TUESDAY, JUNE 21, 2022
This is what he did: As we noted yesterday, he's one of the three most significant logicians in the history of the western world. Or at least, that's pretty much what the leading authority on the widely-ignored topic says.
We're discussing a time span of roughly 2500 years! Let's refresh ourselves:
Kurt Friedrich Gödel (1906 – 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an immense effect upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell, Alfred North Whitehead, and David Hilbert were using logic and set theory to investigate the foundations of mathematics, building on earlier work by the likes of Richard Dedekind, Georg Cantor and Frege.
Aristotle, Frege and Gödel! Many people have heard of the first of the three. The other two, not so much!
Yesterday, we asked you what that somewhat peculiar fact might possibly mean. What does it mean when no one has heard of Frege and Gödel—when no one has heard of the two top logicians of the past (more than) two thousand years?
Today, we'll postpone an attempt at an answer. Instead, let's continue along with the leading authority as we start to learn what Gödel actually did.
According to that overview, Gödel was the most significant logician of the 20th century. Continuing along from the text shown above, this is what he did:
Gödel published his first incompleteness theorem in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. The first incompleteness theorem states that for any ω-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the natural numbers that can be neither proved nor disproved from the axioms. To prove this, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers. The second incompleteness theorem, which follows from the first, states that the system cannot prove its own consistency.
Gödel also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted Zermelo–Fraenkel set theory, assuming that its axioms are consistent. The former result opened the door for mathematicians to assume the axiom of choice in their proofs. He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.
No one has ever heard of Frege, or of Gödel either. Can you start to see why that might be? Does this start to suggest some thoughts about possible flaws, or even shortcomings, in our failed intellectual culture?