• Real Analysis
• 1. Sets and Relations
• 2. Infinity and Induction
• 2.1. Countable Infinity
• 2.2. Uncountable Infinity
• 2.3. The Principle of Induction
• 2.4. The Real Number System
• 2.5. Pi and e are irrational
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 7. The Integral
• 8. Sequences of Functions
• 9. Historical Tidbits
• Java Tools

Example 2.2.6: Logical Impossibilities - The Set of all Sets

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This definition seems to make sense, because a set could be an entity of its own, as well as an element of another set. For example, we could define two sets

• A = { {1}, {1,3}, A }
• B = { {1}, {1,3} }

Then A is a set that is also a member of itself, whereas B is not a member of itself. Therefore, we could consider the set of all those sets that are not members of itself. Call this set S. The above set A would not be an element of S, whereas B is an element of S. While this, albeit strange, does seem to make sense, we might ask:

• Is S an element of itself or not ?

But this question will give a contradiction, because:

• If S is an element of itself, then - since by definition S contains those sets only that are not part of itself - S is not an element of itself. That's not possible.
• If S is not an element of itself, then - since S does contain those sets that are not part of itself - S is a member of S. That's not possible either.

Hence, we have arrived at a logical impossibility, and the set S does indeed not exist.

Interactive Real Analysis, ver. 2.0.1(b)
(c) 1994-2017, Bert G. Wachsmuth
Page last modified: Mar 3, 2017