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## Main Question or Discussion Point

[tex]M\equiv N[/tex] means M and N are elementarily equivalent, that is, for any setence S in the language of M and N, [tex]M\models S \Leftrightarrow N\models S[/tex].

An elementary embedding f:M->N is a mapping f:|M|->|N| between the underlying sets such that, for any formula t(x) and matching tuple y of elements of M, we have that [tex]M\models t(y) \Rightarrow n\models t(f(y))[/tex]

Assume that [tex]M\equiv N[/tex]. Show that there is an ultrafilter U=(I, U) and an elementary embedding [tex]g:N\rightarrow M^U[/tex].

How do I do that?

An elementary embedding f:M->N is a mapping f:|M|->|N| between the underlying sets such that, for any formula t(x) and matching tuple y of elements of M, we have that [tex]M\models t(y) \Rightarrow n\models t(f(y))[/tex]

Assume that [tex]M\equiv N[/tex]. Show that there is an ultrafilter U=(I, U) and an elementary embedding [tex]g:N\rightarrow M^U[/tex].

How do I do that?