Introduction to Higher Mathematics - Lecture 5: Set Theory

In this lecture we discuss the beginnings of set theory, a topic that runs throughout almost every area of mathematics.

[NOTE: Please ignore the part about "sets cannot contain other sets". Since making this video I have included an annotation about how Russell's paradox can be avoided while still allowing sets to be included in other sets by using the standard axioms of Zermelo-Frankel set theory (ZFC).]

[twisty]

Thank you for breaking set theory down so an old lady with a GED can understand, Your teaching skills is the

[rohitslimshady]

i generally do not comment on videos but cant stop myself after watchinng this.. great work this is awesome when i study for 30 minutes in class it feels like days but here i enjoyed this 30 minutes like i enjoy watching my fav programm on tv.great great great work pls upload more

[conradgarcia]

Great lecture. Perfect balance of simplicity and complexity, fun and challenge, insight and foresight. I'm looking forward for more lectures. You give symbols not just its meaning but how it connects to other disciplines. This technique makes the learner not only understand the symbol but how it is connected to other disciplines.

[ramuk]

Well, that's assuming the unrestricted comprehension principle. If you go with the axiom of choice, you have to "choose" if it is a part of your set or not, you could include sets in sets.

[MCPOTOTE]

thanks so much! you cannot imagine how grateful I am for having such a nice course available :)

[roberttatum]

your lectures are fantastic....you are the personification of lucid didactic communication

[mrnettek]

Thank you, sir! I'm learning higher mathematics to help me with software engineering, as well as just gaining general math knowledge. These videos are excellent!

[hilljonathon]

Thanks Mr. Shilito, I think I have learned more from watching your videos then I have from listening to my university lecture...

[MichaelZibulevsky]

Great lesson: Deep, simple and intuitive!

[kstahmer]

There’s a compelling reason to include 0 as a natural number (20:39), namely:

0 = {}
1 = {0}
2 = {0, 1}
3 = {0, 1, 2}
…
w = {0, 1, 2, …}
w + 1 = {0, 1, 2, …, w}
…

Where w represents the ordinal number omega, the set of all natural numbers:


(Currently, I haven’t seen your entire series.  You may have included this in one of your later videos.)

[ManigandanRajasekar]

Is there anyway, we can get the mathematical community to vote for the trollface as the accepted symbol for paradoxes?

[nditafonngeh]

Bill this is a very great lecture. You are simply a wonderful teacher

[geanp]

I just wanted to express how much I love your videos. 

[sloaiza]

just watched this one and I´m hooked. thank you very much.

[BillShillito]

Which "larger sets" are you talking about?

And I do talk about algebraic and transcendental numbers in a later lecture, the one about constructing the real numbers. :)

[BillShillito]

Yeah, I noticed the pointer thing too late ... I'm trying to fix it in all the videos I've been making since I started noticing it. Sorry about that! Glad you enjoy the videos though!

[GudLawdHammercy]

Great video(s)... FINALLY! I GET IT! THANK YOU!! I was most impressed by the methodical and easy-to-understand approach with EXCELLENT EXPLANATIONS of the meaning and use of SHORT-HAND MATH SYMBOLS. If you want to LEARN MATH, or LEARN HOW TO TEACH MATH, then I strongly encourage you to WATCH THESE VIDEOS! GREAT JOB BILL SHILLITO (- If Samuel L. Jackson could type ;-) )

[davidrudolph]

Fantastic videos! Thank you for posting. Please post more. My only crit is the pointer being in the capture, its a little distracting. Still, its cream of the crop. Bravo!

[UberMarauder]

Bill !! Im thrilled with your videos. I am embarking on the same journey, I love logic and set theory and abstract mathematics. Though I have a question, seems to me that your definition "sets cannot contain other sets" is wrong, because what about the power set of say a set A: P(A). This is the set of every possible subset of A, and given a subset of a set is a set itself it cannot be true, given the power set exists, that a set cannot contain other sets in it.

[bristol]

Metaphysics as set theory

I believe that Leibniz-Plato is not really philosophy, but mathematics. In fact, set theory. Unfortunately I am not a mathematician, and need some help. But here are some initial thoughts.

In Leibniz, the physical object in spacetime is represented as a mental point called a monad. Mind is a formalization or form of mathematics in Plato-Leibniz. From set theory we find that a set is a collection of objects that it owns. This ownership I believe is what we call control, perception or creation. That is, a set controls the objects or sets within it hierarchically.

Using the notation < to mean is conceptually contained in (a sidewise U), M as plato's Mind (the One), T as the domain of time, and S as the domain of space, p as a particle is spacetime, and m(p) as its monad, we have the sets

m(p) < S < T < M


The motion of an object in spacetime begins with its birth as a still point in Plato's Mind. That is. as a point in the all-encompassing set of points, Plato's Mind, which is the point containing all other (mental) points. The mental is the mathematical, the formal in terms of set theory. Time and space are quantized, as is experience, so experience is given to us a set of movie frames that we expeience as a movie.

--
Dr. Roger B Clough NIST (retired, 2000).