Goedel’s theorem shows that Hilbert’s program is impossible and that mathematics itself is built upon faith assumptions, the axioms that we find that we have to use. There are no closed, self-defining systems. “logical systems of arithmetic can never contain a valid proof of their own consistency.”
The real field is complete in the sense of Goedel.
“logical systems of arithmetic can never contain a valid proof of their own consistency.”
But you're not talking about peano arithmetic, are you? So why not simply leave Goedel out of this?