Nikolaj Kuntner Nikolaj-K

This is the script for the video at

product and sum notations

Goal: We want to define a transition "$\sum_N\circ\prod_N^{-1}$"

${\mathrm{iter}}(\mathrm{op}, x, 1) := x$ ${\mathrm{iter}}(\mathrm{op}, x, N+1) := \mathrm{op}({\mathrm{iter}}(\mathrm{op}, x, N), x)$

Script and python code discussed in

https://youtu.be/g3-Tmk_e42k

Motivation

1824: Carnot lays foundations of thermodynamics
1865: Clausius introduces entropy

Script for the video discussed at

https://youtu.be/yRXxBsy2Z6c

CZF's characterization of the von Neumann "hierarchy"

In this video we analyze how much we get without LEM or the Powerset axiom. (But with function spaces and the very set theoretical Replacement.) That context has semnatics where all sets are in the surjective image of a subset of ${\mathbb N}$. (Subcountability, see also subquotients.)

This is the script for the video discussed at

https://youtu.be/7gbRTShKvp8

Cardinality individual coded reals

Z_2 reals

Constrain $d_i > 0, s_i\in{0,1}$.

$i\mapsto \tfrac{n_i}{d_i}(-1)^{s_i}$ in ${\mathbb Q}^{\mathbb N}$

Dowload the Codex UI at

https://chatgpt.com/codex/

log in with your OpenAI (I assume you're on a mac here)
create a new empty folder somewhere and then with Codex open a project/thread in there
set it to Gpt-5.3-Codex (maybe Extra High)
(during runtime you might progressively want to say Yes to various permissions - you won't go far apart from downloading python stuff if you don't have it)
start chatting with it a bit, then dump in the following longer prompt at once:

This is the script discussed in the video at

https://youtu.be/gNSv2kWRv88

Gödel arithmetization and the anti-classical Church's thesis

Outlook for this video:

On the arithmetization of proof
On Gödel incompleteness with emphasis on quantifiers
Note on Existence and Disjunction property
Note on the failure of the Least Number Principle

Video where this script is discussed: https://youtu.be/Lsf4eAGvODs

Consider a non-strictly ordered space $X$ with an (absolutely homogenous) absolute value function $|\cdot|: X\to X$ If $B:(X\times X)\to X$ is a symmetric map, then

$B(\max(x, y)-x, \max(x, y)-y) = B(0, |x-y|)$

	<!doctype html>
	<html>
	<head>
	<title>Agentathon Round 1</title>
	</head>
	<body>
	<h1>Agentathon Round 1</h1>

	<p id="round-index">1</p>

	This script is explained at

	https://youtu.be/53lvGfk9ib8


	For all $n$,
	$\int_{-\pi}^\pi\, x^{2n} \left(\dfrac{2}{1 + {\mathrm e}^{\sin(x)}}\right){\mathrm d}x = \int_{-\pi}^\pi x^{2n} \,{\mathrm d}x$

	$\dfrac{2}{1+{\mathrm e}^x} = 1 - \tanh(\frac{1}{2}x)$

	#!/usr/bin/env python3
	"""
	Minimal Wikipedia semantic search wrapper for web integration.

	Install hints (CPU):
	- python, duh
	- pip install -U sentence-transformers faiss-cpu numpy

	Expected index artifacts (see zip):
	- path/to/pages_embeddings.index (main data item, 200 MB)