Different sizes of infinity...?

QuarkHead said:
Those who think that the cardinality of a set, infinite or otherwise has something to do with size are doomed to confusion.
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Not really. It's not intuitive, sure, but that doesn't mean that one is doomed to confusion. One merely needs to have a suitable and sufficient understanding, and the confusion vanishes in a puff of realisation. And, fwiw, cardinality of a set does have "something to do with size". Heck, the cardinality is literally defined as the number of elements in the set. There's another word for that: size.
So the cardinality of the finite set of integers between 1 and 100 (inclusive) is 100. I.e. the set contains 100 distinct elements. This is the size.
For infinite sets, the number of elements is, by definition, infinite, so there needs to be some other means of defining size beyond there being an infinite elements. So the cardinality (size) of inifinte sets (e.g. aleph-null, aleph-one etc) extends the ordinary notion of counting by defining size via one-to-one correspondences.
QuarkHead said:
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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Ah, so in essence you are confused by it, so you assume that cardinality can have nothing to do with size of the set?
Well, thank goodness there are people who do understand it. ;)

The infinite set of non-zero positive integers goes from 1, 2, 3, 4, 5... all the way forever.
The infinite set of reciprocals goes from 1/1, 1/2, 1/3, 1/4, 1/5... all the way forever.
As you can hopefully see, both sets are countable, and there is a fairly obvious one-to-one matching between the sets. Therefore they have the same number of elements in the two sets: aleph-null.
No confusion necessary. ;)
 
Fun Fact:

There are infinitely many different cardinalities, and in fact they form an unending, strictly increasing hierarchy with no maximum. The moment you think you’ve reached “the biggest infinity,” set theory shows you how to build a strictly larger one.

When I studied this in college, we learned that we don't know the cardinality of the set of all infinities. But since then, I have read this doesn't form a proper set, so no cardinality exists.
 
The set of positive integers is infinitely large. And the set of even positive integers is also infinitely large. Even though the former is twice the size of the latter.

Infinity is a concept - not an amount.
 
gmilam said:
The set of positive integers is infinitely large. And the set of even positive integers is also infinitely large. Even though the former is twice the size of the latter.
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No, they're the same size.
 
gmilam said:
Well, both are countable, and there is a fairly obvious two-to-one matching of the sets...
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Both are countable, and there's a clear 1-to-1 matching. Mathematically speaking, they are the same size. Get an infinitely long length of rope. Get another infinitely long rope. Which is longer? Why does it matter if, at regular intervals, one has 1,2,3,4... and one has 2,4,6,8...? They're both the same size.
 
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gmilam said:
The set of positive integers is infinitely large. And the set of even positive integers is also infinitely large. Even though the former is twice the size of the latter.

Infinity is a concept - not an amount.
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Aleph-nought is a concept.

So I guess Infinity is an infinite class of concepts.
 
Sarkus said:
Both are countable, and there's a clear 1-to-1 matching. Mathematically speaking, they are the same size. Get an infinitely long length of rope. Get another infinitely long rope. Which is longer? Why does it matter if, at regular intervals, one has 1,2,3,4... and one has 2,4,6,8...? They're both the same size.
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A natural number line could be regarded as consisting of indivisible units. In contrast, a real number line is infinitely dense -- the "space" between any two of its units can be endlessly partitioned further.

So ultimately (brushing away the obscurantist layers of legalese), it seems to boil down to a claim that a "container" of infinite indivisible units (process of eternally adding more) is smaller than a container of divisible units (process of eternally adding more PLUS process of eternally dividing what's in there). Ergo, the impression that the latter has greater magnitude.

Though technically, I likewise feel that if we could ignore all the distinct identity (counting) labels being assigned to each unit in both kinds of number lines or both kinds of containers, and ignore the two ways that they are increasing in quantity, then all infinities are still going to match up as non-ceasing and perpetually uncompleted processes. But that's going against the establishment that eventually came around to believing slash accepting Cantor's abstract view, so politically we're expected to publicly humor or accommodate that mainstream convention (regardless of personal construals and privately held mutinies).
_
 
C C said:
A natural number line could be regarded as consisting of indivisible units. In contrast, a real number line is infinitely dense -- the "space" between any two of its units can be endlessly partitioned further.
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Yep, the set of integers, being countably infinite, has cardinality of aleph-null, while the set of reals is uncountably infinite, and is equal to 2 to the power of aleph-null. So is larger. Much larger.

With infinite sets there is also, as mentioned previously, the concept of density, which you allude to. Which can, in some situations, give one a more intuitive feel for relative "size", even if it is distinct from the number of elements in the infinite set. But it starts to get complicated, with different ways to look at / measure / interpret density.
 
Damn Google and their ****ing algorithms... this video from Veritasium just popped into my feed, presumably as a result of me looking some things up on Google...
All about Cantor, Zermelo, et al, and, among other things, sizes of infinities. Whodathunkit. ;)
It's a pleasant enough watch, in Veritasium's usual style. Enjoy. Or don't. But it's about Cantor and his work on infinities, so relevant here. :)
 
Sarkus said:
Damn Google and their ****ing algorithms... this video from Veritasium just popped into my feed, presumably as a result of me looking some things up on Google...
All about Cantor, Zermelo, et al, and, among other things, sizes of infinities. Whodathunkit. ;)
It's a pleasant enough watch, in Veritasium's usual style. Enjoy. Or don't. But it's about Cantor and his work on infinities, so relevant here. :)
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The Cantor story is very interesting.
 
