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Fix `\sf` grouping
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LSpice
LSpice
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What's the benefit of adding a well-ordering over all classes to $\sf MK$$\textsf{MK}$?

Working in $\sf ZF + \text { there is a strongly inaccessible cardinal}$$\textsf{ZF} + \text {there is a strongly inaccessible cardinal}$.

Let $\kappa$ be the first strongly inaccessible cardinal, and let $|V_\kappa|= \kappa$$\lvert V_\kappa\rvert= \kappa$, then $(V_{\kappa+1}, \in)$ would be a model of $\sf MK$$\textsf{MK}$.

Now, is it consistent to add that $V_{\kappa+1}$ is non-well-orderable?

If yes, then what's the benefit of having $V_{\kappa+1}$ well-orderable, on the theory $\sf MK$$\textsf{MK}$? I mean what additional axioms in the language of $\sf MK$$\textsf{MK}$ would this confer?

What's the benefit of adding a well-ordering over all classes to $\sf MK$?

Working in $\sf ZF + \text { there is a strongly inaccessible cardinal}$

Let $\kappa$ be the first strongly inaccessible cardinal, and let $|V_\kappa|= \kappa$, then $(V_{\kappa+1}, \in)$ would be a model of $\sf MK$.

Now, is it consistent to add that $V_{\kappa+1}$ is non-well-orderable?

If yes, then what's the benefit of having $V_{\kappa+1}$ well-orderable, on the theory $\sf MK$? I mean what additional axioms in the language of $\sf MK$ would this confer?

What's the benefit of adding a well-ordering over all classes to $\textsf{MK}$?

Working in $\textsf{ZF} + \text {there is a strongly inaccessible cardinal}$.

Let $\kappa$ be the first strongly inaccessible cardinal, and let $\lvert V_\kappa\rvert= \kappa$, then $(V_{\kappa+1}, \in)$ would be a model of $\textsf{MK}$.

Now, is it consistent to add that $V_{\kappa+1}$ is non-well-orderable?

If yes, then what's the benefit of having $V_{\kappa+1}$ well-orderable, on the theory $\textsf{MK}$? I mean what additional axioms in the language of $\textsf{MK}$ would this confer?

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Zuhair Al-Johar
Zuhair Al-Johar
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What's the benefit of adding a well-ordering over all classes to $\sf MK$?

Working in $\sf ZF + \text { there is a strongly inaccessible cardinal}$

Let $\kappa$ be the first strongly inaccessible cardinal, and let $|V_\kappa|= \kappa$, then $(V_{\kappa+1}, \in)$ would be a model of $\sf MK$.

Now, is it consistent to add that $V_{\kappa+1}$ is non-well-orderable?

If yes, then what's the benefit of having $V_{\kappa+1}$ well-orderable, on the theory $\sf MK$? I mean what additional axioms in the language of $\sf MK$ would this confer?