Difference between revisions of "User blog:Zetapology/K(L)"

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m (For any transitive class $W$ and any $\alpha$, $\mathcal{T}_\alpha(W)$ is a transitive class.)
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{{DISPLAYTITLE: K(L)}}
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Deleted Blog Post (old, inconsistent/useless)
 
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== $\mathrm{K}(W)$ and $\mathcal{T}_\alpha(W)$ ==
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=== Definition ===
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For a class $W$:
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*$\mathcal{T}_0(W)=W$
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*$\mathcal{T}_{\alpha+1}(W)=\{x\in V:x\subseteq\mathcal{T}_\alpha(W)\}\cup\mathcal{T}_\alpha(W)$
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*$\mathcal{T}_\beta(W)=\bigcup_{\alpha<\beta}\mathcal{T}_\alpha(W)$ for limit ordinals $\beta$
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*$\mathrm{K}(W)=\min\{\alpha:\mathcal{T}_\alpha(W)=V\}$
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=== Loose Meaning ===
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$\mathcal{T}_\alpha$ 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 transitive class $W$ and any $\alpha$, $\mathcal{T}_\alpha(W)$ is a transitive class.
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$\mathrm{K}(W)$ 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.
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== The $\mathrm{K}(L)\geq\alpha$ Axioms ==
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$\mathrm{K}(L)$ is an interesting subject. Assuming $V=L$, $\mathrm{K}(L)=0$. Interestingly, $\mathrm{K}(L)\neq n$ for any finite number $n>0$.
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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)'''.
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== Known Facts ==
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Here are a list of known facts about $\mathrm{K}(L)$ and $\mathcal{T}_\alpha(L)$:
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*$V_{\omega+\alpha}\subset\mathcal{T}_\alpha(L)$
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*$\mathrm{K}(L)=0\Leftrightarrow$ [[Zero sharp|$V=L$]]
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*$\mathrm{K}(L)\neq\alpha+1$ for any $\alpha$
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*$\mathrm{K}(L)=\omega$
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(This article is to be edited when more facts are known about $\mathrm{K}(L)$).
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Latest revision as of 23:25, 11 October 2017

Deleted Blog Post (old, inconsistent/useless)