Consider the set and use the form of compactness theorem from this thread.
Assume that is true in all infinite models of a theory . Show that there is a finite number such that is true in all models of for which has or more elements.
The sentence for there are at least things for is
Any hint to get started?
My understanding is,
is true in all infinite models of a theory T by assumption.
By compactness theorem, is true in some finite models of a theory T having cardinality k.
However, I think we still have to establish that is true in all finite models of a theory T having cardinality greater than k.
Is this idea wrong?