## Russell's ParadoxEasy to state, yet difficult or impossible to resolve; self contradictory statements or paradoxes have presented a major challenge to Mathematics and Logic. Russell's Paradox can be put into everyday language in many ways. The most often repeated is the 'Barber Question.' It goes like this: In a small town there is only one barber. This man is defined to be the one who shaves all the men who do not shave themselves. The question is then asked,Another popular form of Russell's Paradox is the following: Consider the statement Let's look at this situation as mathematicians do. You may have noticed the remarkable similarity between logical symbols (like Ù for '
In logic a statement that has a single variable, like - c(x) = (x is the largest city in Canada)
- b(x) = (x is bigger than 5)
is called a unary predicate. These unary predicates are used to build sets.
- p(x) = (x is female)
- q(x) = (x is a student at Vanier)
we form a set of all the x's that make the predicate true. - P = { x | x is female }
- Q = { x | x is a student at Vanier }
- P Ç Q = { x | p(x) Ù q(x)}
- P È Q = { x | p(x) Ú q(x)}
- P' = { x | ~p(x) }
We see that set It has been recognized for a long time that the correspondence between logical and set symbols is not one-to-one. For example - p(x) = (x is an adult human of the female sex)
- q(x) = (x is a woman)
are not the same predicates, but they do give rise to the same sets. But, prior to the discovery of Russell's Paradox, just before the turn of the Twentieth Century, it was thought that any predicate gave rise to a set. Unfortunately -- this also is not the case. Let's notice first that most ordinary sets do not contain themselves as elements. For example, if A = {a, b, c} we have a Î A and A Í A, but not A Î A.
R = { x | (x is a set) Ù (x Ï x)}
We have seen that some sets contain themselves as elements, like the set S; while others, like the set A, do not. We collect together all of the sets that do not contain themselves as elements and call this set R. So the set R shown here is just the set of all sets that do
itself as an element?
Look closely! Each answer leads to the opposite conclusion.
To find a way out of this predicament most mathematicians have chosen to outlaw the formation of sets from just This has one major consequence: There can be no truly The study of the Foundations of Mathematics, as constructed during this century, has largely been concerned with trying to make sure that no paradoxes or self contradictions get into Mathematics at the lowest levels. But, can we be certain that there are no, as yet, undiscovered contradictions in Mathematics or Logic? |

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