Taming The infinite/ Chapter 17/ The shape of logic.
On this chapter we deal with those major contradictions, contradictions highly remarked in Hofstadter's writing. Kurt Godel, Cantor, Russel, these great mathematicians all contribute to those major contradiction I speak off. Godel made the claim that there simple cannot be a system that can be both complete and consistent at the same time, generating a paradox. A system that is capable of proving its won consistency is by definition inconsistent. Sounds odd, and that is because we have entered the realms of the paradox. Cantor deals with infinities, how much can we know, and inevitable ended up in a loop, and endless loop of repetition and it is much understandable when we consider what infinity is.
Let us consider this paradox, as Russel exemplifies.
There is a town. In this town all people must remain well shaved, they are very careful in this detail. Each person that lives within the town, must get his beard shaved by a barber, or shave the beard themselves. There is only 1 barber in town though, therefore everybody in town must go see him to get his beard shaved, unless they chose to shave it themselves. Now here lies the paradox, a man either goes to the barber, or shaves it himself... How does the barber shave his beard? ... If the barber shaves it himself, he cannot because the barber has shaved his own beard.
I am paraphrasing somewhat, but the paradox is still there. How can one be and not be at the same time? This contradiction is also symbolic of Godel's theorem and/or the Epidermises paradox i.e. This sentence is false.
Let us consider this paradox, as Russel exemplifies.
There is a town. In this town all people must remain well shaved, they are very careful in this detail. Each person that lives within the town, must get his beard shaved by a barber, or shave the beard themselves. There is only 1 barber in town though, therefore everybody in town must go see him to get his beard shaved, unless they chose to shave it themselves. Now here lies the paradox, a man either goes to the barber, or shaves it himself... How does the barber shave his beard? ... If the barber shaves it himself, he cannot because the barber has shaved his own beard.
I am paraphrasing somewhat, but the paradox is still there. How can one be and not be at the same time? This contradiction is also symbolic of Godel's theorem and/or the Epidermises paradox i.e. This sentence is false.
