## Friday, 9 May 2014

### Conundrums and Puzzles

I enjoy conundrums and puzzles. I'm not talking about trick questions. But real puzzles that task the brain. Take the following two statements 'The following sentence is false. The preceding sentence is true.' Are these two sentences true or false?

Actually they are neither. It is what we call a paradox. For first sentence to be true, second sentence must be false. Which in turn makes the first statement false as well. Thus the question ' true or false' does not work.

In the 'foundations of mathematics.' Russell's paradox, discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction. According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell's paradox.
Symbolically:
$\text{Let } R = \{ x \mid x \not \in x \} \text{, then } R \in R \iff R \not \in R$
In 1908, two ways of avoiding the paradox were proposed, Russell's type theory and the Zermelo set theory, the first constructed axiomatic set theory.

I know another paradox, which is the Canal and River Trust. We think of the trust as a charity for the inland waterways. Yet in reality the canal and river trust is a pseudo charity. The old BW waterways are still being managed by the same old agency, that is registered as a charity. The difference being the CaRT can raise charitable donations. So where is the paradox. The management of British Waterways did not  perform very well. The management of the Canal and River trust is not performing well either. How can two management structures be so piss poor?

Because its the same piss poor management with the same piss poor working practices.

#### 1 comment:

1. Mike.

A bit, well actually very off topic, but do you know anything about the boat that has sunk in the lock on the Tinsley flight?