The Smallest Infinity-Series on Infinity Part 5

[yoyongirlyoyo]

thank you very much! you make everything so simple in detailed explanations!

[SSCCMath142]

Hello, Thanks for the comment! I've been pretty busy since I did these videos a few years ago. I plan to get back to it and add a few more this December if I can figure out how to log back in under Davidson1956. (I have several hundred instructional videos for my classes under a different log in.) I'd like to do a simple explanation on relativity, and also a series on musical acoustics, for starters. Jon

[Thetarget1]

@SSCCMath142 That would be really great!

[kooky323]

Professor Davidson:

Enjoyed all of your videos! Will you be posting more of a different nature?

[srfriggen]

I feel like I'm learning from a very smart Phillip Seymour Hoffman

[at1with0]

Great videos. btw, I found an explicit formula (and its "inverse") whose domain is the natural numbers and range is the set of rational numbers, and it is onto (a surjection). The usual proof that Q is no larger than N involves making a table of rational numbers and counting them diagonally to convince the reader that every rational number will be counted. The N onto Q video should have a link in the description to the formulas that map N onto Q.

[SACkings44]

We just went over this in my math class the other day, my professor explained it a little differently however. He showed that the interval 0 to 1 has an infinite number of real number correspondents very interesting!

[TheJimzhe842]

nice video, and a lovely mind-boggling problem for me to ponder

[StitchedMusik]

Wait. How does that work out, if you're using the fraction duplicates? If you took out all the duplicate fractions, wouldn't there be less fractions than counting numbers? Please clarify. Because we aren't using duplicates of the counting numbers.

[SSCCMath142]

Great question! But it still works. If we erase all the duplicates, we can still match up the unduplicated fractions in a one to one correspondence with the counting numbers. Rather odd this infinity thing, eh?

[StitchedMusik]

@SSCCMath142 oh, right. And also, for the larger infinities, how are they uncountable? I can't see how aleph-null is smaller than aleph-one.

[StitchedMusik]

My favorite video. Ever. :)

[pinballwyz]

Thank you so much for this wonderful tutorial. A math professor with a knack for teaching is a blessing. As to the 1,000 or more hits, I propose that it will achieve an infinite number of hits...it will simply take an infinite amount of time. Lock 'em up in the Hilbert Hotel!

[SSCCMath142]

Thank you very much for these very kind and witty comments!--Jon Davidson, author of these videos

[gen6k]

its greater by being bootstrapped into an array in an arena that must be greater than the case.

[RiaanEloff]

I realize that I may be challenging mathematics and how mathematicians see numbers, still I see a flaw: Saying that there is a larger or smaller infinity is paradoxical in itself. There cannot be "more" or "less" in an INFINITE group. The very meaning of the word contradicts this. Infinite "HAS NO END", thus it CANNOT be a rational "set" in the way humans understand "sets" or "groups of things". Comparing endless to endless - they're both endless...NO END, no useable "set" in that sense.

[m8u8s8i8c]

Sir, its not seeming to be 1-1 correspondence .. I mean a bijective function between N = set of all naturals and Q+= set of all positive rationals. few numbers (infinitely more) such as 1/2 = 2/4 = 3/6 .... 3/1=6/2=9/3. ....are repeated in your list you did not omit - see CANTOR's Diagonalization VIDEO on YouTUBE! Educate me (but avoid mistakes!) ...

[frantully]

How is 2 over one the same as one over two?

[kbjami]

Please make more videos. There are very informing.

[kbjami]

@kbjami Okay I'm a dumb ass.

Please make more videos.

They're very informing. Please excuse my awful grammar mistake

[PhRiZoN]

I don't know much about mathematics, obviously, I'm still in school, but to say that one infinity is larger than, or smaller than another infinity doesn't make sense, to me. To say that one is smaller is to say that there are end points which determine it's length compared to a larger infinity. If they are both infinity how could one possibly be larger than another?

[paulio2293]

this guy should make more videos (for more advanced mathematicians), this is what youtube should be used for. its an awful shame that this video has 1,000 views whereas the latest Ke$ha song has 49,000,000 views. Knowledge is the essence of human existence

[PearyHD]

@paulio2293 Couldn't have been said better.

[nowenterpsie]

Again, get your precepts out for all to see:

Mathematicians such as this seem to treat Infinity as a 'number', or a 'size' here, and this is the basic and fundamental flaw in all of this logic. Infinity is a non-number, closely related to Zero or None. By definition a 'number' or any 'size' is bounded by finite-ness.

We are waking up.

[intermender]

who's this we anyway you keep talking about. he is showing his reasoning (clue, progressive series of vids, and enough reference for use to follow-up); while you are just trolling every vid with the same undemonstrated, unreferenced, pompous comment. Why don't you do your own set of videos instead of trolling, big head' or at least give a reference apart from 'i'm clever coz i say so you plebs'.

[nowenterpsie]

Maybe i'm a 'big head', but i really didn't mean to condescent, so the 'plebs' reference is not really fair. Anyway...

Did you actually hear what I was trying to say before reacting so harshly to my comment?

[intermender]

Sorry, it seemed to me you were saying he wasn't going through his course synthesising transparently step by step his arguments and referencing his sources for us to check asssumptions; in comparison you offered a flat out unsubstantiated statement and judgement with no reference or demonstration.

So it looked to me like you were a some kind of creationist troll as typical on youtube trying to spoil a great set of vids and spread unhappiness by invoking irrelavent spirituality.

[19916718514]

thank you!!

[greenkid336600]

It's worth pointing out that getting the 1-to-1 correspondence necessitates counting only the reduced fractions, since indexing fractions such as 2/2, 3/3, 2/4, 3/6 etc. will result in mapping (using the index presented in your video) 1, 5, 13 etc. to the fraction 1/1, and similarly for other fractions.

[HandicapReborn]

I wish there were more videos like these, that fed the mind rather than the millions of videos that are destroying it.

[CogitoErgoCogitoSum]

Explain something to me. How is there a 1-1 ratio between counting and rational numbers. It seems to me that every rational number is counted an infinite number of times. The diagonal from top left to bottom right are all equal/reducible to 1. Therefore, 1 alone, is counted infinitely many times when its paired with a counting number. The same is true for 2/3 and 4/6 and 10/15, etc, all are 2/3. We are counting a single rational number infinitely many times when pairing it to a counting number.

[Davidson1956]

Great question! But we call it a 1-1 correspondence, not a 1-1 ratio. Applied to this example, it means that each number in the counting number set is paired off uniquely with each number in the diagonalization set of the rational numbers. And you're absolutely right that Cantor took the set of rational numbers and greatly expanded its size by including all the unreduced fractions, yet each one can still be labeled with a unique counting number. Mind boggling!

[Aliuar]

I think the issue with the number line extending to 2 versus extending to 1, is a fundamental issue with Vectors.

The vector line that makes up 1 and 2, are actually extensions of an infinite number of multiplication of the vector length. So realistically the fundamental reality here is that of a vector and splicing vectors into measurable lengths. In other-words, they are the same hehe.

[Aliuar]

Let me add one thing, as to expanding the Vector, there is always the inverse action (getting smaller). So logically as you expand you can divide (make smaller).

[nothingstopskings]

Does your name mean "I think therefore I think I am"?

[CogitoErgoCogitoSum]

Indeed it does. At least, it was my poor attempt to construct that statement. I dont know Latin, and so undoubtedly mis-phrased it.