One of my favorite subjects.

Hope there was no permanent damage from the mind blowing experience.
infinite.png

Cantor began with an 'infinite' random list of horizontal sequences formed from {a, b}. He then assigned a coordinate grid to each symbol shown as row and column.
He selects the diagonal symbol from each sequence to form sequence D.
He then interchanges the symbols to form E.
He then claims there is no place for E in the list since it will conflict with D, thus the list is incomplete.
The two rows D and E show both coexisting in the horizontal orientation, contradicting his claim.

The problem is in the familiar illustration above.
When D and E are both in the horizontal orientation, the row coordinates are different.
If D is in the diagonal orientation, the coordinates for D and E are the same.
He is expecting two sequences with different orientations to occupy the same coordinate space while having different symbols.

For the example in #12
inf sequence.gif
View the set of real numbers as sequences of integers, partitioned into 10 subsets from R0 to R9. The decimal point is ignored.
The modified diagonal sequence must be in the subset R8.
Each branch point contains the set R, shown with the closest two values for clarity.

The problem, the diagonal sequence is not compatible with the horizontal sequences.
At each intersection the integer can't have 2 different values simultaneously.
Cantor's diagonal argument was contrived to get the results he wanted.
 
I think you are missing the point.
You start with an infinite list of unique infinite sequences.
Then, taking the diagonal (D) and flipping each element simply means you guarantee that the new sequence you are making (E) is different to every other sequence in the list.
E is different to row 1 because the first element is different.
E is different to row 2 because the second element is different.
E is different to row 3 because the third element is different.
Forever.
D isn't important, it's just part of the process to generate E.
The diagonals and coordinates are just there to guide the construction of E.

i.e. the initial list was infinite, and we might map it against the natural numbers (so, count the list), row 1, row 2, ... but we came up with a valid possible member (E) of the set which won't be in that list. The set isn't countable.
 
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pzkpfw said:
I think you are missing the point.
You start with an infinite list of unique infinite sequences.
Then, taking the diagonal (D) and flipping each element simply means you guarantee that the new sequence you are making (E) is different to every other sequence in the list.
E is different to row 1 because the first element is different.
E is different to row 2 because the second element is different.
E is different to row 3 because the third element is different.
Forever.
D isn't important, it's just part of the process to generate E.
The diagonals and coordinates are just there to guide the construction of E.

i.e. the initial list was infinite, and we might map it against the natural numbers (so, count the list), row 1, row 2, ... but we came up with a valid possible member (E) of the set which won't be in that list. The set isn't countable.
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That is the question, ‘what is the point’?
For D the 6th symbol is ‘b’ with coordinates (6,6), with diagonal orientation.
For E the 6th symbol is ‘a’ with coordinates (6,6), with horizontal orientation.
Both symbols a and b cannot occupy the same coordinates simultaneously.

Cantor interpreted this as no location for E, when it is mixing of incompatible orientations.
 
Infinite literally means without a boundary, and is not a number.
The universe reveals itself as a collection of finite discrete objects.
Boundaries allow measurement of discrete objects, thus an infinite object can't be measured.
Imagine a stick of infinite length.
A beam of light from the near end could reflect from a mirror at the far end.
Can't measure the stick since it has no far end!
That would include the most basic measurement, counting.

#3
Why doesn't the manager of Hilbert's hotel just put the guest in room (n+1) and not bother the other guests?

#18
" It is not a number, but rather a concept used to describe an endless process or magnitude, such as the infinite sequence of numbers"
Sounds like
"What we call infinite is only the endless possibility of creating new objects no matter how many exist already" (Poincaré quoted from Kline 1982).

Any Peano type process can be applied indefinitely to create larger integers, but not a largest integer.
Two different integers can be compared to determine the larger. Comparing one integer to all integers to determine the largest is impossible.
Sounds like
"Where the nonsense starts is with our habit of thinking of a large number as closer to infinity than a small one".(Ludwig Wittgenstein).

How can a continuous line be divisible?

To have a list with no end and be complete is a contradiction of terms.

How do you approach infinity? I have not seen any method so far.
 
phyti said:
That is the question, ‘what is the point’?
For D the 6th symbol is ‘b’ with coordinates (6,6), with diagonal orientation.
For E the 6th symbol is ‘a’ with coordinates (6,6), with horizontal orientation.
Both symbols a and b cannot occupy the same coordinates simultaneously.

Cantor interpreted this as no location for E, when it is mixing of incompatible orientations.
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You are confusing the method of construction with what it's showing.
Nobody is trying to put two thing in any coordinate, (6,6) or other.
The orientation has nothing to do with it.

Try without cells and diagonals:

We're making a set of strings of letters a and b.
Say row 1 is "aaaaaaaaa..."
You can make a string (E) that's different by flipping the first letter: "baaaaaaaa..."
But there are more rows.
Say row 2 is "baaaaaaaa..."
OK, we change (E), flipping the second letter: "bbaaaaaaa..."
We've now guaranteed that (E) is different to both row 1 and row 2.

Ad infinitum; there are infinite sequences of letters in the list, yet we can make sure that (E) isn't in the list.
 
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