Different sizes of infinity...?

Killjoy

Killjoy

Propelling The Farce!!
Valued Senior Member
The Man Who Stole Infinity

In an 1874 paper, Georg Cantor proved that there are different sizes of infinity and changed math forever. A trove of newly unearthed letters shows that it was also an act of plagiarism.

***

In their 1872 papers, though, Cantor and Dedekind ( FYI - Richard Dedekind, the man Cantor is accused of plagiarizing) had found a way to construct a number line that was complete. No matter how much you zoomed in on any given stretch of it, it remained an unbroken expanse of infinitely many real numbers, continuously linked.

***

...It all started with a proof he published in 1874. In that proof, Cantor showed that there are different sizes of infinity, putting to bed the notion that infinity was merely a piece of mathematical trickery.
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This was all very interesting stuff - notwithstanding that the bits about philosophizing about precisely what a number is sounds ever so 19th century German.

My main question, however - is how the Devil can there be "different sizes" of infinity ?
I mean - if it's infinity, seems to a poor, dumb proletarian dork-ass like your humble narrator that infinity's pretty much got no size, because it's beyond size.

I suppose that's why I refer to any mathematics which requires "rune-like symbols" to write its equations to be the work of the Devil.:biggrin:

Still, if any or our resident worthies more well versed in this sort of thing might be able to explain it, or point out where I might find info that would, it would indeed be appreciated.
 
My high level gloss over the details example:

There are infinite Integers, and infinite Naturals, and infinite Primes. You can map (not match :) ) these to come up with unique 1 to 1 pairings "forever", and this shows they have the same cardinality. Their "size of infinity" is the same. e.g. map even Naturals to +ve Integers and odd Naturals to -ve Integers: (1,-1),(2,1),(3,-2),(4,2),...

But now look at the Reals. There are infinite numbers just between 1 and 2. You cannot find a mapping (bijection) from Natural or Integer to Real. e.g. What would even be the "next" Real number after 1 to map with the Natural 2? Reals have a whole 'nother level of infinity.
 
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Hilbert's Hotel is the classic example.

  • The Setup: Imagine a hotel with a countably infinite number of rooms (numbered 1, 2, 3, 4, ...), and every single room is currently occupied by a guest.
  • The Problem: A new guest arrives looking for a room. In a normal, finite hotel, there would be "No Vacancy."
  • The Solution: Because the hotel is infinite, the manager can simply ask the guest in Room 1 to move to Room 2, the guest in Room 2 to move to Room 3, and so on (guest n moves to n+1).
  • The Result: Room 1 becomes vacant for the new guest, despite the hotel being "full"
Takeaways:
  • Infinity + 1 = Infinity: Adding to an infinite set does not make it larger.
  • Infinite New Guests: If an infinite bus arrives, Hilberts manuever provides infinite space for the new arrivals.
  • Veridical Paradox: It is a true statement that feels false—it shows that "full" does not mean the same thing in the world of infinity as it does in the finite.
 
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OK, apparently there are at least some things that our giant-brained WoW has not "learned and moved beyond".
 
pzkpfw said:
My high level gloss over the details example:

There are infinite Integers, and infinite Naturals, and infinite Primes. You can map (not match :) ) these to come up with unique 1 to 1 pairings "forever", and this shows they have the same cardinality. Their "size of infinity" is the same. e.g. map even Naturals to +ve Integers and odd Naturals to -ve Integers: (1,-1),(2,1),(3,-2),(4,2),...

But now look at the Reals. There are infinite numbers just between 1 and 2. You cannot find a mapping (bijection) from Natural or Integer to Real. e.g. What would even be the "next" Real number after 1 to map with the Natural 2? Reals have a whole 'nother level of infinity.
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Aiee, Chee-wa-wa !

This seems to me to represent differing interpretations of the term infinity rather than that the "infinite set" of each type of number is of a different size. Guess I'm hung up on the term infinity itself.
 
DaveC426913 said:
Hilbert's Hotel is the classic example.

  • The Setup: Imagine a hotel with a countably infinite number of rooms (numbered 1, 2, 3, 4, ...), and every single room is currently occupied by a guest.
  • The Problem: A new guest arrives looking for a room. In a normal, finite hotel, there would be "No Vacancy."
  • The Solution: Because the hotel is infinite, the manager can simply ask the guest in Room 1 to move to Room 2, the guest in Room 2 to move to Room 3, and so on (guest n moves to n+1).
  • The Result: Room 1 becomes vacant for the new guest, despite the hotel being "full"
Takeaways:
  • Infinity + 1 = Infinity: Adding to an infinite set does not make it larger.
  • Infinite New Guests: If an infinite bus arrives, Hilberts manuever provides infinite space for the new arrivals.
  • Veridical Paradox: It is a true statement that feels false—it shows that "full" does not mean the same thing in the world of infinity as it does in the finite.
Click to expand...
I've either heard or read this example somewhere. It's the notion that there are "greater" or "lesser" (larger and smaller ?) infinities that's boggling my poor ol' Rebel Ape's mind.
 
