Thread: Foundations of mathematics - something is troubling me

One of the main reasons I study mathematics is because of the certainty of it. Given certain assumptions, one can deduce whether or not something is true.

Recently, I've just discovered that the foundations of mathematics is not self-consistent. There is a possibility that there may exist a contradiction in the system. But if there is a possibility that a contradiction may exist, then why should we be sure that the theorems we prove from the axioms are true?

Also, when we're talking about consistency in systems here, we're talking about set theory right? For example, for the field axioms, that system of axioms must be consistent, no?

This is really bothering me. All help would be appreciated!

Originally Posted by JG89

One of the main reasons I study mathematics is because of the certainty of it. Given certain assumptions, one can deduce whether or not something is true.

Recently, I've just discovered that the foundations of mathematics is not self-consistent. There is a possibility that there may exist a contradiction in the system. But if there is a possibility that a contradiction may exist, then why should we be sure that the theorems we prove from the axioms are true?

Also, when we're talking about consistency in systems here, we're talking about set theory right? For example, for the field axioms, that system of axioms must be consistent, no?

This is really bothering me. All help would be appreciated!

emakarov can correct me if I'm wrong, but I'm not aware of a contradiction in, say, ZF set theory or symbolic logic as of now. We do know that while first-order logic is complete, second-order logic is incomplete. But that's not the same question as consistency.

Have a look at this page.
It will give you a bit more on the scope and meaning of your question.

Originally Posted by JG89

Recently, I've just discovered that the foundations of mathematics is not self-consistent.

You misunderstood something you read.

Originally Posted by JG89

There is a possibility that there may exist a contradiction in the system.

What system? There are lots of systems. Perhaps you are referring to the second incompleteness theorem that informs us that certain systems cannot prove their own consistency. But the second incompleteness theorem does not suggest that any particular system is inconsistent.

Originally Posted by JG89

Also, when we're talking about consistency in systems here, we're talking about set theory right?

Any system can be considered for consistency or inconsistency. Formal set theories are among the many systems.

Originally Posted by JG89

the field axioms, that system of axioms must be consistent, no?

I know of no evidence that they are not.

I think you need to study this matter more methodically. It seems to me that you're going from bits and pieces of information about this. To get a clear grasp on these matters I would first learn the predicate calculus, then a certain amount of set theory, then some basic mathematical logic.