Define a formal tautology as a statement where by the nature of its atomic components there exists no truth-value assignment where it is not true. A contingent statement is a statement that is true by facts of the world. A statement is necessarily true if the statement is true in all possible worlds. A necessarily true statement is not contingent and a contingent statement is not necessarily true. A formal tautology is necessarily true. But a necessarily true statement is not always a formal tautology.


Are there mathematical theorems which are contingent? Are all mathematical theorems necessarily true? 



Are all mathematical theorems necessarily true?
Apart from the storm of comments, let me just try to answer the question. 


There are several ways in a which a mathematical theorem
can be contingent.


* First, the independence phenomenon in set theory shows the striking ubiquity of
contingency in mathematics. For example, the Continuum Hypothesis is true
is some set-theoretic
worlds and false in others (and we can control it
exactly). There are hundreds of other examples of
statements with the same independence status---they are
true in some worlds and false in others. The method of
forcing has been used to spectacular effect in proving
many of these independence results.
* The Incompleteness phenomenon of Goedel can be used to show that
whether a statement is provable or not from a given
axiom system (in classical logic) can be contingent.
Specifically, the Incompleteness theorem says that no
theory T, if consistent, can prove its own consistency.
Thus, if ZFC is consistent, then there are models of ZFC
in which ZFC is thought to be inconsistent. In such a
model, ZFC is thought to prove any statement at all! But
in our world, not all these statements will be theorems.
Thus, in this sense, even the question of whether a given statement is a theorem or not
can be contingent.
* The large cardinal hierarchy in set theory provides
numerous examples of statements transcending the
consistency strength of weaker statements. If large cardinals are consistent, then there are some set-theoretical universes in which ZFC proves that there are no inaccessible cardinals and other universes where it does not.


There are also several ways in which contingency is ruled
out.


* First, one of the most important properties of a proof
system is *soundness*,
which means that any statement provable in the system from
true hypotheses will remain true. Of course, this is an
expected feature of any proof system worthy of the name. A
*theorem* is a statement having a proof in such a system. Once we have adopted a
given proof system that is sound, and the axioms are all
necessarily true, then the theorems will also all be necessarily
true. In this sense, there can be no contingent theorems.
* Second, one of the profound achievements of Goedel was
his *Completeness theorem*, which states that any
statement that holds in all models of a given first order
theory T, actually has a proof from the theory. For
example, every statement in the language of group theory
that happens to be true in all groups, actually has a
finite proof from the group axioms (using any of several
proof systems). This is far from obvious, and I find it profound.
But it answers a dual version of a question you might
have asked, which I find interesting, namely:
Is every necessary truth a theorem?
The answer is Yes, and this is just what the Completeness
theorem expresses.


These last two points together explain that if one takes the possible worlds to be all models of a given theory, then the necessary truths are precisely the theorems of that theory.


