TML: The Infinities In Between (2 of 2)

[hmodywakid]

number 8 in 0:35 is wrong it should be phi not {phi}

[thewiffenpoof]

I think there's a slight inaccuracy here. The continuum hypothesis is that the cardinality of c = aleph1. Aleph1 is an ordinal and mathematicians are not sure how big it is (whether or not it is equal in size to the continuum). So you are correct that the power set of N is the continuum or 2^aleph0 but it is an open question (in fact the continuum hypothesis) whether or not 2^aleph0 = aleph1. Aleph1 is just the next larger (ordinal) cardinal infinity whatever that is.

[kaosmostly]

Great video.

And super groovy music at the end.

Is it you playing?

If so, you should call your band Aleph and the Infinities.

If it's not you, tell us where to find more of this music.

Thanks

[CellarAcademic]

@kaosmostly If I create a band, I'll use that name and give you credit. The song in the vid is called "Soul Thing" by Tony Newman

[2eelShmeal]

I would think that there are an infinite # of infinities between the alephs. I guess the question is how you show that mathematically.

Whaddaya think? Please forgive my ignorance on the subject...

I just feel that fractal geometry will be key to explain it perhaps. How far out in left field am I?

[Arlesterc]

Excellent. I would be very interested in seeing you 'explain' Godel's theory - I haven't seen one yet and I've looked for a while.. If you have already done that/or know of one you could recommend I would appreciate a link to it.

[gen6k]

why dont u just do an inverted power set thats a vacuum between the two but have antinumbers instead of reals.

[ILoveYouToThe9999999]

I found a rule that makes the reals countable.

[DJFishlips]

let me guess: this margin is too small to contain it :)

[ILoveYouToThe9999999]

No but it needs to be proved.

[MrMush4Brains]

An earlier comment is correct - Aleph-one means the second size of infinity, and since no one has proven the Continnum Hypotheis, we can't say that the cardinality of the continuum is aleph-one. You mean ro say beth-one. Still a good video.

[DanRYoung]

You've presented a very clear video examining some of the astounding ideas Cantor gave us. I have yet to take a class on set theory, but your video is good inspiration to consider doing so. Thanks!

[CellarAcademic]

Thanks much. Be warned however. Set theory can be mind-numbing. It's essentially 90% philosophy (the scary kind of philosophy).

[siddisidam]

@CellarAcademic Yeah....this is actually the hidden dimension of every day numbers.

Numbers look simple but hold oceans beneath them. Simple logic such as "every number whose digits add to a total of 9 is divisible by 9", and many more are not just mathematical principles. They show the extreme networking of nature and its foundation. Also, a step towards the grand unification theory.

[owchywawa]

Why wouldn't someone assume that all numbers are infinite; that all numbers are actually subsets of infinity? There are infinite points between the numbers 1 and 2 for example. Would not then 2 simply be a set of infinity?

[CellarAcademic]

You could do that if you found something useful in the process. Of course, in that case, it would be difficult to call it math since the number 2 has had a very simple, concrete, and finite interpretation ever since humans were counting their sheep

[owchywawa]

Yeah. I was thinking about Zeno's paradox at the time. To define 2 as infinity would be pretty useless if we are talking about sheep. I was thinking about space and time.

[ancalagon12321]

Much kudos to you for spreading the light of set theory to the benighted land of youtube. However:

Aleph(n) (for n > 0) means the nth infinite cardinal after Aleph(0). What you're calling Aleph(n) is actually known as Beth(n).

It's *not* a case of having to choose between different authors' notations. Rather, the use of the Aleph notation has been set in stone since Cantor's day and *no-one* who understands it uses it your way.

(Sorry to make this pedantic comment!)

[CellarAcademic]

Well, the 'author' was my undergrad professor and I took the 'notation' from his notes; I've looked around and it turns out you're right that no one else uses Cantor's notation that way.

Of course, What's in a name? (you must be familiar with THAT author) As long as the idea is properly communicated for the YT community, then all should be fine. And yes, our discussion here is probably too pedantic for the YT community, but thanks for bringing it up.

Cheers

[ancalagon12321]

There's something to be said for that view.

Still, the Aleph function is pretty fundamental. I think what you're doing is a bit like a lecture on particle physics where the 'neutron' is really the proton.

Properly introducing the Alephs means covering the ordinals first, and Zermelo's theorem. So I guess it'd require at least one more vid.

I'm vaguely planning to make a video explaining 'zero sharp', but perhaps only at the end of a sequence of Aleph(1) preliminaries.

[CellarAcademic]

Good luck with the zero sharp vid. If you post it on YouTube, keep in mind the type of audience that will see it. There's a reason for the lack of videos involving set theory, foundations, Godel, etc. looking forward to it though

[ancalagon12321]

Much kudos to you for spreading the light of set theory to the benighted land of youtube. However:

Aleph(n) (for n > 0) means the nth infinite cardinal after Aleph(0). What you're calling Aleph(n) is actually known as Beth(n).

It's *not* a case of having to choose between different authors' notations. Rather, the use of the Aleph notation has been set in stone since Cantor's day and *no-one* who understands it uses it your way.

(Sorry to make this pedantic comment!)

[himkdm]

Thanks, that really helped me explore the hypothesis.

[kautilya33]

I have a simple view of set theory. I believe that any grammar which can handle ZF theeoy has to have four types of sentences: theorems, falsehoods, introversions, and profundities. What Goedel has shown is that there are introversions in set theory, and what Cohen has shown is that the continuum hypothesis is a profundity of ZF theory. I believe that there are an infinite number of profundities yet to be dicovered.

The clarity of your lectures I find admirable.

[dasklavierleben]

Hey good job!! You should think about becoming a professor for a living. I think you could pull it off if you don't mind large audiences.

[dasklavierleben]

no wait, you are becoming a professor aren't you? That's why you're going for a PhD im assuming. nevermind I thought you were like 21-ish. But thats ok, I'm 19 and everyone thinks I'm like 14.

[CellarAcademic]

Yes, I've already taught for 12 semesters, but I do look younger, and I act way younger, glad you liked the lecture

[mightybazooka]

the mathematics we have isnt powerful enough

that is some crazy ass statement....

(mind boggeling terms)

[oscartrellesm]

A little errata in the power set of {1,2,3} with the empty set.

[CellarAcademic]

I see, {Ø} is reserved for the set containing the empty set, standard notation would just have either Ø or {}. That is quite specific, but I probably shouldn't be playing fast and loose with the notation. Thanks for the heads up, and for watching.

[NoFearmin]

Nice job

[CellarAcademic]

Ta V much NoFearmin

[jemsut]

Nice videos, one of the best covering the subject. I particulary like how you emphasised the need for a bijection as the key concept, something other people gloss over.

[CellarAcademic]

thanks, it was intended mainly for undergraduate math majors so it had to be accessible, but I couldn't completely dispense with the formality either, glad you enjoyed it

[CellarAcademic]

Axioms are statements which are generally assumed to be true. For example, say you have infinitely many sets, each having at least one element. Some mathematicians think it's okay start off the proof by taking 1 element from each set and working with it (since each set has at least one to work with). But others think you need a rule to tell you which element to take. It's hard to prove something like this, so if you want to do it, you assume what they call the "axiom of choice".

[clyniclymsdup]

thanks ;)

[clyniclymsdup]

what types of axioms are too fundamental to prove?