Are there an infinite number of infinities?

[donglepuss]

TL;DR Summary

Are there an infinite number of infinities?

How many whole numbers are there?
infinity.
How many tenths of whole numbers are there?
ten times infinity.
How many hundredths of whole numbers are there?
100 times infinity.
How many millionths of whole numbers are there?
1,000,000 times infinity
How many decimal numbers are there?
infinity times infinity

 

[FactChecker]

TL;DR Summary: Are there an infinite number of infinities?

How many whole numbers are there?
infinity.
How many tenths of whole numbers are there?
ten times infinity.
How many hundredths of whole numbers are there?
100 times infinity.
How many millionths of whole numbers are there?
1,000,000 times infinity
How many decimal numbers are there?
infinity times infinity

Good question. Your argument is a good start, but there are different sizes of infinity, called "cardinality". They are indicated by $\aleph_0, \aleph_1, ...$ You can't use regular arithmetic when infinity is involved. You have to use one-to-one (1-1) mappings. If there is a 1-1 mapping between two infinite sets, then they have the same cardinality. Otherwise, they have different cardinality.
The cardinality of the natural numbers is $\aleph_0$.
Consider the natural numbers, 1,2,3,4,... and the even natural numbers, 2,4,6,8,... They have the same cardinality because the mapping $n \rightarrow 2n$ is a 1-1 mapping between the natural numbers and the even numbers.
You can define a 1-1 mapping between the natural numbers and an infinite number (cardinality $\aleph_0$) of infinite sets (cardinality $\aleph_0$).
On the other hand, Cantor has shown that there is no 1-1 mapping between the natural numbers and the real numbers in the interval [0.1]. The real numbers in [0,1] is a set with a larger cardinality. It is a larger type of infinity, $\aleph_1$ (EDIT or larger).

 

[TeethWhitener]

Generally, when people want to discuss sizes of sets, they use bijections. Basically, if you can think of a function from one set to another such that each input has one output and each output corresponds to one input (i.e., the function has an inverse), then the sets are the same “size.” This is called the cardinality of the set. So for instance:
$A=\{1,2,3\}$ and $B=\{4,5,6\}$ have the same cardinality because there is a function $f:A\mapsto B$ that maps each input of $A$ to one, and only one, output of $B$, namely $f(a)=a+3$.

For infinite sets, the same rules apply. Because we have a function from the set of all integers to the set of tenths of integers (namely, $f(a)=\frac{a}{10}$), which has an inverse from the set of tenths of integers to integers (namely $f^{-1}(b)=10b$), we say that the two sets have the same cardinality.

In fact, it should be clear that this argument works for any finite decimal expression. They therefore have the same cardinality as the set of integers. A more complicated argument can show a bijection between the integers and all rational numbers. However, the set of real numbers has a strictly larger cardinality than the set of integers, which was a cornerstone result in modern mathematics first discovered by Georg Cantor.

 

[Mark44]

TL;DR Summary: Are there an infinite number of infinities?

How many whole numbers are there?
infinity.
How many tenths of whole numbers are there?
ten times infinity.
How many hundredths of whole numbers are there?
100 times infinity.
How many millionths of whole numbers are there?
1,000,000 times infinity

It doesn't work this way. The cardinality of each set listed above is $\aleph_0$ (Aleph-null). So each set above is the same "size" as the others.

How many decimal numbers are there?
infinity times infinity

If by this, you mean how many real numbers there are, now you're talking about a different "size" of infinity.

 

[Office_Shredder]

The real numbers in [0,1] is a set with a larger cardinality. It is a larger type of infinity, $\aleph_1$.

Do you realize how contentious the last two characters of your post are :)

https://en.m.wikipedia.org/wiki/Continuum_hypothesis

 

[TeethWhitener]

Do you realize how contentious the last two characters of your post are :)

https://en.m.wikipedia.org/wiki/Continuum_hypothesis

The good news is, whether you agree or disagree with @FactChecker , you’re both right.

 

[FactChecker]

Do you realize how contentious the last two characters of your post are :)

https://en.m.wikipedia.org/wiki/Continuum_hypothesis

Yes. Thanks. I thought that would be too much to mention. But I think I can make it better without getting tangled up in it.

 

[MidgetDwarf]

What is your level of mathematics? There is a free legally to download book that goes through this material in detail. But it may or may not be suitable for you.

 

[jbriggs444]

The various responders here have spoken about Cardinality. This is one way to speak about the "size" of sets. It has the advantage of being quite inclusive in terms of the sets it can operate on. It has the disadvantage that it does not make fine-grained distinctions between sets that intuitively seem like they should have different sizes.

