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LSpice
LSpice
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This MathOverflow question by Trevor Wilson The MathOverflow question A weak (?) form of Shelah cardinals by Trevor Wilson defines weakly Shelah cardinals as follows:

A cardinal $\kappa$ is weakly Shelah if for all $f : \kappa \to \kappa$ there is some $\alpha < \kappa$ that is closed under $f$ and there is some elementary embedding $j : V \to M$ (where $M$ is a transitive class) such that $\operatorname{crit}(j) = \alpha$ and $j(\alpha) > \kappa$ and $V_{j(f)(\kappa)} \subset M$.

I would like to add a requirement that $j(f) \upharpoonright \kappa = f$ but as a commentcomment by Sean Cox on this question made me realize, it is not clear that that definition is equivalent to Trevor Wilson's definition. Are the definitions equivalent?

This MathOverflow question by Trevor Wilson defines weakly Shelah cardinals as follows:

A cardinal $\kappa$ is weakly Shelah if for all $f : \kappa \to \kappa$ there is some $\alpha < \kappa$ that is closed under $f$ and there is some elementary embedding $j : V \to M$ (where $M$ is a transitive class) such that $\operatorname{crit}(j) = \alpha$ and $j(\alpha) > \kappa$ and $V_{j(f)(\kappa)} \subset M$.

I would like to add a requirement that $j(f) \upharpoonright \kappa = f$ but as a comment by Sean Cox on this question made me realize, it is not clear that that definition is equivalent to Trevor Wilson's definition. Are the definitions equivalent?

The MathOverflow question A weak (?) form of Shelah cardinals by Trevor Wilson defines weakly Shelah cardinals as follows:

A cardinal $\kappa$ is weakly Shelah if for all $f : \kappa \to \kappa$ there is some $\alpha < \kappa$ that is closed under $f$ and there is some elementary embedding $j : V \to M$ (where $M$ is a transitive class) such that $\operatorname{crit}(j) = \alpha$ and $j(\alpha) > \kappa$ and $V_{j(f)(\kappa)} \subset M$.

I would like to add a requirement that $j(f) \upharpoonright \kappa = f$ but as a comment by Sean Cox on this question made me realize, it is not clear that that definition is equivalent to Trevor Wilson's definition. Are the definitions equivalent?

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Arvid Samuelsson
Arvid Samuelsson
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Is this strengthening of the definition of weakly Shelah cardinals equivalent to being weakly Shelah?

This MathOverflow question by Trevor Wilson defines weakly Shelah cardinals as follows:

A cardinal $\kappa$ is weakly Shelah if for all $f : \kappa \to \kappa$ there is some $\alpha < \kappa$ that is closed under $f$ and there is some elementary embedding $j : V \to M$ (where $M$ is a transitive class) such that $\operatorname{crit}(j) = \alpha$ and $j(\alpha) > \kappa$ and $V_{j(f)(\kappa)} \subset M$.

I would like to add a requirement that $j(f) \upharpoonright \kappa = f$ but as a comment by Sean Cox on this question made me realize, it is not clear that that definition is equivalent to Trevor Wilson's definition. Are the definitions equivalent?