What are the problems and theorems that are derived from it?
In: 80
It basically says that any logic system breaks apart when you force it to talk about itself.
For example, “this statement is false”, “what would Pinocchio’s nose do if he says “my nose will now grow”, what to do if you are compelled to accept the mission “mission : reject this mission” and so on.
Of course Godel did it with formal proof for any formal system of logic that can define and perform addition and multiplication of numbers.
Is it final exam week already?
If you’re interested in learning more about this, there is a great book called Gödel’s Proof by Ernest Nagel and James R. Newman that explains some of the context around the theorems. It’s nice and short too (like 130 pages?)
[This Veritasium video](https://www.youtube.com/watch?v=i1oEHCKpE4U) has a great ELI5-ish introduction to the subject. I highly recommend it.
Math is supposed to be a reasoning system of pure thought where everything has a logical explanation.
Except you can’t have a logical explanation for literally everything. As any 5-year-old knows, you can always ask “Why? Why? Why?” again and again, and eventually you get to a point where the explanation is “Because that’s just the way it is” (or you get into some kind of loop, which is illogical circular reasoning).
In math, people have been aware of how to solve the “Why? Why? Why?” problem for thousands of years. You say “Well let’s start with some obvious things that we all agree are just true and need no explanation,” one classic example is “Given a line and a point not on the line, there is exactly one parallel line through the point.”
These basic assumptions are *axioms*. When you prove something, you prove it from the axioms. In other words, every chain of “Why’s” eventually stops at an axiom — a statement we all agreed at the beginning is true and needs no explanation.
Well…you’re *supposed* to prove things from the axioms. Over the centuries people started getting.. sloppy. And in the mid-to-late 1800’s and early 1900’s serious cracks were showing:
– The biggest issue was Russell’s paradox in set theory. “Consider the set of all (THINGS) that (HAVE PROPERTY)…” was regarded as an acceptable mathematical statement, but Russell eventually discovered this kind of statement could lead to logical contradictions (“S is the set of all sets that don’t contain themselves. Does S contain itself?”)
– Calculus (in use for ~200 years) needed a major “patch”. Specifically, people realized sloppy thinking was hiding behind the phrase “As Δx gets infinitely small…” and eventually “patched” the possible logical “hole” with limits and epsilon-delta arguments.
– In classical geometry (in use for ~thousands of years), people realized the parallel postulate (the point/line thing I mentioned above) *isn’t* “a thing that we can all agree is true.” If you do geometry on a sphere or other surface, you can get a perfectly good mathematical field of study (logically sound and practically interesting) where the parallel postulate is false.
These were very unpleasant surprises. Logical flaws invading the most basic assumptions of long-settled fields of study? Alternate sets of axioms describing alternate mathematical realities? Quite shocking and unsettling.
Mathematicians hoped that, with enough work, you could fix the problems once and for all:
– Create a “proof checking procedure”, you can specify all your proofs in an unambiguous language and run a mechanical process to be sure there are no logical flaws or sloppy language hiding anywhere
– Do a “consistency check” of the axioms themselves, i.e. prove that your chosen set of axioms can’t lead to logical contradictions
– Do a “completeness check” of the axioms, i.e. prove your axioms can prove every true statement and disprove every false statement.
– Find one “good” (consistent and complete) set of “axioms to rule them all”, i.e. be used as a foundation for building any of the different possible “mathematical realities”
Godel’s Incompleteness Theorem basically completely shattered that hope. It basically says that (1) consistency checks are logically impossible and (2) most interesting axiom sets fail the completeness check. Basically every [1] mathematical world has statements that are true but cannot be proved.
On the bright side, people were thinking about an unambiguous language to describe mathematics and using machines to manipulate symbols. At a time when the industrial revolution was in full swing and humanity’s mechanical abilities were increasing by leaps and bounds. So these insights ultimately led to building the first computers, which over the following decades evolved into the world-shaping (or in the words of Marc Andreessen, world-eating) tech industry.
[1] Technically, Godel’s Incompleteness Theorem only applies if you have a consistent set of axioms that creates a mathematical world that allows addition and multiplication of positive integers. It’s still possible that a very simple or logically inconsistent mathematical world can be complete. These worlds are perhaps theoretically interesting to logicians, but practically speaking are useless to working mathematicians for most problems.
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