You forget the hypotheses of the theorems.
First these are not a theorems
about math. These are theorems about formal systems with the capability of
reproducing Peano arithmetic. This is a large portion of math but this is not
all of math. These theorems don't apply to Boolean algebra for
example.
Second, if you don't have consistency, you don't have math. The
incompleteness theorems work because consistency is assumed in the hypotheses.
Without an assumption of consistency the proofs mean nothing.
You wrote
literally:
maths in its very foundations is built on incompleteness
and
inconsistency.
Maybe you wrote something that means more than
you intended, but it is not possible to build math on inconsistency. You can't
be confident your proofs mean anything when the conclusions are inconsistent.
What Gödel's second incompleteness theorem said is you can't prove the
consistency of Peano arithmetic from within the same system. This doesn't mean
it isn't consistent, just that the proof is unavailable from within the same
system. This is an incompleteness theorem because it is about a proof that can't
be done. This is not about (in)consistency itself.
FWIW Gentzen has managed
to prove
the consistency of first order arithmetic using transfinite
induction.
What Gödel proved is not that you cannot have a solid
foundation underpinning mathematics. He proved that you cannot build this
foundation and prove this it is solid (ie consistent) using the methods
envisioned by David Hilbert. The possibility of having a solid foundation you
can't prove to be solid is left open. Then empirical evidence of solidity may be
available. If after a few decades of searching for the inconsistencies none are
being found, mathematicians may have grounds to believe their foundations are
consistent.
Mathematicians can't imagine a system of logic for writing some
foundations of mathematics which won't be subject to the methods of Gödel. But
how can we know whether this is a failure of human imagination or a fundamental
limitation of mathematics? Most mathematicians think it is a fundamental
limitation on the grounds that all attempts to find a work around have failed.
This is drawing conclusions from strong empirical evidence. This is different
from proven mathematical knowledge.
As Gentzen has illustrated, it may still
be possible to discover some principles of logic or mathematics which will
surprise us. Gentzen proof has its own assumptions and I am not familiar with
it. I don't know how big of a work around Gödel it offers. But I think it hints
that we should be careful not to overstate what Gödel theorems actually mean.
They are very strong theorems but they are not without limits. And I also think
you tend to overstate them because you are not familiar with their limits.
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