

The videos below are part of a remotely delivered course on the
fundamentals of mathematics: proofs, logic, sets and numbers. The course
covers the construction of the real numbers from the axioms of set theory
and there is material for a follow up seminar on why quintic equations
are, in general, not solvable by radicals. These video presentations
follow my text
for this course. The names of the videos should be selfexplanatory.
For the "license", click here.
For mathematics presentations in general, the pause button is your best
friend. But I must emphasize this statement for proof classes, such as
this one. Because material can be presented rapidly with slides, there is
no slowing down as when I write stuff by hand on the board. That means you
really need to pause the videos frequently to make sure that you can
digest the points that are being made. The best approach is to first read
the
book and then follow the presentations. (That actually goes for any
mathematics class, including those presented face to face.)
The videos were shot consecutively without retakes in continuous
sessions from 22109 to 22409 and on 3709 and 3809. In some videos
the slides have "nonlethal typos", but (thankfully) I did not find fatal
flaws on the slides. Of course, the typos that I found were eliminated in
the book and on the posted slides. Some reshoots could marginally improve
the videos, but I do like the rather natural "continuous office hour"
atmosphere. No conversation is ever flawless, and it can be fun to catch
the teacher misspeaking as well as finding the few typos on the
slides. (Actually, at this level, with previous reading of the text,
that can be considered educational. The more you are attuned to finding
mistakes, including the teacher's, the more you will learn.)
The videos were produced with tegrity. For best performance, they
should be downloaded and then run with Internet Explorer as the default
browser. The slides were produced with the LaTeX beamer package.
Introduction.
This is an overview of the course and of my
philosophy for the course, the text and the videos.
Video.
(15:52min,
58MB)
Slides.
Logic
 Statements
Video. ( 7:49min,
29MB)
Slides.
 Implications
Video.
(17:24min, 67MB)
Slides.
 Conjunction, Disjunction and Negation
Video. (25:53min,
98MB)
Slides.
 Special Focus on Negation
Video. (14:53min,
55MB)
Slides.
 Variables and Quantifiers
Video.
(15:11min, 59MB)
Slides.
 Proofs
Video.
(37:40min,143MB)
Slides. Dinner
Time. Disclaimer.
 Using Tautologies to Analyze Arguments
Video. (10:30min,
41MB)
Slides.
 Russell's Paradox
Video. (10:35min,
41MB)
Slides.
Set Theory
 Sets and Objects
Video. (
8:14min, 31MB)
Slides.
 The Axiom of Specification
Video. (14:31min,
54MB)
Slides.
 The Axiom of Extension
Video. (17:37min,
66MB)
Slides.
 The Axiom of Unions
Video. (14:50min,
54MB)
Slides.
 The Axiom of Powers; Relations and Functions
Video.
(35:16min,130MB)
Slides.
 The Axiom of Infinity; Natural Numbers
Video.
(29:54min,112MB)
Slides.
Number Systems I: Natural Numbers
 Arithmetic With Natural Numbers
Video.
(31:24min,116MB)
Slides.
 Ordering the Natural Numbers
Video.
(30:51min,115MB)
Slides.
 A More Abstract Viewpoint: Binary Operations
Video.
(28:00min,105MB)
Slides.
 Induction
Video.
(30:52min,114MB)
Slides.
 Sums and Products
Video.
(34:47min,127MB)
Slides.
 Divisibility
Video.
(35:44min,129MB)
Slides.
 Equivalence Relations
Video. (14:51min,
58MB)
Slides.
 Arithmetic Modulo m
Video. (21:02min,
80MB)
Slides.
 Public Key Encryption
Video.
(30:47min,118MB)
Slides. Exercise Break.
Disclaimer.
Number Systems II: Integers
 Arithmetic With Integers
Video. (17:44min,
67MB)
Slides.
 Groups and Rings
Video.
(27:14min,102MB)
Slides.
 Finding the Natural Numbers in the Integers
Video. (16:46min,
63MB)
Slides.
 Ordered Rings
Video.
(33:51min,128MB)
Slides.
 Division in Rings
Video.
(54:26min,205MB)
Slides. Braid
Break. Disclaimer.
 Countable Sets
Video.
(36:25min,133MB)
Slides.
Number Systems III: Fields
 Arithmetic With Rational Numbers
Video.
(13:09min, 49MB)
Slides.
 Fields
Video. (25:45min,
95MB)
Slides.
 Ordered Fields
Video. (
9:41min, 37MB)
Slides.
 A Problem with the Rational Numbers
Video.
(15:12min, 59MB)
Slides.
 The Real Numbers
Video.
(55:14min,206MB)
Slides.
 Uncountable Sets
Video.
(23:10min, 88MB)
Slides.
 The Complex Numbers
Video.
(15:15min, 58MB)
Slides.
 Solving Polynomial Equations
Video.
(41:28min,153MB)
Slides.
 Beyond Fields: Vector Spaces and Algebras (my confusion about
quaternions became an exercise for the book)
Video. (21:48min,
85MB)
Slides.
Unsolvability of the Quintic by Radicals
 Irreducible Polynomials
Video.
(25:32min, 97MB)
Slides.
 Field Extensions and Splitting Fields
Video.
(22:04min, 85MB)
Slides.
 Uniqueness of the Splitting Field
Video1.
(38:08min, 146MB)
Video2.
(25:56min, 99MB)
Slides.
 Field Automorphisms and Galois Groups
Video.
(22:04min, 86MB)
Slides. The
remaining videos will be uploaded when this site migrates to a larger
server.
 Normal Field Extensions
Video.
(35:12min, 132MB)
Slides.
 The Groups S_{n}
Video1. (37:44min,
142MB)
Video2. (16:29min,
63MB)
Slides.
 The Fundamental Theorem of Galois Theory and Normal Subgroups
Video1.
(28:57min, 133MB)
Video2.
(24:26min, 93MB)
Video3.
(39:48min, 152MB)
Slides.
 Consequences of Solvability by Radicals
Video1.
(27:22min, 104MB)
Snack Time Disclaimer.
Video2.
(21:33min, 83MB)
Slides.
 Abel's Theorem
Video. (25:36min,
96MB)
Slides.
More Axioms
 The Axiom of Choice, Zorn's Lemma and the WellOrdering
Theorem
(Video not
available.)
Slides.
 Ordinal Numbers and the Axiom of Substitution
Video.
(17:13min, 64MB)
Slides.
 Cardinal Numbers and the Continuum Hypothesis
Video.
(35:52min, 133MB)
Slides.
"License".
Obviously, if I post something, I want people to use it. So do it. If
you like the videos, try the book. If you are a teacher, feel free to use
the videos and the slides in classes. The goal is to get people to
do better in mathematics.
One caveat: If you want to create an online course with the videos,
note that I have already done so. Please consider sending your students to
my course :)
Slides for a presentation on
online delivery (focused on Differential Equations)
Disclaimer.
Braid Break, Dinner Time, Exercise Break and Snack Time were privately
recorded by the author. No tax dollars were wasted to produce these small
diversions from the rather challenging matter of this course.
