The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules... One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question that can at all be formally expressed in these systems. It will be shown below that this is not the case, that on the contrary there are in the two systems mentioned relatively simple problems in the theory of integers that cannot be decided on the basis of the axioms.

All generalizations, with the possible exception of this one, are false.

I don't believe in empirical science. I only believe in a priori truth.

But every error is due to extraneous factors (such as emotion and education); reason itself does not err.

...a consistency proof for [any] system ... can be carried out only by means of modes of inference that are not formalized in the system ... itself.

The axiomatic method is very powerful

Nothing new had been done in Logic since Aristotle!