• 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 2.3.11:

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This is a paradox, because who will shave the barber of Seville himself ?

• if the barber does not shave himself, he - according to definition - must shave himself, because he shaves all who do not shave themselves
• if the barber does shave himself, he - according to definition - can not shave himself, because the barber only shaves those who do not shave themselves

This is actually and example of an invalid recursive definition: In the definition the barber appears twice: one as the defined entity "barber of Seville" and again as a member of all men living in Seville. Such a recursive definition is invalid, and in fact leads to a contradiction in this case.

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