Section 2.6: Problem 8 Solution
Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
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.
If not, then for every
there is some
and a model
of
of size
such that
is false in
. But then,
is true in all such
’s, and, by Theorem 26A,
has an infinite model
, implying that
is false in the infinite model
of
. Contradiction.