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A Quick Overview Of GöDel Escher Bach
Godel Escher Bach is a book which was written by Douglas Hofstadter in 1979. If 'book' simply means 'read', perish the thought. This is not a book you kick back with.
Godel Austrian Mathematician and Logician studied at the University of Vienna - then moved on to the Princeton institute for Advanced Study Incompleteness theorems statements which in Mathematics refer to themselves. I'm not provable (therefore either true but not provable OR provable then it's false - if you can derive that sentence you system is inconsistent. If true then not provable if the system is consistent then - comes from Liar paradox). 2nd Incompleteness Theorem No consistent system can prove its own consistency. You can only prove it if it's inconsistent If T is consistent then G is not derivable in T
How does a self come out of things that have no selves - how to go from meaningless to becoming something that refers to itself. Tools for doing this used in the book:
Isomorphism - a loose meaning, not a strict mathematical sense
Recursion - construct the next number by the sum of the previous two - you define a thing in terms of itself like Fibonacci numbers which fractals are created through recursion. Fractal mandelbrot coined the word
Paradoxes (Veridical - true, but seem paradoxical at first like Zeno, Falsidical, Antinomy Liar's Paradox This sentence is not true, Russel's once you form a set ask is that set a member of itself. if it is then it's not if it's not then it is. Russle's set is provably both a member of itself and not a member of itself - exactly the same as the Liar Paradox/ The Barber's Paradox - he shaves all people and only people who don't shave themselves - does he shave himself or not. Bertrand Russell the sets which aren't members of themselves
Infinity you can't match every real number to an integer - there are different degrees of infinity, infinitely many different infinities
Formal Systems - the Mu puzzle. Bag of 3 letters. Rules 1. If we have an I we can tack a u on MIU. 2nd rule miu. Starting from MI using 4 Rules of Inference can you get MU Axiom MI trying to prove theorem of mu theorem is a string which results at the end of a derivation (proof)
How do we go from meaningless to meaningful