Bertrand Russell and Alfred Whitehead wanted to produce a founding Bible that built up the rules of Mathematics from the ground-up. So they began with Peano’s work and titled their opus Principia Mathematica. [1910; Latin titles are always dangerous, vide Wittgenstein and his Tractatus Logico Philosophicus.]
The celebrated 3-volume opus was among other much grander ambitions, an attempt to resolve the conflicts between the observing Subject and his inclusion or exclusion in a Set.
The issue never found a resolution [Russell and Whitehead had to use a variety of sequenced exceptions to hold the logic together]. Until Kurt Godel came along.
[No point in repeating here what is well in the public domain. The story of the Godel iceberg is common knowledge among the mathematically-inclined and there is a large readable literature out there on Godel, Russell’s Paradox, Principia Mathematica and the rest. I’ll limit myself to the outlines.]
In 1952, Harvard University honored the Logician Kurt Godel with an award that read: [For] ‘The discovery of the most significant mathematical truth of the century.’
The award was for Kurt Godel’s seminal paper: ‘On formally undecidable propositions of Principia Mathematica and related systems.’
Godel proved that the consistency of any Formal System cannot be proven using the methods of that System without simultaneously violating the basis of consistency of that System itself [There were other issues such as ‘Completeness’. But this will do for now.]
Can you sniff the Self-Loop?
A Theorem which itself drew upon the work of Jules Richard and others and went forward in seminal revelations that have hugely influenced modern computer, information and cognitive theories [as with the work of Alan Turing and numerous less famous mathematicians].
In a footnote to his paper Godel notes: ‘Any Epistemological Antinomy, such as the ‘Liar’s Paradox’ could be used for a similar proof’. Godel’s resolution relies on the form of a famously curious shout by Epimenides the Cretan who declared: ‘All Cretans are liars!’
‘All Cretans are liars’ is close. It would be closer if, unless you happen to be a Cretan yourself, it read: ‘All men and women are liars’, for then it would include you, the observer, in any interpretation of this claim. A full-blooded Self-Eating Expression.