Different Sets of Infinity

[Hegel]

Are there differnt sets of infinity? Is the infinate number of integers smaller that the infinate number of real numbers? After all you have an infinate set of numbers in between the integers for real numbers?

[Sundog]

Yes, there are. Congratulations if you came up with that by yourself.

A real mathematician will jump in here any second, but I do know that the "infinity of the integers" is known as "aleph-null". "aleph-prime" is, I believe, the infinity of the real numbers. And after that we'll leave it to the experts. ;)

[Soapy Sam]

While awaiting the arrival of the onshift mathematician, you mught google
Hilbert or Cantor

[Ziggurat]

Originally posted by Hegel
Are there differnt sets of infinity? Is the infinate number of integers smaller that the infinate number of real numbers? After all you have an infinate set of numbers in between the integers for real numbers?

As mentioned above, yes, there are different sorts of infinity. The real numbers are a "smaller" infinity than the real numbers, but maybe not for quite the reason you think. There are an infinite number of rational numbers between any two integers, and yet the infinity of rational numbers is exactly the same as the infinity of real numbers. Strange, no? The reason these are the same infinity is you can do a one-to-one mapping from one set to the other and back again. Ponder how they do that for a little while if you want a challenge - though the answer isn't hard to understand if you're given it. Perhaps even stranger is that you can map the 1-dimensional real number set R^1 onto the 2D plane of real numbers R^2 - one dimensional vs two dimensional (or even higher) doesn't make a difference to the order of infinities. Similarly, you can make a 1-to-1 map of the real numbers between 0 and 1 to the entire set of real numbers, and vice versa. There are also infinities larger than the set of real numbers - any takers on how to describe the next set up?

[Sundog]

Originally posted by Ziggurat

There are also infinities larger than the set of real numbers - any takers on how to describe the next set up?

Hmm. Transcendental numbers? Irrationals? I used to know...

[Ziggurat]

Originally posted by Sundog

Hmm. Transcendental numbers? Irrationals? I used to know...

Irrationals form the bulk of the real numbers, those two infinities are exactly the same.

Hint: you need to go beyond just numbers (and I don't mean letters).

[Stimpson J. Cat]

Ziggurat,

There are also infinities larger than the set of real numbers - any takers on how to describe the next set up?

The superset of reals.

The superset of a set A is the set whose elements are all of the subsets of A.

For example, the superset of {1, 2, 3} would be

{ Null, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} }

where Null is the empty set.

For a finite set with n elements, the superset has 2^n elements.

For an infinite set, the superset has the next higher cardinality. The superset of the integers has the same cardinality as the reals. The superset of the reals has the next higher cardinality, and so on.

As a side note, when I say next higher cardinality, I mean the next higher cardinality whose existence can be derived from the axioms of arithmetic. It turns out that having other cardinalities between them is neither inconsistent with, not implied by, the axioms of arithmetic. This is an example of Godel's incompleteness theorem.


Dr. Stupid

[slimshady2357]

Originally posted by Stimpson J. Cat


As a side note, when I say next higher cardinality, I mean the next higher cardinality whose existence can be derived from the axioms of arithmetic. It turns out that having other cardinalities between them is neither inconsistent with, not implied by, the axioms of arithmetic. This is an example of Godel's incompleteness theorem.


Dr. Stupid

The Continuum Hypothesis (that the reals were the next higher cardinality) was one of Hilbert's 20 unanswered questions originally, wasn't it?

Godel and another fellow, whose name escapes me, showed that The Continuum Hypothesis was not inconsistent with ZF set theory, but neither was it's negation.

As Stimpson said, it's an example of a statement that cannot be proven or disproven from within the system (in this case ZF set theory).

What I always find interesting exaplaining to people is not only that the Reals are a higher infinity than the integers, but that the natural numbers are the same level of infinity as the integers!

So there are the same 'number of members' in the set {1, 2 ,3,...} as there are in the set {...-3, -2 ,-1, 0, 1, 2, 3,...} which can seem counter intuitive.

Adam

[Cabbage]

For an infinite set, the superset has the next higher cardinality. The superset of the integers has the same cardinality as the reals. The superset of the reals has the next higher cardinality, and so on.

