Every last Sunday of April, I have been organizing a small workshop called BBD. We discuss logic, math, and science on a blackboard (this year, it was actually on a blackboard unlike the past 3 years!)

The main theme was paradox. A paradox is a puzzling contradiction; using some sort of reasoning one derives two contradicting conclusions. Consistency is an essential quality of a reasoning system, that is, it should not be able to produce contradictions by itself. Therefore, true paradoxes are hazardous to the fundamentals of being rational, but fortunately, most paradoxes are only apparent and can be resolved. Today (April 27th, 2014), we had several paradoxes presented:

Memming: I presented the Pinocchio paradox, which is a fun variant of the Liar paradox. Pinocchio’s nose grows if and only if Pinocchio tells a false statement. What happens when Pinocchio says “My nose grows now”/”My nose will grow now”? It either grows or not grows. If it grows, he is telling the truth, so it should not grow. If it is false, then it should grow, but then it is true again. Our natural language allows self-referencing, but is it really logically possible? (In the incompleteness theorem, Gödel numbering allows self-referencing using arithmetic.) There are several possible resolutions, such as, Pinocchio cannot say that statement, Pinocchio’s world is inconsistent (and hence cannot have physical reality attached to it), Pinocchio cannot know the truth value, and so on. In any case, a good logical system shouldn’t be able to produce such paradoxes.

Jonathan Pillow, continuing on the fairy tale theme, presented the sleeping beauty paradox. Toss a coin, sleeping beauty will be awakened once if it is head, twice if it is tail. Every time she is awakened, she is asked “What is your belief that the coin was heads?”, and given a drug that erases the memory of this awakening, and goes back to sleep. One argument (“halfer” position) says since a priori belief was 1/2, and each awakening does not provide more evidence, her belief does not change and would answer 1/2. The argument (“thirder” position) says that you are twice more likely to be awakened for the tail toss, hence the probability should be 1/3. If a certain reward was assigned to making a correct guess, the thirder position seems to be correct probability to use as the guess, but do we necessarily have matching belief? This paradox is still under debate, have not had a full resolution yet.

Kyle Mandi presented the classical Zeno’s paradox where your intuition on infinite sum of finite things being infinite is challenged. He also showed Gabriel’s horn where a simple (infinite) object with finite volume, but infinite surface area is given. Hence, if you were to pour in paint in this horn, you would need finite paint, but would never be able to paint the entire surface. (Hence its nickname: painter’s paradox)

Karin Knudson introduced the Banach-Tarski paradox where one solid unit sphere in 3D can be decomposed into 5 pieces, and only by translation and rotation, they are reconstructed into two solid unit spheres. In general, if two uncountable sets A, B are bounded with non-empty interior in $R^n$ with $n \geq 3$, then one can find a finite decomposition such that each piece in A is congruent to the corresponding piece in B. It requires some group theory, non-measurable sets, and the axiom of choice (fortunately).

Harold Gutch told us about the Borel-Kolmogorov paradox. What is the conditional distribution on a great circle when points are uniformly distributed on the surface of a sphere? One argument says it should be uniform by symmetry. But, a simple sampling scheme in polar coordinate shows that it should be proportional to cosine of the angle. Basically, the lesson is, never take conditional probabilities on sets of measure zero (not to be confused with conditional densities). Furthermore, he told us about a formula to produce infinitely many paradoxes from Jaynes’ book (ch 15) based on ill-defined series convergences.

Andrew Tan presented Rosser’s extension of Gödel’s first incompleteness theorem with the statement $R$ that colloquially says “For every proof of me, there’s a shorter disproof.” For any consistent system T that contains PA (Peano axioms), there exists an $R_T$, which is neither provable nor disprovable within T. Also, by the second incompleteness theorem, the consistency of PA (“con(PA)”) implies $G_{PA}$, which together with Gödel’s first incompleteness theorem that $G_{PA}$ is neither provable nor disprovable within PA, implies that PA augmented with “con(PA)” or “not con(PA)” are both consistent. However, the latter is paradoxical, since it appears that a consistent system declares its own inconsistency, and the natural number system that we are familiar with is not a model for the system. But, it could be resolved by creating a non-standard model of arithmetic. References: [V. Gitman’s blog post and talk slides][S. Reid, arXiv 2013]

I had a wonderful time, and I really appreciate my friends for joining me in this event!