There are other notions which can be used for the "size" of a set, including the size of infinite sets. These may place restrictions on the sorts of sets whose sizes they compare. But, in return, they can make fine grained distinctions which better respect one's intuition or allow for numeric calculations such as integration.

Asymptotic density: If we have a subset of the natural numbers, we can ask about how dense this subset is as we go farther and farther out on the number line. You take the limit of the number of set members less than or equal to $n$ compared to the number of natural numbers less than or equal to $n$ (which is simply $n$) as $n$ increases without bound.

Measure theory: A general framework for schemes that assign a non-negative numeric "measure" to subsets of a given complete set. Measures within this framework include things such as the Lebesgue Measure. The Wiki article for Measure theory lists many other measures. Some quite boring (the Dirac measure - omg). Some not.

 

[FactChecker]

What is your level of mathematics?

One of my pet peeves is that so many people put nothing in their profile to tell us what their background is. (Or maybe they just block it from being viewed.)

 

[bob012345]

Can the basic summary question be answered with a yes or no? Yes or no?

 

[phinds]

Can the basic summary question be answered with a yes or no? Yes or no?

In terms of this thread it really doesn't matter since the question was based on a fallacious assumption about the cardinality of subsets of aleph null.

 

[FactChecker]

Can the basic summary question be answered with a yes or no? Yes or no?

Judging by the misconceptions of the OP, a simple "yes", "no" answer would be seriously misinterpreted.

 

[FactChecker]

The answer to the summary question is that there are an infinite number of distinct infinities: $\aleph_0, \aleph_1. \aleph_2, \aleph_3, ...$. Explaining that might require something more.

 

[bob012345]

Judging by the misconceptions of the OP, a simple "yes", "no" answer would be seriously misinterpreted.

It seems straightforward to ask if it is possible to make an infinite number of sets that each contain an infinite number of members not all exactly the same.

 

[FactChecker]

It seems straightforward to ask if it is possible to make an infinite number of sets that each contain an infinite number of members not all exactly the same.

If you want to answer it your way, go ahead. But how can you be sure that is what the OP is asking. I think that he believes all the infinities he lists are different, but they are all the same cardinality except the last one. The original post is just full of errors that should not be answered with a simple "yes" or "no".

 

[phinds]

I'm not sure the OP even wants an answer. He has opened 11 threads and on 3 of them he has responded once, with none on the other 8. So, he generally asks a question and then just wanders off.

 

[FactChecker]

Here is a set of an infinite union of infinite sized sets:
$\{\{1_1, 2_1, 3_1, ....\}, \{1_2, 2_2, 3_2, ....\}, \{1_3, 2_3, 3_3, ...\}, ...\}$
All elements are distinct because the index is different. It is the same size of infinity (i.e. has the same cardinality), $\aleph_0$, as the set of natural numbers, {1,2,3,...}.

A 1-1 mapping from the natural numbers to the set (i.e. counting the set) is
$(1, 1_1), (2, 2_1), (3, 1_2), (4, 3_1), (5, 2_2), (6, 1_3), (7, 4_1). (8, 3_2), (9, 2_3), (10, 1_4), ....$

 

[bob012345]

If you want to answer it your way, go ahead. But how can you be sure that is what the OP is asking. I think that he believes all the infinities he lists are different, but they are all the same cardinality except the last one. The original post is just full of errors that should not be answered with a simple "yes" or "no".

Your answer in post #14 satisfies my curiosity which I did not see until after I posted #15. Thanks.

 

[Drakkith]

Here's a favorite video of mine from Vsauce that might help the OP understand a bit better what mathematicians mean when they talk about infinity:

 

[Infrared]

One easy way to see that there are infinitely many different infinite cardinalities is to show: For any set $X$, its powerset $2^X$ has larger cardinality.

 

[Drakkith]

How many tenths of whole numbers are there?
ten times infinity.
How many hundredths of whole numbers are there?
100 times infinity.
How many millionths of whole numbers are there?
1,000,000 times infinity
How many decimal numbers are there?
infinity times infinity

*Note that the following is my understanding and I am not a mathematician. My explanation and terminology is probably not exactly correct, but I'm hoping to get across the general idea.

Remember that you can't multiply anything by infinity, as infinity is not a number.
What you're referring to with all these examples is something more like a 'density'.
That is, given some finite span of real numbers (basically some continuous section of a number line), you'll find fewer odd numbers than whole numbers, fewer multiples of 10 than odd numbers, more tenths of whole numbers than whole numbers, more hundredths of whole numbers than whole numbers, etc. In other words, the amount of whole numbers per unit of number line is, say, X. The number of odd or even numbers per unit of number line is about 1/2X. The number of tenths per unit of number line is 10X.

