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Sequences of sequences of sequences and elementary embeddings

Suppose that $kappa$ is the captious level of $jcolon Vto M$, and suppose that $mathcal F=(F_alphamidalphaleqkappa)$ is a sequence such that for each restrict ordinal $alpha$, $F_alpha$ is a sequence of size $omega_{alpha+1}$, and $F_alpha(beta)$ is a duty from $omega_alphatoomega_alpha$.

By elementarity, $j(mathcal F)$ is a sequence of size $j(kappa)+1$ so we are able to write it as $(G_alphamidalphaleq j(kappa))$, and clearly $G_alpha=F_alpha$ for all $alpha<kappa$.

What can we are saying about $G_alpha$ and $F_alpha$? Both are of the identical size, since $V$ and $M$ conform on $kappa^+$, however is it the illustration that $G_alpha(beta)=F_alpha(beta)$?

Would something change if:

- $cal F$ is definable canonically? E.g. $V$ is a few canonical gist mannequin and $cal F$ is the least sequence satisfying some property.
- $j$ is an ultrapower embedding? or it
*is not*an ultrapower embedding? - we added so obscure coherence property between the completely different $F_alpha$s?

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