 # Sequences of sequences of sequences and elementary embeddings Answer

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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.

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:

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

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