Conmathematical Resolution of Russell's Paradox

Written by Dr Kedar Joshi


Russell's Paradox -

'A paradox uncovered by Bertrand Russell in 1901 that forced a reformulation of set theory. One version of Russell's paradox, known asrepparttar barber paradox, considers a town with a male barber who, every day, shaves every man who doesn't shave himself, and no one else. Doesrepparttar 127623 barber shave himself ? The scenario as described requires thatrepparttar 127624 barber shave himself if and only if he does not ! Russell's paradox, in its original form considersrepparttar 127625 set of all sets that aren't members of themselves. Most sets, it would seem, aren't members of themselves - for example,repparttar 127626 set of elephants is not an elephant - and so could be said to be "run-of-the-mill". However, some "self-swallowing" sets do contain themselves as members, such asrepparttar 127627 set of all sets, orrepparttar 127628 set of all things except Julius Caesar, and so on. Clearly, every set is either run-of-the-mill or self-swallowing, and no set can be both. But then, asked Russell, what aboutrepparttar 127629 set S of all sets that aren't members of themselves ? Somehow, S is neither a member of itself nor not a member of itself.'

( See David Darling : The Universal Book of Mathematics, 2004 )

Conmathematical Resolution -

The term 'Conmathematics' means conceptual mathematics ( invented by Dr. Kedar Joshi ( b. 1979 ), Cambridge, UK ). It is a meta - mathematical system that definesrepparttar 127630 structure of superultramodern mathematics. It essentially involves a heavy or profound conceptual approach which is in striking contrast withrepparttar 127631 traditional symbolic or set theoretic approach.

The NSTP Theoretical Resolution of Zeno's Paradoxes

Written by Dr Kedar Joshi


Zeno of Elea's (b.490 BC) arguments against motion precipitated a crisis in Greek thought. All of these, concerning motion, have had a profound influence onrepparttar development of mathematics. They are described in Aristotle's great work 'Physics' and are presented as four arguments inrepparttar 127622 form of paradoxes, stated below :

1. The Racecourse or Dichotomy Paradox -

There is no motion because that which is moved must arrive atrepparttar 127623 middle of its course before it arrives atrepparttar 127624 end. In order to traverse a line segment it's necessary to reachrepparttar 127625 halfway point, but this requires first reachingrepparttar 127626 quarter - way point, which first requires reachingrepparttar 127627 eighth - way point, and so on without end. Hence motion can never begin.

This problem isn't alleviated byrepparttar 127628 well - known infinite sum + + 1/8 ... = 1 because Zeno is effectively insisting thatrepparttar 127629 sum be tackled inrepparttar 127630 reverse direction. What isrepparttar 127631 first term in such a series ?

( See David Darling : The Universal Book of Mathematics, 2004. )

2. Achilles andrepparttar 127632 Tortoise -

This is perhapsrepparttar 127633 most famous ofrepparttar 127634 Zeno's paradoxes.

The slower when running will never be overtaken byrepparttar 127635 quicker; for that which is persuing must first reachrepparttar 127636 point from which that which is fleeing started, so thatrepparttar 127637 slower must necessarily always be some distance ahead. Thus, Achilles, however fast he runs, will never catchrepparttar 127638 plodding Tortoise who started first. And yet, of course, inrepparttar 127639 real world, faster things do overtake slower ones.

3. The Arrow -

An arrow cannot move at a place at which it is not. But neither can it move at a place at which it is. But a flying arrow is always atrepparttar 127640 place at which it is. That is, at any instant it is at rest. But if at no instant is it moving, then it is always at rest.

4. The Moving Blocks or Stadium -

Suppose three equal blocks, A , B, C, of width 1, with A and C moving past B atrepparttar 127641 same speed in opposite directions. Then A takes one time, t, to traverserepparttar 127642 width of B, but halfrepparttar 127643 time, t/2, to traverserepparttar 127644 width of C. But these arerepparttar 127645 same length, 1. So A takes both t and t/2 to traverserepparttar 127646 distance 1.

( See Simon Blackburn : Dictionary of Philosophy, 1996 )

The German set theorist Adolf Frankel ( 1891 - 1965 ) is one of many modern mathematicians ( Bertrand Russell is another ) who have pointed out that 2,000 years of attempted explanations have not cleared awayrepparttar 127647 mysteries of Zeno's Paradoxes : "Although they have often been dismissed as logical nonsense, many attempts have also been made to dispose of them by means of mathematical theorems, such asrepparttar 127648 theory of convergent series orrepparttar 127649 theory of sets. Inrepparttar 127650 end, however,repparttar 127651 difficulties inherent in his arguments have always come back with a vengeance, forrepparttar 127652 human mind is so constructed that it can look at a continuum in two ways that are not quite reconcilable."

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