I think it’s interesting that if you list the sequential digits of pi as follows:
1 - 3
2 - 1
3 - 4
4 - 1
5 - 5
6 - 9
As this list grows, it should be correct to say the number of numbers in the left column will be equal to the number of numbers in the right column.
I studied a math course in college that claimed this refers to the order of infinity.
In other words, both of these lists contain an infinite number of numbers but some lists containing an infinite number of things can be classified as having a greater order or a lesser order.
I hope you will agree that the number of digits in Pi is equal to the number of rational integers because the above list equates those the size of those two lists - or in other words … the number of numbers in those two lists are the same.
If you compare the list of integers to the list of those integers squared, it looks like this:
1 - 1
2 - 4
3 - 9
4 - 16
5 - 25
Once again, the number of numbers in the left column will be equal to the number of numbers in the right column.
But how can it be said the two lists will have the same number of numbers when the right hand column will grow at an exponential rate and the numbers in that list will always be much larger than the numbers in the left column?
Both colums are of the same “order”. One can say that is one order of infintiy. But there are several other orders which produce an infinite list that contains a greater number that these two lists.
I’m just racking my brain trying to remember some examples of those kinds of lists. But I went to college in 1967 and that was 50 years ago. Can you believe that? I want to cry.
It was so long ago that I can’t remember any good examples of different orders of infinity. But I bet someone here will be able to tell us about that.
Dang! I tried to correct some of the typos in the above post but the system editor would not allow me to do that because I ran out of time. I hope you will understand my meaning regardless of the typos.
I’d like to take this opportunity to thank my competitors, my fans, my parents, and my physics teacher who, with his patience and engaging lectures, has helped me to the success and intellectual curiosity I still have to this day
Wouldn’t the list of all lists that does contain itself have one more element than that? Or am I mistaking what the antecedent for the final clause is?
Call me a pervert, but I clicked on that and was suprised to find
The set of reals is larger than the set of rationals (which is equal in size to the set of integers), and the set of curves in the plane is larger still. Although there are infinitely many distinct infinities, these three infinite cardinalities are the only commonly discussed. It is “undecideable” whether or not there is some mysterious 4th infinity smaller than the set of curves, yet distinct from the infinities of reals and rationals.
Almost but not quite. There are indeed higher orders of infinity, but you can’t have an infinite list that contains a greater (cardinal) number of items than the list of positive integers (or of squares, or primes, or other examples you gave). A list, by definition, has a first item, a second item, a third item, etc. So the items in the list can be matched up one-to-one with the counting numbers (positive integers).
By contrast, the set of all real numbers—or even just the set of all real numbers between zero and one—cannot be listed. Cantor proved this: any (infinite) list of these real numbers is going to have some numbers that don’t appear anywhere on the list.
Here’s a good article (or series of articles; it takes several pages) on the issues involved.
I’d like to ask you if you know that excellent info because you studied math or if you found it by looking in Wiki.
I’m asking you this because if you have studied math and know this info, I’d love to be able to ask you some questions about it if that’s OK with you. Either in this thread or in PMs if anyone thinks that would be a form of thread jacking.
Thudlow Bank too. Sorry I posted the above paragraph before I read your post. More excellent info. Thank you very much!
Programmers from the 1980s to maybe late 1990s will remember** Ralf Brown’s Interrupt List**, a massive compilation of every software and hardware use of IBM PC-spec interrupts. It was essential for programming any kind of application or utility that used hardware interrupts, because it could conflict with existing uses. The last copy I had around in text file form showed a printing format of well over 600 pages.
I first studied math like this 50 years ago, but there are many on the Board much more knowledgeable than me, including, I’m sure, Thudlow Boink. In retirement I do try to refresh and advance my math knowledge on-line but what I posted off the top of my head above is very well-known to every mathematician. (It’s not “important” to the vast majority of math, but is very famous.)
You’re welcome to PM me a question, but no guarantee I can PM an answer back.
BTW, I clicked on the thread because I’ve been interested in lists for 55+ years – as a very young lad I once tried to memorize all the streets in a large city :smack: – but I couldn’t think of any interesting very long lists for the thread.
I just read something in the article Thudlow Boink recommended above and it made me LOL.
Definition: A set is infinite if we can remove some of its elements without reducing its size.
I’m not sure why. But I thought this was very funny. Perhaps it’s because of some famous earlier mathematicians (like Gauss) who used to tell people they must not even discuss actually infinite sets.
After he was proven wrong, I don’t think I’ve ever heard any mathematicians try to convince others that certain topics must never be discussed.
Can I ask you to check something in the article you recommended?
It’s fairly early on in Cantor’s proof. I think I found an error in it. Of course it can’t be an error in the proof since it’s held up for more than one hundred years. The error - if it really exists - must be in the transcription from the original to the internet version.
It occurs fairly early on the 2nd or 3rd page. It’s in Figure 5 titled, “Making a new real number”.
Do you see the 3rd position of the 3rd number is 8? The number is …
0.498310123
But the 3rd position of the result is also 8 and that is not supposed to happen.
Do you think that is really an error? If so, I will notify the author that it should be changed.
But I don’t want to do that unless someone will confirm I’m correct.
Every time I try to reverse search a phone number I am inundated with hits from sites that simply list all possible combinations. They have no information on the numbers at all, they just list them so that you’ll get suckered into clicking the link, thus raising their hit list, seeing their ads, and then moving on. Cha-ching! for them, but there are so many that it’s now virtually impossible to reverse search a number without a paid subscription somewhere.
Anyway, rant over, but however many possible combinations there are for a 10 digit number, that’s how many are on their evil list; almost certainly the highest I should think.
