No matter how many articles I read on this subject I cannot comprehend how it proves what it proves. I do well with words and rhetorics, philosophy and science – but as soon as you add numbers my mind goes blank. Not very helpful when those fields often rely on equations and models for explanations and proof. I can somewhat understand equations if explained in a simple or cohesive way – but if at all possible analogies or just word-centric explanations would be very helpful.
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Self referential paradoxes are one of the oldest kinds of paradoxes known to man.
The famous “everything I say is a lie” comes to mind, or “this statement is false.”
Bertrand Russell used a variation of this to construct his famous paradox in set theory. Many, many different paradoxes and brain teasers boil down to self referential statements. Keep this in mind, and I’ll walk you through the history.
Now in mathematics, things are proven to be true via inference from other things known to be true. Pick any theorem in an introductory calculus textbook, and you can keep asking “why” or “how do we know that is true.” Eventually, this chain must end somewhere.
The place it ends is called axioms. Axioms aren’t really “true” or “false” rather “useful” or “not useful.” Euclid included several axioms in his description of geometry, most famous his fifth postulate. Mathematicians tried for centuries to prove the fifth postulate from the other, simpler axioms, but failed. In the 19th century, it was discovered that taking a different axiom than Euclid’s fifth resulted in different geometries. These geometries, elliptical and hyperbolic, were actually used in the development of general relativity and form part of our modern understanding of cosmology.
Between Cantor’s paradoxes of different sized infinites, the discovery of new geometries, Riemann and Cauchy placing calculus on firmer ground, and Russell’s paradox, mathematicians became increasingly concerned with their foundations, the axioms as we call them.
Since the axioms are a freely chosen set of base truths, we can construct them how we please. There are certain features we desire.
1.) Sufficiently strong to do arithmetic with. If your axioms aren’t at least sufficient to describe adding natural numbers, it likely isn’t going to be useful. There would be little you could do with such a limited system. The technical term here is “Peano arithmetic” after a mathematician named Giuseppe Peano, who listed the first set of axioms for the natural numbers.
2.) Consistent. From your axioms, you should not be able to derive both a statement and it’s negation. Once you allow contradictions in, principle of explosion results in anything in principle being proven. We only want our axioms to be able to prove true statements.
3.) Complete. Any question that we can ask in terms of this axiomatic system should be able to be answered with a “yes” or a “no.”
What Godel did was use a clever way of constructing a self referential paradox to show that this is impossible. Any system of axioms that is at least powerful enough to perform elementary arithmetic of natural numbers must have either the possibility of contradictions, or statements that are true but unprovable.
For an understandable construction of Godel’s theorem, there is a veritasium video on YouTube. But the above is the “how, when, and why” so to speak.
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