Nikolaj-K

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:

Nikolaj-K

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

Nikolaj-K

This script is explained at

https://youtu.be/53lvGfk9ib8


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

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

Nikolaj-K

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

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

Proof

Nikolaj-K

#!/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)

Nikolaj-K

This file starts with Exercises, which ends at exercise 10 and is followed by "Notes: Applied Optimization".

Search for "Exercise 9" for the Regression cost function minimization exercises.

Greetings Nikolaj Kuntner

---

Exercises - Applied Optimization

Nikolaj-K

Script for the video

https://youtu.be/LfiIiSPg3Ms

$f$ ... Lebesgue-integrable over (all of) ${\mathbb R}$,

$\int_{-\infty}^\infty f\big(x + d\big),{\mathrm d}x = \int_{-\infty}^\infty f\big(x\big),{\mathrm d}x$

$\int_{-\infty}^\infty f\big(x - \dfrac{1}{x}\big),{\mathrm d}x = \int_{-\infty}^\infty f\big(x\big),{\mathrm d}x$

Nikolaj-K

Probability Theory 1 – Script skeleton

• 2024W-Prob1__script.pdf: sections 1–5 (up to and including 5. Lebesgue-Integral)
• 2025W-Prob1-Collection.pdf: sections 0–3 (up to and including 3. Das Lebesgue-Integral)

Notes:

• Warning: I dropped \mathcal, since it doesn't render in gist
• Likewise, _\# became _H
• The items have rough importance rankings
• roughly "important" vs "neutral" and "random"; I'll derank a few later

Nikolaj-K

The relevant content starts at page 17 in the script. The chapter ends on page 25. Note that the exercises are different ones from the listing in the script.

==== Exercise's context ====

We consider the general minimization problem $p^{\ast} := \inf_{x \in M \subseteq X} f(x),$ where $X \subseteq \mathbb{R}^n$, $f:X\to\mathbb{R}$ is continuous,

Nikolaj-K

"""
Code and prompt used in the video
https://youtu.be/xcE_0azawvM
"""

"""You'll get two tasks
First part:
Consider a joint distirbution of two random variables X_k with k in {a, b} (i.e. two random variables X_a and X_b),
over a finite outcome sets of size n_a and n_b.
Explain what the marginal distributions are.