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Here’s Mathematician KP Hart’s Math Question and Answer for Monday, October 19th!

A barber shaves all men that do not shave themselves. Must he shave himself?

Barber paradox shows the relativity of the logic/rationality that is so revered by science. Together with, among others, Goedel’s incompleteness theorem this pleads for room for intuition in science (what Kurt Goedel also seems to have pursued).

This paradox has its own Wikipedia page and on that page it is argued that this is not so much a paradox as an untenable assumption about the village in question. Contrary to what the poser of the question claims this `paradox’ shows the power of first-order logic: one can translate the statement into a logical formula and verify that the statement is never satisfiable, that its negation is always satisfied and thanks to Gödel’s Completeness theorem we know that that negation has a formal derivation.

The original paradox, due to Russell, shows, ironically, that it was the intuition that let down the initial developers of Set Theory. That paradox caused a thorough revision of Set Theory’s foundation in logical terms.

Many books and web pages have been written about Gödel’s Incompleteness Theorems and I am not going to add another long essay to this collection. The content of these theorems, especially the first one, is much more technical than many people think; my advice to anyone who really wants to know what these results say is: take a book on Mathematical Logic and study it up to and including the incompleteness theorems; I learned logic from an earlier edition of this book.

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Read all of KP Harts math questions here!

KP Hart Full

About Dutch Mathematician KP Hart: In the beginning of this year the Dutch government opened a website, The Dutch Science Agenda, where everyone could post questions that they thought were of scientific interest. This was an attempt to involve the whole country in determining what the Dutch science agenda should be in the coming years.

I looked through the questions and searched for terms like `mathematics’, `infinity’ … to see what mathematical questions there were and I noticed various questions that already have answers (and have had for a long time). On a whim I decided to post answers to those questions, in Dutch. For your edification I will translate these posts into English.

Follow KP Hart on Twitter here!

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