This used to be a vexing problem, but today most mathematicians take the "pragmatic" approach -- and simply ignore it.

A system is inconsistent if it can prove both a proposition and its negation. Clearly, this is not a healthy state of affairs to be in: having proved a proposition and its negation, any other proposition can be proved! Around the turn of the 20th century, it was thought to be helpful to prove that "mathematics" (or more precisely set theory, which is the language that most other mathematics could be cast into) is indeed consistent.

Gödel's second theorem dashed that hope, forever. It states that no "sufficiently complex" system which is consistent can prove its consistency (here "sufficiently complex" is not very complex -- being able to define arithmetic is more than enough). Obviously if the system is inconsistent (and able to formulate the claim that it is consistent), then it can prove it is consistent, since it can prove anything. So if set theory could prove itself consistent, we'd be in trouble -- it would be inconsistent.

But most mathematicians have stopped worrying about all this. A system can be proved consistent by embedding it in a larger system and proving consistency there. (Of course, this assumes the larger system is itself consistent, so this is not a rigourously justifiable procedure). For example, Peano arithmetic has a model (e.g. in set theory), so we call it consistent. The problem is that set theory is too comprehensive to be embeddable in anything much larger, which would still contain self-evident truths (even set theory has very subtle propositions, like the continuum hypothesis, or the axiom of choice). But we can still claim we believe set theory is consistent, because we can imagine (parts of) it. Here imagination has replaced model-building, but it is the best we can hope to do.

The rival approach is to embed set theory in a larger, simpler system. This system would need to have self-evident "combinatorial" axioms, so we'd know (intuitively) that it is indeed consistent. And it would suffice for building a model for set theory. With any luck, it would also settle questions of the truth of set theory's "disreputable axioms".

Log in or register to write something here or to contact authors.