The Outer Limits of Reason, Yanofsky

Review, notes, and reflections on The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us by Noson S. Yanofsky

Language Paradoxes

• Liar paradox - I am lying. This sentence is false.
• “This sentence is false” has five words.
• “Yields falsehood when preceded by its quotation” yields falsehood when preceded by its quotation.
• Barber paradox - all who do not shave themsleves go to the barber, all that do shave themselves do not go to the barber; whod oes the barber go to? Self-referential.
• Homological/autological - adjectives that describe themselves. Heterological - adjectives that don’t. Is heterological heterological? Heterological is only heterological iff it is not heterological.
• Reference-book paradox: a list of all reference books that do not list themsleves. Does this list list itself?
• Russell’s paradox - does the set of sets that do not contain themselves contain itself?
• Yabloi’s paradox: \(K_n := K_i\) is false for all \(i > n\). So which sentence is true and which is false? This is a contradiction without self-reference.
• All integers are interesting by contradiction - let the smallest noninteresting number be \(N\). Therefore \(N\) is interesting.
• Berry phrase - “the least number not expressible in fewer than eleven words”.
• Richard paradox. Let phrases describing real numbers between 0 and 1 be Richard phrases. What is the Richard phrase that is different from all Richard phrases?

Philosophical Conundrums