Nikolaj-K

These are the notes for the video of the paper discussed in

https://youtu.be/YdKVqfKbXxg

# Paper
  * https://arxiv.org/abs/1712.01815
  * https://arxiv.org/pdf/1712.01815.pdf

# Autors
https://en.wikipedia.org/wiki/David_Silver_(computer_scientist)

Nikolaj-K

Text for the video

https://youtu.be/CwKsskImZUw

=====Comprehension, Specification, Separation, Predicativity=====
====About====
There's no clear-cut accepted definition of "predicative", but here's a informal description:
"A definition is said to be **impredicative** if it generalizes over a totality to which the entity being defined belongs."

Predicativity is about "from below" specification or even construction.

Nikolaj-K

My Master thesis

http://othes.univie.ac.at/16190/

discussed in the video

https://youtu.be/sX2VmeVSw_U

====Overview====

Nikolaj-K

Text for the video at

https://youtu.be/IYBY0qe01x8

=====Galilean group cohomology=====
====Lagrangians====
All

$L = \tfrac{m}{2}\left(\frac{{\mathrm d}}{{\mathrm d}t}q_t\right)^2 + \frac{d}{dt}\lambda(t, r(t))$

Nikolaj-K

The file discussed in this video:

https://youtu.be/f_8myEDBD9A

First order logic up to strong set theory axioms. Axioms with constructive interpretation first.

====Preliminaries====
===Axiomatization strategy===
$\bullet$ Propositional logic (Give Lambda term justification)
$\bullet$ Predicate logic

Nikolaj-K

Document used in the video

https://youtu.be/_sttXg1w474

====The Continuum Hypothesis and its consequences====
===have been a disaster for integration theory===

Let $I:=[0,1]$ be the real unit interval.
Consider the ordinal $\beta$ with $|\beta|=|I|$ and $b\colon I\to\beta$ the corresponding bijection.

Nikolaj-K

"""
This is the code used in the video:

https://youtu.be/KFobEtBQ_bE
"""

import matplotlib.pyplot as plt

# Just some pretty-print routines
def print_strings(strings):

Nikolaj-K

This is the text used in the videos on topoi here:

Predicates in Sets: https://youtu.be/Ysp7P4wW-zE
Subobject classifier in Sets: https://youtu.be/uqBhy1c4MZc
Defining topoi here: https://youtu.be/Bdn64edr4Ng

=====The internal logic of a topos of sets=====
Note: We do first-order logic (constructive) set theory here, i.e. we work with axioms for a membership-predicate $\in$ over all terms.
So $y$ is merely another set parameter in $x\in y$ any such expression might be assigned a true value for any $x,y$.
We don't do category theory proper, axiomizing concatenation $\circ$, but we'll speak about it on this page too.

Nikolaj-K

==== Motivation ====
Things that are Heyting algebras:
$\bullet$ $\mathcal P\{0\}$ resp. $1\to\Omega$
$\bullet$ Finite distributive lattices
$\bullet$ Every Boolean algebra
$\bullet$ Open sets of a topological space
$\bullet$ Open elements of an interior algebra
Equivalently
$\bullet$ Lindenbaum-algebra (all equivalence class of sentences that imply each other in the theory with connectives as operations) of IPC

Nikolaj-K

Text file going with the video

https://youtu.be/E6wr0MtbKrY


A         ... Predicate
A(x)      ... One reading: "A is satisfied by x."
              Here: "The state x forces A to hold."

Modalities: Model necessity, knowledge, believe, persistence, obligation, ...