Gödel and Death

14 Jan, 2026

just went to a sketch club meetup that was so bad i am rethinking what life and death mean.

Gödel

Some guy proved some things about math about a hundred years ago. I find this thematically interesting.

Axiomatic Systems

Most math is just a bunch of games where you start with

start with Obviously True Things (called Axioms... although they can just be anything you wanna start with) and
use Deduction Rules (usually just basic logic)
to Prove
new statements (called Theorems)

Obviously if you start with fucked up axioms, or use fucked up deduction rules, you get some weird theorems out at the end.

Anyway, most of math is just playing with this sort of game. People generally use a special set of axioms called ZFC and some pretty normal looking deduction rules... and basically all of math follows. People keep finding new things to prove.

Some folks use different axioms or do some other weird stuff. Good for them!

Anyway, turns out you can study these sorts of axiomatic systems themselves. And....

they're kinda broken.

(First) Incompleteness Theorem

Let's say you like (natural) numbers. Like 0,1,2,3,4... Those.

And you like facts about numbers, such as:

1+1=2
$1 + 2 + \dots + n = \frac{n (n + 1)}{2}$
Every integer has a prime factorization
There are no non-trivial solutions to $a^{n} + b^{n} = c^{n}$ when $n \geq 3$ .

These are nice.

Let's say you want to start proving things about numbers. Basically just... doing math.

Well, you probably start with some axioms like the Peano axioms and some basic logical deduction rules, and see what you get!

Gödel proved about a hundred years ago that any system of axioms and nice rules for deducing theorems about numbers is fucked up in at least one of the following ways:

It is incomplete (there are true things about numbers the system will Never be able to prove!)
It is inconsistent (the system will prove false things, and "prove" that they are true!)

(There are a bunch of details I don't know or care about, so I'm probably saying it wrong)

Gödel sentence

The proof involves interpretations...

Gödel sets up a system of interpretation where

Statements in the Axiomatic System get translated to Statements about Numbers

This means some statements about numbers can be interpreted in two ways:

As just plain old number facts (might be true might be false)
As claims about a particular axiomatic system.

And the statement is true as just a number fact exactly when it's true as a claim about an axiomatic system!

Given some axiomatic system $A$ , Gödel then constructs a very special statement about numbers $S_{A}$ . Like many number statements, it's a statement with two interpretations...

Just something about numbers
A claim about $A$ ... that $A$ cannot prove $S_{A}$ .

So if $S_{A}$ is true... then $A$ cannot prove $S_{A}$ . And if $S_{A}$ is false... then $A$ can prove $S_{A}$ (i.e., $A$ says wrongly that $S_{A}$ is true).

The proof hinges on a self contradicting statement that lives at different layers of interpretation.

$S_{A}$ is called a Gödel sentence.

Interpretations

Sometimes I think most life is a kind of Gödel sentence. A self-negation, a contradiction that lives at multiple layers of interpretation.

On one layer, just a claim about numbers.

On another layer, a claim of some impossibility.

These layers break each other when held together.

Some statements can be a Gödel sentence for different systems while meaning something harmless or nonsensical in another system.

I think about the many layers of interpretations in the world. Many are deeply incompatible. Many are self denying.

When someone holds no meaning in life and death... what kind of life is this... what meaning could it have besides like a kind of Gödel sentence. Someone for whom life has no meaning... what does their life mean?

Art and Death

I think life and death should be different.

I think it should mean something when someone lives.

I think it should mean something when they die.

I think to many people it doesn't.

What if life weren't death.

(I think it isn't to the dead and dying!)