I commend you on your open inquiring mind.

Now, I'm not sure if you realized but, G�del's proof doesn't necessarily have to shred the last hope of perfection of mathematics.
in fact, it opens up many doors for new things. This means that our knowledge will always have new properties to be found, new constants, etc.

Now, you are right in that the basic laws of math can be broken and do become inconsistent under certain conditions. (i.e. you can create matter and energy from no where) But, even when they do break down they still obey some other set of rules on top of that, and I can assure you there will be ways to account for those variations.
So, you're right, the absolution of math is fallible. But you can say that the absolution of the absolution is always perfect. Always. And i find a calming peacefull feeling in that sort of thing.

Now as for those interesting paradoxes, from what I experienced, most paradoxes are either trying to make sense from nonsense or nonsense from nonsense. When there is a paradoxical statement, i know that at least one of two things are true: Either one or both of the statements simply cannot be.
For example, if you've ever heard that reality is a paradox. well, thats not right, something doesnt have to have someone to percieve it for it to exist, whether you know about it or not.

As for the 'pi is rational', that's saying that you can divide any number by 1 and say it's rational. which is true. but it has to be proven that pi is not transcendal before you can divide it by anything. And it's been proven to be transcendal. :)

umm, i'm fresh out for now but your welcome to ask about more stuff if you'd like.

If you've made it this far and ive kept your attention then you must be on somewhat the same level as me.
I am willing to offer a freebie, to anyone who can tell me how to use a Jacobian transform properly and to adequately explain it's interpretation and uses.

another interesting thought to chew on: the limit of reality is perfection, and vice versa. but in a weird abstract way, everything is already perfect and as it should be.

There are many illusions in reality that are yet to be overcome. don't give up on it.