I doubt that anyone can show you how to prove Goedel's incompleteness theorem- its very long and very deep. There are a number of books written on it alone.
Yes, Goedel's theorem says that any system of axioms (large enough to encompass the natural numbers) is either inconsistent (in which case it's useless) or incomplete (in which case it's not perfect!).
Saying that a system is incomplete means there exist some theorem, statable in terms of the system which can be neither proven nor disproven. Of course, you could always add that theorem itself as an axiom but then you would have some other theorem that can be neither proven nor disproven.
One cannot prove, in absolute terms, that most of the systems used in mathematics are consistent- it is possible to prove, for example, that Euclidean geometry is consistent if and only if hyperbolic geometry is, or that Euclidean geometry is consistent if and only if algebra is, which is consistent if and only if the natural numbers are, which is true if and only if set theory is consistent....
Since there are one heckuva lot of things we don't know HOW to prove, most mathematicians are willing to live with the knowledge that they can't prove EVERYTHING!
It applies only to logical systems based on axioms. That includes all of mathematics. It does not apply to physics or other sciences that are based on experimental evidencel.
It is conceivable that, if a particular axiomatic system used to model physics were inconsistent, the mathematical foundation would be in trouble. The experimental evidence, of course, would still be valid. I will point out that there is no evidence that any important mathematical system is inconsistent.
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