Bernd Schröder

Fundamentals of Mathematics


Curriculum Vitae
Class Materials
Research Interests



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 self-explanatory. 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 re-takes in continuous sessions from 2-21-09 to 2-24-09 and on 3-7-09 and 3-8-09. 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 re-shoots 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.


This is an overview of the course and of my philosophy for the course, the text and the videos. 

Video.    (15:52min, 58MB)                Slides.             


  1. Statements
    Video.   ( 7:49min, 29MB)              Slides.
  2. Implications
    Video.   (17:24min, 67MB)              Slides.
  3. Conjunction, Disjunction and Negation
    Video.   (25:53min, 98MB)              Slides.
  4. Special Focus on Negation
    Video.   (14:53min, 55MB)              Slides.
  5. Variables and Quantifiers
    Video.   (15:11min, 59MB)              Slides.
  6. Proofs
    Video.   (37:40min,143MB)              Slides.
    Dinner Time.
  7. Using Tautologies to Analyze Arguments
    Video.   (10:30min, 41MB)              Slides.
  8. Russell's Paradox
    Video.   (10:35min, 41MB)              Slides.

Set Theory

  1. Sets and Objects
    Video.   ( 8:14min, 31MB)              Slides.
  2. The Axiom of Specification
    Video.   (14:31min, 54MB)              Slides.
  3. The Axiom of Extension
    Video.   (17:37min, 66MB)              Slides.
  4. The Axiom of Unions
    Video.   (14:50min, 54MB)              Slides.
  5. The Axiom of Powers; Relations and Functions
    Video.   (35:16min,130MB)              Slides.
  6. The Axiom of Infinity; Natural Numbers
    Video.   (29:54min,112MB)              Slides.

Number Systems I: Natural Numbers

  1. Arithmetic With Natural Numbers
    Video.   (31:24min,116MB)              Slides.
  2. Ordering the Natural Numbers
    Video.   (30:51min,115MB)              Slides.
  3. A More Abstract Viewpoint: Binary Operations
    Video.   (28:00min,105MB)              Slides.
  4. Induction
    Video.   (30:52min,114MB)              Slides.
  5. Sums and Products
    Video.   (34:47min,127MB)              Slides.
  6. Divisibility
    Video.   (35:44min,129MB)              Slides.
  7. Equivalence Relations
    Video.   (14:51min, 58MB)              Slides.
  8. Arithmetic Modulo m
    Video.   (21:02min, 80MB)              Slides.
  9. Public Key Encryption
    Video.   (30:47min,118MB)              Slides.  
    Exercise Break.   Disclaimer.

Number Systems II: Integers

  1. Arithmetic With Integers
    Video.   (17:44min, 67MB)              Slides.
  2. Groups and Rings
    Video.   (27:14min,102MB)              Slides.
  3. Finding the Natural Numbers in the Integers
    Video.   (16:46min, 63MB)              Slides.
  4. Ordered Rings
    Video.   (33:51min,128MB)              Slides.
  5. Division in Rings
    Video.   (54:26min,205MB)              Slides.
    Braid Break.
  6. Countable Sets
    Video.   (36:25min,133MB)              Slides.

Number Systems III: Fields

  1. Arithmetic With Rational Numbers
    Video.   (13:09min, 49MB)              Slides.
  2. Fields
    Video.   (25:45min, 95MB)              Slides.
  3. Ordered Fields
    Video.   ( 9:41min, 37MB)              Slides.
  4. A Problem with the Rational Numbers
    Video.   (15:12min, 59MB)              Slides.
  5. The Real Numbers
    Video.   (55:14min,206MB)              Slides.
  6. Uncountable Sets
    Video.   (23:10min, 88MB)              Slides. 
  7. The Complex Numbers
    Video.   (15:15min, 58MB)              Slides.
  8. Solving Polynomial Equations
    Video.   (41:28min,153MB)              Slides.
  9. 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

  1. Irreducible Polynomials
    Video.   (25:32min, 97MB)              Slides.
  2. Field Extensions and Splitting Fields
    Video.   (22:04min, 85MB)              Slides.
  3. Uniqueness of the Splitting Field
    Video1.   (38:08min, 146MB)           Video2.   (25:56min, 99MB)               Slides.
  4. Field Automorphisms and Galois Groups
    Video.   (22:04min, 86MB)              Slides.
    The remaining videos will be uploaded when this site migrates to a larger server.
  5. Normal Field Extensions
    Video.   (35:12min, 132MB)              Slides. 
  6. The Groups Sn 
    Video1.   (37:44min, 142MB)            Video2.   (16:29min, 63MB)              Slides. 
  7. The Fundamental Theorem of Galois Theory and Normal Subgroups
    Video1.   (28:57min, 133MB)            Video2.   (24:26min, 93MB)            
    Video3.   (39:48min, 152MB)            Slides. 
  8. Consequences of Solvability by Radicals
    Video1.   (27:22min, 104MB)          Snack Time  Disclaimer.         
    Video2.   (21:33min, 83MB)               Slides. 
  9. Abel's Theorem
    Video.   (25:36min, 96MB)              Slides.

More Axioms

  1. The Axiom of Choice, Zorn's Lemma and the Well-Ordering Theorem
    (Video not available.)               Slides.
  2. Ordinal Numbers and the Axiom of Substitution
    Video.   (17:13min, 64MB)              Slides.
  3. Cardinal Numbers and the Continuum Hypothesis
    Video.   (35:52min, 133MB)              Slides.


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 on-line course with the videos, note that I have already done so. Please consider sending your students to my course  :)

Slides for a presentation on on-line delivery (focused on Differential Equations)


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.