It's not really number theory, it's just numbering.

I really liked this video by Veritassium.

https://youtu.be/HeQX2HjkcNo

I had read that before but I’ve still not looked at how he actually did it.

Huh, I had heard a lot about Godel's Incompleteness Theorem, but I never realized that he proved it by turning mathematical statements into unique numbers, thus connecting the study of formal logic into number theory.  Brilliant.

SimpleStacker

I feel like truth is similar to simultaneity. The more you look into it, the more you find it to be localized, yet at the same time the whole universe seems to be a simultaneous event. We all know there is a *truth,* but we can't quite see it all at once.

lrm_btc

Yes, math is akin to legal system: a human invention that approximates reality only up to a certain precision. 2+2 is not 4 when you try to apply it on elementary particles scale or for intergalactic travel. There our ordinary math breaks down, so it is not an exact science.

SwapMarket

At 25, Kurt Gödel proved there can never be a mathematical “theory of everything.” Columnist Natalie Wolchover explores the implications.

In 1931, by turning logic on itself, Kurt Gödel proved a pair of theorems that transformed the landscape of knowledge and truth. These “incompleteness theorems” established that no formal system of mathematics — no finite set of rules, or axioms, from which everything is supposed to follow — can ever be complete. There will always be true mathematical statements that don’t logically follow from those axioms.

I spent the early weeks of the Covid pandemic learning how the 25-year-old Austrian logician and mathematician did such a thing, and then writing 

 in fewer than 2,000 words. (My wife, when I reminded her of this period: “Oh yeah, that time you almost went crazy?” A slight exaggeration.)

But even after grasping the steps of Gödel’s proof, I was unsure what to make of his theorems, which are commonly understood as ruling out the possibility of a mathematical “theory of everything.” It’s not just me. In 

 (a classic 1958 book that I heavily relied upon for my account), philosopher Ernest Nagel and mathematician James R. Newman wrote that the meaning of Gödel’s theorems “has not been fully fathomed.”

 refers to the subjective qualities of our experience: what it’s like for Alice to see blue or for Bob to feel delighted. Qualia are “the ways things seem to us,” as the late philosopher Daniel Dennett put it. In these essays, our columnists follow their curiosity, and explore important but not necessarily answerable scientific questions.

Maybe not, but six decades have passed since then. Where are we with these ideas today? Recently, I asked logicians, mathematicians, philosophers, and one physicist to discuss the meaning of incompleteness. They had plenty to say about the implications of Gödel’s strange intellectual achievement and how it changed the course of humanity’s unending search for truth.

math

science

> **At 25, Kurt Gödel proved there can never be a mathematical “theory of everything.” Columnist Natalie Wolchover explores the implications.**
>
> In 1931, by turning logic on itself, Kurt Gödel proved a pair of theorems that transformed the landscape of knowledge and truth. These “incompleteness theorems” established that no formal system of mathematics — no finite set of rules, or axioms, from which everything is supposed to follow — can ever be complete. There will always be true mathematical statements that don’t logically follow from those axioms.
>
> I spent the early weeks of the Covid pandemic learning how the 25-year-old Austrian logician and mathematician did such a thing, and then writing [a rundown of his proof](https://www.quantamagazine.org/how-godels-proof-works-20200714/) in fewer than 2,000 words. (My wife, when I reminded her of this period: “Oh yeah, that time you almost went crazy?” A slight exaggeration.)
>
> But even after grasping the steps of Gödel’s proof, I was unsure what to make of his theorems, which are commonly understood as ruling out the possibility of a mathematical “theory of everything.” It’s not just me. In [*Gödel’s Proof*](https://nyupress.org/9780814758014/godels-proof/) (a classic 1958 book that I heavily relied upon for my account), philosopher Ernest Nagel and mathematician James R. Newman wrote that the meaning of Gödel’s theorems “has not been fully fathomed.”
>
> **I**n philosophy, **“qualia”** refers to the subjective qualities of our experience: what it’s like for Alice to see blue or for Bob to feel delighted. Qualia are “the ways things seem to us,” as the late philosopher Daniel Dennett put it. In these essays, our columnists follow their curiosity, and explore important but not necessarily answerable scientific questions.
>
> Maybe not, but six decades have passed since then. Where are we with these ideas today? Recently, I asked logicians, mathematicians, philosophers, and one physicist to discuss the meaning of incompleteness. They had plenty to say about the implications of Gödel’s strange intellectual achievement and how it changed the course of humanity’s unending search for truth.
>
> [**...read more at quantamagazine.org**](https://www.quantamagazine.org/what-do-godels-incompleteness-theorems-truly-mean-20260518/)