The essence of the G\u00f6del idea is to encode not just the facts but also the \u2018facts about the facts\u2019 of the formal system being examined within the framework of the system being examined.\u00a0 This meta-mathematics technique allowed G\u00f6del to prove simple facts like \u20182 + 2 = 4\u2019 and hard facts like \u2018not all true statements are axioms or are theorems \u2013 some are simply out of reach of the formal system to prove\u2019 within the context of the system itself.\u00a0 The hard facts come from the system talking about or referring to itself with its own language.<\/p>\n
As astonishing as Godel\u2019s theorem is, the concept of paradoxes within self-referential systems is actually a very common experience in natural language.\u00a0 All of us have played at one time or another with odd sentences like \u2018This sentence is false!\u2019.\u00a0 Examined from a strictly mechanical and logical vantage, how should that sentence be parsed?\u00a0 If the sentence is true then it is lying to us.\u00a0 If it is false, then it is sweetly and innocently telling us the truth.\u00a0 This example of the liar\u2019s paradox<\/a> has been known since antiquity and variation of it have appeared throughout the ages in stories of all sorts.<\/p>\n Perhaps the most famous example comes from the original Star Trek television series in an episode entitled \u2018I Mudd\u2019. In this installment of the ongoing adventures of the starship Enterprise, an impish Captain Kirk defeats a colony of androids that hold him and his crew hostage by exploiting their inability to be meta.<\/p>\n