# Section 2.3: Problem 1 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

Show that for a proper initial segment
of a wff
, we have
.

Here, for any wff
, by
we denote an initial proper segment of
.

For an atomic formula
consists of an
-place predicate symbol (
) followed by at most
complete terms (
) and, possibly, an initial proper segment of the subsequent term (
). Overall,
. Then, by induction.

For
,
is one of the following,
(
),
(
),
(
),
(
).

For
,
is one of the following,
(
),
(
),
(
),
(
),
(
),
(
).

For
,
is one of the following,
(
),
(
),
(
).