On May 2nd 2015, I organized yet another BlackBoardDay, this time in New York City (on Columbia University campus, thanks Evan!).

I started the discussion by tracing the history of modern mathematics back to Gottlob Frege (Vika pointed out the axiomatic tradition goes back to Euclid (300 BC)).

• Gottlob Frege’s Begriffsschrift, the first symbolic logic system powerful enough for mathematics (1879)
• Giuseppe Peano’s axiomatization of arithmetic (1889)
• David Hilbert’s program to build a foundation of mathematics (1900-1920s)
• Bertrand Russell’s paradox in Frege’s system (1902)
• Kurt Gödel was born! (1906)
• Russell and Whitehead’s Principia Mathematica as foundation of mathematics (1910)
• Kurt Gödel’s completeness theorem (for first-order logic) (1930)
• Kurt Gödel’s incompleteness theorem (of Principia Mathematica and related systems) (1931)

Victoria (Vika) Gitman talked about non-standard models of Peano arithmetic. She listed the first-order form of Peano axioms which is supposed to describe addition, multiplication, and ordering of natural numbers $\mathbb{N} = \{0, 1, 2, \cdots \}$. However, it turns out there are other countable models that are not natural number and yet satisfy Peano axioms. She used the compactness theorem, a corollary of completeness theorem (Gödel 1930), that (loosely) states that for a consistent first-order system, if any finite subset of axioms has a model, then the system has a model. She showed that if we add a constant symbol ‘c’ (in addition to 0 and 1) to the language of arithmetic, and a set of infinite axioms which is consistent with the Peano axioms: {c > 0, c > 1, c > 2, … }, then using the compactness theorem, there exists a model. This model is somewhat like integers $\mathbb{Z}$ sprinkled on rational numbers $\mathbb{Q}$, in the sense that (…, c-2, c-1, c, c+1, c+2, …) are all larger than the regular $\mathbb{N}$, but then 2c is larger than all of that. Then there are also fractions of c such as c/2, and so on. This is still countable, since it is a countable collection of countably infinite sets, but this totally blew our minds. In this non-standard model of arithmetic, those ‘numbers’ outside $\mathbb{N}$ can be represented as a pair in $\mathbb{Q} \times \mathbb{Z}$, but actual computation with those numbers turn out to be non-trivial (and often non-computable). Ashish Myles talked about the incompleteness theorem, and other disturbing ideas. Starting from the analogy of liar’s paradox, Ashish stated that arithmetic (with multiplication) can be used to encode logical statements into natural numbers, and also write a (recursive) function that encapsulates the notion of ‘provable from axioms’. The Gödel statement G roughly says that “the natural number that encodes G is not provable”. Such statement is true (in our meta language) since if it is false, there’s a contradiction. However, either adding G or not G as an axiom to the original system is consistent. Even after including G (or “not G”) as an axiom to Peano arithmetic, there’ll be statements that are true but not provable! Vika gave an example statement that is true for natural number but is not provable from Peano axioms: all Goodstein sequence terminates at 0. At this point, we were all feeling very cold inside, and needed some warm sunshine. So, we continued our discussion outside:

Kyle Mandli talked about Axiom of Choice (AC), which is an axiom that is somewhat counter intuitive, and independent of the Zermelo-Fraenkel (ZF) set theory: Both ZF with AC and ZF with not AC are consistent (Gödel 1964). We discussed many counter intuitive “paradoxes” as well as usefulness of AC in mathematics.

Diana Hall talked about an counter intuitive bet: suppose we have a fair coin, and we are tossing to create a sequence. Would you bet on seeing HTH first or HHT first? At first one might think they are equally likely. However, since there’s a sequence effect that makes them non-equal! Unfortunately, due to time constraints we couldn’t talk about Uygar planed: “approximate solutions to combinatorial optimization problems implies P=NP”, hopefully we’ll hear about it on BBD11!