Killjoy said:
Aiee, Chee-wa-wa !

This seems to me to represent differing interpretations of the term infinity rather than that the "infinite set" of each type of number is of a different size. Guess I'm hung up on the term infinity itself.
Click to expand...

This may do more harm than good, and I apologise, but in my head I visualise it as "dimensions":

Integers, Naturals, Primes etc can go off in a single line, "1 dimension":
0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, ... --->
1, 2, 3, 4, 5, 6, 7, ... --->
2, 3, 5, 7, 11, ... --->

But with Reals (simplified list here), you get that and also the infinite values between each of the whole values:
1, 2, 3, 4, 5, 6, 7, ... --->
|...
|1.0000000001
|...
|1.999999999000001
|...
V

I sort of see that as "2 dimensions", which helps me see it as a different class of infinity vs the "1 dimension".

(Not real math terms.)
 
pzkpfw said:
This may do more harm than good, and I apologise, but in my head I visualise it as "dimensions":
Click to expand...
Here's the trouble with that. Dimensions are generally "degrees of freedom" and orthogonal, meaning one can change independtly of the other.

But Reals really are on the same number line. You can subtract pi from root 2 and get a different number on the same number line.
 
DaveC426913 said:
Here's the trouble with that. Dimensions are generally "degrees of freedom" and orthogonal, meaning one can change independtly of the other.

But Reals really are on the same number line. You can subtract pi from root 2 and get a different number on the same number line.
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I don't think dimensions help that much unless you want to consider the universe itself.
Number lines that stretch out infinitely are hard enough, especially when you start comparing and mapping them.
 
Killjoy said:
The Man Who Stole Infinity


This was all very interesting stuff - notwithstanding that the bits about philosophizing about precisely what a number is sounds ever so 19th century German.

My main question, however - is how the Devil can there be "different sizes" of infinity ?
I mean - if it's infinity, seems to a poor, dumb proletarian dork-ass like your humble narrator that infinity's pretty much got no size, because it's beyond size.

I suppose that's why I refer to any mathematics which requires "rune-like symbols" to write its equations to be the work of the Devil.:biggrin:

Still, if any or our resident worthies more well versed in this sort of thing might be able to explain it, or point out where I might find info that would, it would indeed be appreciated.
Click to expand...
If you can get your hands on "1,2,3 Infinity," by George Gamov, I would give it a go. It investigates some of these concepts in a fun way.
 
Just to add, it is worth reading about how Cantors work was received and how this affected his mental health over the years.
 
There's a neat argument to show that the set of real numbers is uncountably infinite. i.e. that real numbers cannot be put into a one-to-one correspondence with the natural numbers.

It's an argument by contradiction.

Suppose that the real numbers can be counted (this demonstration will actually show that they can't). If they can be counted, then we can, in principle, make a big list of all of them. Here are a few of the real numbers between 0 and 1, in no particular order, which make up part of the infinite hypothetical list of all numbers between 0 and 1:
Code: 
0.7892345000000...
0.1111111111111...
0.5724320000001...
0.2121212121212...
0.3333333333333...
0.1999999999999...
Now, remember that, hypothetically, I could have posted the entire list of all the real numbers between 0 and 1.

Now I'm going to do something tricksy with the numbers on my list. I'm going to add 1 to the first digit of the first number in the list, and add 1 to the 2nd digit of the second number, and add 1 to the 3rd digit of the third number, and so on. If the digit happens to be a nine, I'll replace it with a zero. After going through this process, my list looks like this:

0.8892345000000...
0.1211111111111...
0.5734320000001...
0.2122212121212...
0.3333433333333...
0.1999909999999...

Finally, I'm going to write down one more number, using the digits that I just changed. So, in this example, the number I'm writing down starts with
Code: 
0.823240...............
and so on.

Now here's the clever bit:

The "new" number I just wrote down is a number between 0 and 1. It certainly differs from the first number on my original list in the first decimal place. It certainly differs from the second number on my list in the second decimal place. And so on.

In other words, the new number differs in at least one digit from every single number I wrote down on the original list. But the new number is still a number between 0 and 1.

There is a contradiction here, because my original assumption was that I could write down all of the numbers between 0 and 1 on my list. But the above procedure guarantees that no matter how many numbers I include on my list, I can still generate a new one that isn't on the list - even if the original list has a countably infinite number of numbers on it originally.

