In 1931, Kurt Godel issued a paper titled “On Formally Undecidable Propositions in Principia Mathematica and Related Systems I”. His theorem (paraphrased) stated that,
“All consistent axiomatic formulations of
number theory include undecidable propositions.” [1]
This was a bombshell to Bertrand Russell and Alfred
North Whitehead, who had created the ultimate mathematical work, the Principia
Mathematica. In fact, Russell had encountered a dominant paradox, the “set of all
sets” paradox, which would not allow the complete resolution of set theory.
“The implication [of Godel’s 2nd
theorem] is that all logical systems of any complexity are, by
definition, incomplete; each of them contains, at any given time, more true
statements than it can possibly prove according to its own defining set of
rules.”[2]
The impact of Godel’s theorem was that a consistent
system cannot verify itself. It
requires a higher order system to verify it.
And this leads to the infinite regress, where the higher order system
cannot verify itself either, and requires an even higher order system to verify
it. The end cannot be reached.
“And it
[Godel’s 2nd theorem]
has been taken to imply that you'll never entirely understand yourself,
since your mind, like any other closed system, can only be sure of what it
knows about itself by relying on what it knows about itself.”[3]
Thus number theory, and by extension mathematics,
logic, and rational thought, cannot verify themselves. An infinite series of higher order systems is required. So mathematics, logic and rational thought
cannot be known to valid – except intuitively.
Once again, intuition is a fundamental requirement,
if mathematics, logic, and rational thought are not to be rejected as
“undecidable” or “unknowable”.