# Section 2.6: Problem 2 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

Let
and
be theories (in the same language) such that (i)
, (ii)
is complete, and (iii)
is satisfiable. Show that
.

If
, then
, hence,
, and
.

Note. Another way to look at the problem, as suggested in the commentary to the book by the author, is to recall Exercise 8 of Section 2.2, which says that for a set of sentences
such that for every sentence
,
or
, and every model
of
,
iff
. In our case, we just note that every model
of
is a model of
, and, hence, by the result of the exercise, if
then
, then
for every model
of
, then
, and
.