However, when working with infinite amounts, the actual amount of all of these is the same. A somewhat counterintuitive notion, I admit, but when dealing with infinities we must remember that we never run out of numbers. Yes, there are 10 whole numbers for every multiple of ten, but I can count every multiple of ten with a whole number and never run out of whole numbers or multiples. So it doesn't really make sense to say that there are MORE whole numbers than multiples of ten, as I can pick any whole number, no matter how big, no matter how large, and I can ALSO find that many multiples of ten.

In math terminology, we say that the set of multiples of ten can be put into a one-to-one correspondence with the natural numbers. Which basically just means I can say 10 is the 1st multiple, 20 is the 2nd multiple, 30 is the 3rd multiple, etc.

This is in contrast to the full set of real numbers, which CANNOT be put into a one-to-one correspondence with the natural numbers. If I say 1.1 corresponds to 1, 1.01 corresponds to 2, 1.001 corresponds to 3, and so forth, then I'll use every single natural number just making a set of numbers counting where the 1 is placed after the decimal place. There aren't any left over to label 1.2 for example. Or 2.1. Or 123.554. Or any other number. There are literally more real numbers than there are natural numbers, fractions, multiples of any whole number, tenths of whole numbers, or any other similar set that I could put in a one-to-one correspondence with the naturals.

We have special terms in math for this phenomenon. We say that anything that can be put in a one-to-one correspondence with the natural numbers (labeling things with 1, 2, 3, etc) is countable. You can count them if you had an infinite amount of time. You can order them. There's a way to clearly distinguish which ones come first, or which ones come next in the list, or how many are in some finite span. Two is the first even number between 1 and 9. Four is the next even number. Between 1 and 11 there are five even numbers.

But anything that can't be put into a one-to-one correspondence with the natural numbers, such as the real numbers, is uncountable. You can't count them. You can't order them. You can't say which number comes first, or which number comes next after 1.1, or which one comes after 12.55. The number 3.5 isn't the 2nd number after 3. Or the 14th. Or the millionth. Or any other number. There aren't four reals between 1 and 2, or eight, or a googolplex. There are uncountably many. Not just an infinite amount, but an uncountably infinite amount.

 

[jbriggs444]

But anything that can't be put into a one-to-one correspondence with the natural numbers, such as the real numbers, is uncountable. You can't count them. You can't order them.

It is a bit strong to say that you cannot order an uncountable set.

The real numbers have a standard ordering which is a "total order". That is to say that given any pair of unequal elements, you can say which one is greater, which is less and whether they are equal. This order relation (by definition of "total order") is transitive, reflexive and antisymmetric. Further, in the case of the real numbers, this order respects the mathematical operations of addition, subtraction, multiplication and division. Which is to say that the real numbers, like the rational numbers, are an "ordered field".

A better statement would have been that the standard order on the real numbers is not a sequential order. That is, given any element, there is neither a next higher real number nor a next lower real number. You seem to clarify that this is actually your intended meaning.

Alternately, perhaps you were after a constructible well ordering.

Given the Axiom of Choice, one can prove the Well Ordering Theorem. This theorem states that for every set, there is an ordering for that set that is not just a total order, but also a Well Order. A well order is an ordering such that every non-empty subset has a least element. A well ordered set has a useful property: For every element in the set (except the last, if any) there is a uniquely defined next element. The next element is the least element of the non-empty subset of elements greater than it.

A well order has the uncomfortable property that an element may not necessarily have a defined next smaller element. Every element of an uncountable well ordered set is a predecessor to exactly one other element. But many elements may not be the successors of any other element. So that leaves one's intuition somewhat unsatisfied. There is something a bit asymmetric about the notion.

A well order (which must exist, given the axiom of choice) for an uncountable field will not be the standard ordering. It will not respect the operations of addition, subtraction, multiplication and division. So you might be reluctant to accept it. But it does count as a total order. So, given the axiom of choice, you can order every uncountable set.

 

[Drakkith]

It is a bit strong to say that you cannot order an uncountable set.

Yeah, I felt like I was on some shaky ground with that one, though not for the reasons you brought up in your post.

A better statement would have been that the standard order on the real numbers is not a sequential order. That is, given any element, there is neither a next higher real number nor a next lower real number. You seem to clarify that this is actually your intended meaning.

Indeed. I figured I'd get into some trouble with the nitty gritty details. I've never even heard of 'total order' before. Thanks for clearing this up.