Take the 2-minute tour ×  The following is an excerpt from Paul
Cohen's "The Discovery of Forcing", pp 1091, in which he
explains why we do not want to add new ordinals to a countable
transitive model $M$ when extending it using forcing:  Suppose
$M$ were a countable model. Up until now we have not discussed
the role countability might play. This means that all the sets
of $M$ are countable, although the enumeration of some sets of
$M$ does not exist in $M$. The simplest example would be the
uncountable ordinals in $M$. These of course are actually
countable ordinals, and hence there is an ordinal $I$, not in
$M$, which is countable, and which is larger than all the
ordinals of $M$. Since $I$ is countable, it can be expressed as
a relation on the integers and hence coded as a set $a$ of
integers. Now if by misfortune we try to adjoin this $a$ to $M$,
the result cannot possibly be a model for ZF. For if it were,
the ordinal $I$ as coded by $I$1 would have to appear in $M(a)$.
However, we also made the rigid as