>>86
Mathematicians are trying to solve nonsense problems, they created themselves by some wrong definition of "consistency" in a poorly defined framework, that rests on some crazy axioms (see ZFC), purpose of which a layman wont get without tracing full 20st century history of mathematics, which is full of hacks and silly conventions.
I'm sorry, but to me consistency makes perfect sense, all these terms are well-defined. It makes sense within computer science, first order logic and the various arithmetical theories.
I do partially agree that the syntax of some parts of math is a bit messy and it could be done in a more easily parsable manner, but this doesn't change the fact that the semantics would still stay the same even if you chose a better syntax (look at some theorem provers if you want examples of more "parsable" syntaxes).
I'm not an expert on Set Theory, but as far as I understand it, originally there was the simpler naive set theory, with rather intuitive axioms, but it proved inconsistent (see Russell's paradox), so they had to choose some axioms that limit what sets are and how they can be defined. The axioms themselves may seem a bit strange at first as some of them do appear non-constructive. If that bothers you, you could look at a constructive or iterative approach of building sets, one example that I like is described in the first chapters of Boolos' "Logic, Logic, Logic" - it takes a few simple axioms which are intuitive enough (as opposed to the ones in ZFC which make you wonder where they appeared from) and then it shows how sets are built in it and also shows that most ZF axioms are theorems in that particular theory! You might not like it because it has infinity in it, but it can't be avoided, even with only natural numbers: the process of listing the numbers obtain from 0 and a successor function never terminates - the list/set is infinite (countably). Instead, just understand what infinity is and where it comes from and then you can even look at things like ordinals which are well-defined (constructively even, and a computer/theorem prover can talk about them too, despite being 'infinite'!).
It's consitent, when it's parts are consistent, together with their composition.
How do you know the parts are consistent?
Consistency in math means that starting with some axioms and some logic (inference rules), you will never prove any syntactically valid statements in that language to be BOTH true and false. It's fairly simple and well-defined. You can even write a program that could find an error in some inconsistent axiomatic system (of course, for most modern ones, this is highly unlikely and your theorem prover which is searching for an inconsistency will never halt if the theory is consistent, but you'll never know this!).
Like the Banach-Tarski Theorem, which postulates that given single orange you can transform it into two oranges, by the sole power of applying Set Theory axioms. Behold The Wonder of Infinity's Creation!
I don't find that paradox that much stranger than saying that the cardinality of some continous real interval like [0,1] has the same as the cardinality of all real numbers, or that there are as many even natural numbers as natural numbers or as rational numbers, yet there are more reals than naturals. Infinity is counterintuitive like that, but if you can show a bijective function between 2 sets, then their cardinality is the same. It may be strange, but as I said before, there is no reason why you should set a higher bound for natural numbers, and if you do, you're only inviting trouble for your theories (I don't even know of a single consistent ultrafinitist theory, but maybe you know more about them than me). Not that anyone forces you to work with infinity directly, computable functions only work with finite numbers, just that one has to acknowledge that there's a countable infinity of them (both computable functions and natural numbers).