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.
A Self-Eating Expression is meant to be unwound sequentially in the ‘Backward Step’. It is not to be stared at as a static logical complexity [which is the way it has been treated in the professional literature]. That just sinks you deeper into the swamp.
The ‘Liar’s Paradox’ is a paradox only because we don’t know if Epimenides is a liar or not. He is neither. This might be more transparent if: ‘All Cretans are liars’ is replaced with: ‘All Words are false’ which denies you the ability to extract yourself out of the loop and pronounce on its truth or falsity.