As a side note, when I say next higher cardinality, I mean the next higher cardinality whose existence can be derived from the axioms of arithmetic. It turns out that having other cardinalities between them is neither inconsistent with, not implied by, the axioms of arithmetic. This is an example of Godel's incompleteness theorem.
There are frequent misconceptions about the continuum hypothesis, and this doesn't seem entirely accurate to me.

It can be proven that the cardinals are well ordered--that there's always a next higher cardinal. The first transfinite cardinals go:

aleph-0, aleph-1, aleph-2,...

and it is known there are no cardinals between them.

What's not known is just what the cardinality of the reals is. It's consistent for it to be aleph-1--that's the continuum hypothesis. It's also consistent for it to be (just about) any other possible cardinality--the reals could have cardinality aleph-2, aleph-3, aleph-googolplex, or maybe even have infinitely many transfinite cardinals before it, such as aleph-(omega-1).

The next higher cardinal is always there (provably so). We know that the power set (what you were calling superset) of a set always has a larger cardinality, but we don't know that it's the next highest (that would be the generalized continuum hypothesis)--it could be much bigger.

[Martin]

Originally posted by Cabbage

There are frequent misconceptions about the continuum hypothesis, and this doesn't seem entirely accurate to meI don't think you and Stimpy actually disagree here. Seems to me that you're using the labels aleph-n to describe all possible cardinalities, and he's reserving them for those which can be shown to exist.

[Ziggurat]

For the not quite so mathematically inclined, one way of thinking of the superset of reals is the set of all functions on a line, including functions with infinitely many discontinuities.

[Cabbage]

I don't think you and Stimpy actually disagree here.
It might be a subtle distinction, but there is a distinction. Stimpy said:For an infinite set, the superset has the next higher cardinality. (...) As a side note, when I say next higher cardinality, I mean the next higher cardinality whose existence can be derived from the axioms of arithmetic.
Now, it's a fact that aleph-one is the cardinality next higher than aleph-naught, this comes from the (ZFC) axioms of set theory. An example of a set of cardinality aleph-one would be the set of all non-isomorphic ways to well order the natural numbers.

It's not a fact that the power set has the next higher cardinality. The natural numbers have cardinality aleph-naught. It's certainly consistent for the power set of the natural numbers to have cardinality aleph-one (the next highest cardinal), but it's also consistent for that cardinality to instead be aleph-2 or aleph-5478594 or aleph-(omega-1), as I mentioned above.

In other words, Stimpy's quote gives the impression that the phrase "next highest cardinal" is open to question--maybe there are more cardinals in-between, and it wasn't the "next highest cardinal" after all?

However, that's not the case--there is a next highest cardinal, with no cardinals in-between--that's beyond dispute. What's in dispute is whether the power set actually gives you that next highest cardinal--in fact, it might skip the next highest cardinal and give you something much larger.

[Stimpson J. Cat]

Cabbage is, or course, correct. My statement

For an infinite set, the superset has the next higher cardinality. (...) As a side note, when I say next higher cardinality, I mean the next higher cardinality whose existence can be derived from the axioms of arithmetic.

is poorly phrased, and not technically correct. What I should have said is that by "the next higher cardinality" I mean the next higher cardinality that corresponds to a set which we can actually specifically define within the context of the axioms of arithmetic.

If we have a set of cardinality N, then we can construct a set with higher cardinality 2^N, by taking the power-set. We can never actually construct a set with cardinality between N and 2^N, though, because to do so would be to prove by construction that the continuum hypothesis is false.

Cabbage is also correct that I got the terms super-set and power-set mixed up. Good thing I provided the definition, huh? :p


Dr. Stupid

[Kullervo]

Originally posted by slimshady2357


The Continuum Hypothesis (that the reals were the next higher cardinality) was one of Hilbert's 20 unanswered questions originally, wasn't it?

Godel and another fellow, whose name escapes me, showed that The Continuum Hypothesis was not inconsistent with ZF set theory, but neither was it's negation.

Adam Paul J. Cohen I believe.

I have another question. From Mathnet I find that the number of algebraic numbers is countably infinite (http://mathworld.wolfram.com/CountablyInfinite.html) but that the number of irrationals is not - sqrt(2) being algebraic as I recall, for instance. Therefore the density of the continuum is due to the presence of the transcendentals. Yet, it seems to be really difficult to prove that a number is transcendental. Is there a reason for this?

[T'ai Chi]

Here is a link to the Mathworld entry on infinity: http://mathworld.wolfram.com/Infinity.html just for further information.