The Brandon Five
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In science, the agreed on premises have been validated by experimentation, observation, publication and subsequent peer review. For example, behavior described by the Ideal Gas Law is a premise that is agreed to by the scientific community.
Agreed upon...hmm, why would that be necessary when dealing with "facts"?
There is a whole sub-discipline of philosophy called epistemology that is dedicated to these questions.
Getting back to Gödel. There are two parts to his Incompleteness Theorem:
https://en.wikipedia.org/wiki/Kurt_Gödel
In 1931 and while still in Vienna, Gödel published his incompleteness theorems in Über formal unentscheidbare Sätze der "Principia Mathematica" und verwandter Systeme (called in English "On Formally Undecidable Propositions of "Principia Mathematica" and Related Systems"). In that article, he proved for any computable axiomatic system that is powerful enough to describe the arithmetic of the natural numbers (e.g. the Peano axioms or Zermelo–Fraenkel set theory with the axiom of choice), that:
These theorems ended a half-century of attempts, beginning with the work of Frege and culminating in Principia Mathematica and Hilbert's formalism, to find a set of axioms sufficient for all mathematics.
- If the system is consistent, it cannot be complete.
- The consistency of the axioms cannot be proven within the system
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