mathcal{T}_\alpha(W)$
  • $\mathcal{T}_\beta(W)=\bigcup_{\alpha<\beta}\mathcal{T}_\alpha(W)$ for limit ordinals $\beta$
  • $\mathrm{K}(W)=\min\{\alpha:\mathcal{T}_\alpha(W)=V\}$

Loose Meaning

$\mathcal{T}_\alpha$ is a function on a class (generally transitive) $W$ such that $W \subseteq\mathcal{T}_\alpha(W)$. For almost every transitive class $W$, $\mathcal{T}(W)$, which is the union of all $\mathcal{T}_\alpha(W)$, is $V$. However, this is not the case when $W$ is a transitive almost universal class. It is a generalization of $V_\alpha$ to all classes. For any class $W$ and any $\alpha>0$, $\mathcal{T}_\alpha(W)$ is a transitive class. Thus if $W$ is transitive, then $\mathcal{T}_\alpha(W)$ is transitive for every $\alpha$.

$\mathrm{K}(W)$ is a function on a transitive class $W$ that measures how "close" it is to $V$. Interestingly, as stated above, there are transitive $W$ which are very close to $V$ (i.e. almost universal) yet their "distance" from $V$ does not exist because $\mathcal{T}_\alpha(W)=W$ for every $\alpha$. These cases should not be considered when using this function as they are indisputably outliers.

The $\mathrm{K}(L)\geq\alpha$ Axioms

$\mathrm{K}(L)$ is an interesting subject. Assuming $V=L$, $\mathrm{K}(L)=0$. Interestingly, $\mathrm{K}(L)\neq n$ for any finite number $n>0$.

It could be stated as an axiom that $\mathrm{K}(L)\geq\alpha$; with $\mathrm{K}(L)\geq 1\Leftrightarrow 0^{\#}$. The axiom that $\mathrm{K}(L)$ does not exist should be called The Axiom of Total Inconstructibility (ATI).

Known Facts

Here are a list of known facts about $\mathrm{K}(L)$ and $\mathcal{T}_\alpha(L)$:

  • $V_{\omega+\alpha}\subset\mathcal{T}_\alpha(L)$
  • $\mathrm{K}(L)=0\Leftrightarrow$ $V=L$
  • $\mathrm{K}(L)\neq\alpha+1$ for any $\alpha$
  • $\mathrm{K}(L)=\omega$

(This article is to be edited when more facts are known about $\mathrm{K}(L)$).