The inevitable conclusion is that the list of all numbers between 0 and 1 can't be countably infinite. It is, in fact, impossible to "count" (or list) all of the numbers between 0 and 1, even if one were able to count to "infinity".

Conclusion: There is an uncountable infinity of real numbers between 0 and 1 that is demonstrably larger than any countable infinity.

I don't know about you, but this proof blew my mind the first time I came across it.
 
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James R said:
There's a neat argument to show that the set of real numbers is uncountably infinite. i.e. that real numbers cannot be put into a one-to-one correspondence with the natural numbers.

It's an argument by contradiction.

Suppose that the real numbers can be counted (this demonstration will actually show that they can't). If they can be counted, then we can, in principle, make a big list of all of them. Here are a few of the real numbers between 0 and 1, in no particular order, which make up part of the infinite hypothetical list of all numbers between 0 and 1:
Code: 
0.7892345000000...
0.1111111111111...
0.5724320000001...
0.2121212121212...
0.3333333333333...
0.1999999999999...
Now, remember that, hypothetically, I could have posted the entire list of all the real numbers between 0 and 1.

Now I'm going to do something tricksy with the numbers on my list. I'm going to add 1 to the first digit of the first number in the list, and add 1 to the 2nd digit of the second number, and add 1 to the 3rd digit of the third number, and so on. If the digit happens to be a nine, I'll replace it with a zero. After going through this process, my list looks like this:

0.8892345000000...
0.1211111111111...
0.5734320000001...
0.2122212121212...
0.3333433333333...
0.1999909999999...

Finally, I'm going to write down one more number, using the digits that I just changed. So, in this example, the number I'm writing down starts with
Code: 
0.823240...............
and so on.

Now here's the clever bit:

The "new" number I just wrote down is a number between 0 and 1. It certainly differs from the first number on my original list in the first decimal place. It certainly differs from the second number on my list in the second decimal place. And so on.

In other words, the new number differs in at least one digit from every single number I wrote down on the original list. But the new number is still a number between 0 and 1.

There is a contradiction here, because my original assumption was that I could write down all of the numbers between 0 and 1 on my list. But the above procedure guarantees that no matter how many numbers I include on my list, I can still generate a new one that isn't on the list - even if the original list has a countably infinite number of numbers on it originally.

The inevitable conclusion is that the list of all numbers between 0 and 1 can't be countably infinite. It is, in fact, impossible to "count" (or list) all of the numbers between 0 and 1, even if one were able to count to "infinity".

Conclusion: There is an uncountable infinity of real numbers between 0 and 1 that is demonstrably larger than any countable infinity.

I don't know about you, but this proof blew my mind the first time I came across it.
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Not the first time I have seen this but it still blows my mind.
 
Killjoy said:
My main question, however - is how the Devil can there be "different sizes" of infinity?
Click to expand...

Infinity can't be a completed situation, otherwise it is finite (regardless of how ludicrously large the proposed definite quantity is). That's just basic avoidance of contradiction. So infinite entails a constant or intermittent slash contigent process of adding more (a changing enumeration).

But you can replace the idea of indivisible "counting units" with perpetually divisible ones, and thereby acquire abstract, finite-appearing borders that an infinite process of division is occurring between. That abstract "space" you define is finite, but how many times you can partition it is not.
  • In their 1872 papers, though, Cantor and Dedekind had found a way to construct a number line that was complete. No matter how much you zoomed in on any given stretch of it, it remained an unbroken expanse of infinitely many real numbers, continuously linked.

    [...] Infinity Comes in Different Sizes: In fact, Cantor showed, there are more real numbers packed in between zero and one than there are numbers in the entire range of naturals.
And thereby from that you can extract a more concrete slash "picture" scenario of how two different approaches to measurement -- one with indivisible units that extend forever and another that carves those units up limitlessly -- can be construed as distinct processes that vary in magnitude (one conceivable as larger than the other).
_
 
Something that I admittedly have always found hard to get my mind around...INFINITY, ETERNITY
Infinity is the concept of something that is boundless, endless, or without limit, represented by the symbol

. It is not a number, but rather a concept used to describe an endless process or magnitude, such as the infinite sequence of numbers

or the points on a continuous line.
Wikipedia +3 Yeah, I understand the meaning of the definition, but getting my mind around the "concept"?
 
Those who think that the cardinality of a set, infinite or otherwise has something to do with size are doomed to confusion.

For example, take the set of all strictly non-zero positive integers. This is HUGE right? It is a line that goes on forever, right?

Now take the reciprocal of each. These are all contained in the tiny interval between 0 and 1, but of necessity have the cardinality ("size") as the entire set.

Does that make any sense to you?
 
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