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.

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.

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.

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.

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.

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.

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.