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Question: 8 russels paradox2 let us try to form set of...

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(8) Russels paradox2. Let us try to form set of all sets and call it M. Note that, as M is a set, it is an element of itself! That is to say, we have ME M. (Hmmm... ). Therefore, it makes perfect sense to consider the set of all sets which do not contain themselves as elements, i.e. R={X | X is a set and XEX} There are two possibilities: either R contains R as an element, or R does not contain R as an element Show that actually none of these two possibilities is occurs, and therefore obtain a contradiction. Remark. This paradox shows that the concept of set can lead to logical contradictions when not applied with sufficient care. This problem can be avoided by creating a set theory which is free from such contradictions. The most widely used and accepted set theory is called Zermelo-Fraenkel set theory, abbreviated by ZF, which is usually extended to include the axiom of choice, and then Named after Bertrand Russell (1872-1970), British philosopher, logician, mathematician, historian, writer, social critic, itical activist. In 1950 he received the Nobel Prize in Literature in recognition of his varied and signi cant writings in which he champions humanitarian ideals and freedom of thought denoted by ZFC. Today ZFC is the standard form of axiomatic set theory and as such is the most common foundation of mathematics.3

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