How does Cantor's concept of infinity differ from the extended real numbers?

[Plaster]

Is set theory, counting off members, really indicative of any order of difference when neither set has a 'size' ? If my hotel has infinite rooms, I could accommodate both the real people and the integer people together!

It seems to me, Cantors version of infinity inherantly assumes an end at some point. The fact that one infinite set includes infinite members (even an infinity of infinite members), does not make that set more infinite. It's a mixing of absolute and relative that is surely wrong ?

 

[Eynstone]

You can't accommodate reals in countably many rooms - check out Cantor's diagonalisation process.

It seems to me, Cantors version of infinity inherantly assumes an end at some point. The fact that one infinite set includes infinite members (even an infinity of infinite members), does not make that set more infinite. It's a mixing of absolute and relative that is surely wrong ?

It's not - two sets are equal iff there is a bijection between them.

 

[icantadd]

What is an infinite member of a set?

 

[icantadd]

Typically an element is either in a set or not. Are you talking about a "fuzzy" membership function?

 

[Hurkyl]

Is set theory, counting off members, really indicative of any order of difference when neither set has a 'size' ?

What do you mean by counting?


The notion of cardinality is very useful, so we use it. Furthermore, cardinality is a linear ordering, and on finite sets coincides with the notion of size as a natural number.

 

[Plaster]

You can't accommodate reals in countably many rooms - check out Cantor's diagonalisation process.

You can't accommodate integers in countable rooms, unless you ignore the fact that counting is a process that tends towards an end.
[/QUOTE]

 

[Plaster]

What is an infinite member of a set?

For the sake of counting, let's say the set of real numbers is composed of individual sets that contain all real numbers between one integer and the next. Each of these member sets is countable against the set of integers, even though each member in one set is itself infinite.

 

[Plaster]

The notion of cardinality is very useful, so we use it. Furthermore, cardinality is a linear ordering, and on finite sets coincides with the notion of size as a natural number.

I do understand the usefulness of cardinality in some linear circumstances. If someone would agree that cardinality should be defined as "order of relative growth", I would accept different infinities.

It's this insistance that this equates in some way to "size" that I can't accept, despite the fact size is nearly always the way people subconsciously interpret cardinality.

 

[Hurkyl]

I do understand the usefulness of cardinality in some linear circumstances. If someone would agree that cardinality should be defined as "order of relative growth", I would accept different infinities.

What the heck is "order of relative growth"?

Cardinality has a very precise and unambiguous definition in terms of bijections and such.

If you have an intuitive notion of "size", and that notion doesn't agree with what cardinality is, then when talking about cardinality, you should completely ignore your intuitive notion of size.

(also, you should recognize that your intuitive notion of "size" and my intuitive notion of "size" need not be the same)


Now, I posit that if your notion of "size of a set" is anything other than cardinality, then your notion is wrong^*. Well, not necessarily wrong, but misplaced -- your notion of size probably includes a lot of extra structure you've built up above and beyond the core notion of set.

For example, someone might think that there are more integers than there are even integers. Why? Because the evens are contained in the integers. This is a perfectly reasonable fact to observe, but it is not a fact about sets! Instead, it is a fact about two sets and a particular relationship between them.

Another explanation for the same thought is because the proportion of integers between -N and N is 1, but the proportion of even integers between -N and N converges to 1/2 as N goes to $+\infty$^**. But again, this isn't a fact about sets -- it's a fact about the asymptotic behavior of a particular function related to a particular embedding of our sets in the integers.



Incidentally, IMO it's perfectly reasonable to intuit cardinality as measuring a kind of complexity.



*: In a certain technical sense, cardinality is the only property a set can have.

**: The extended real number $+\infty$ has nothing to do with sets and cardinality

 

[Mark44]

You can't accommodate integers in countable rooms, unless you ignore the fact that counting is a process that tends towards an end.[

I don't think you understand how "countably many" is defined in mathematics. A set is countably infinite if it can be put into a one-to-one relationship with the positive integers.

 

[Plaster]

It think my understanding of size is simplistic - I'll try and substitute complexity for cardinality as hurkyl suggests,and see how I go.

Thanks all

 

[lavinia]

It seems to me, Cantors version of infinity inherantly assumes an end at some point. The fact that one infinite set includes infinite members (even an infinity of infinite members), does not make that set more infinite. It's a mixing of absolute and relative that is surely wrong ?

I think the idea of end is implicit in the idea of infinity. To me it generalizes the idea of a limit point of a Cauchy sequence. for instance the number 1 is the limit of the infinite sequence 1/2 3/4 7/8 ...and geometrically appears at the "end" of the sequence. From this point of view 1 is defined as the end point of an infinite succession of fractions.

I think of Aleph zero in the same way, as the limit point of the infinite succession of integers. In this case though, there is no metric to make the integers into a Cauchy sequence so just think of it as denoting the entire process of counting. As such it can be thought of as coming after all of the integers.

 

[Hurkyl]

I think the idea of end is implicit in the idea of infinity. To me it generalizes the idea of a limit point of a Cauchy sequence. for instance the number 1 is the limit of the infinite sequence 1/2 3/4 7/8 ...and geometrically appears at the "end" of the sequence. From this point of view 1 is defined as the end point of an infinite succession of fractions.

You're thinking of the element $+\infty$ of the extended real numbers, not about cardinalities.

The resemblance with $\aleph_0$ is only superficial -- while it is true that the two ordered sets

$$\mathbb{N} + \{ +\infty \} = \{0, 1, 2, 3, \cdots ; +\infty \}$$​

and

$$\mathbb{N} + \{ \aleph_0 \} = \{0, 1, 2, 3, \cdots ; \aleph_0 \}$$​

are order-isomorphic, (+ meaning concatenation, ; meaning the thing to the right doesn't have anything to do with the dots on the left), that's pretty much the only thing that $+\infty$ and $\aleph_0$ share.

The extended real numbers $\pm \infty$ are like ends because they really are meant to be -- e.g. they are the endpoints of the closed, compact interval $[-\infty, +\infty]$. You might find it interesting to read about projective geometry too.

The infinite cardinal numbers, on the other hand, do not measure geometric information. There are "limit cardinals", such as $\aleph_\omega$ which is the limit of the sequence $\aleph_0, \aleph_1, \aleph_2, \cdots$ where the indices range over the natural numbers, but then there is always $\aleph_{\omega + 1}$. These limits don't represent geometric information anyways. (unless you adopt a rather loose meaning of 'geometric')

 

[Hurkyl]

For the OP: what do you think of ordinal numbers? In the same analogy where cardinal numbers measure size, the ordinal numbers are what you count with.

Rather than try and describe what ordinal numbers are and why they resemble counting, I will just demonstrate transfinite counting.


Way 1: I will count all of the nonnegative integers first, then the negative ones.

At step 0, I have counted nothing yet
At step 1, I have counted {0}
At step 2, I have counted {0,1}
At step 3, I have counted {0,1,2}
...
At step N, I have counted everything in {0,1,2,...,N-1}
...
----------------------
At step $\omega$, I have counted every natural number.^*
At step $\omega + 1$, I have counted every natural number, and {-1}
At step $\omega + 2$, I have counted every natural number, and {-1,-2}
At step $\omega + 3$, I have counted every natural number, and {-1,-2,-3}
...
At step $\omega + N$, I have counted every natural number, and {-1,-2,-3,...,-N}
...
---------------------
At step $\omega + \omega = \omega 2$, I have counted every integer.

($\omega 2$ and $2 \omega$ are different ordinal numbers -- ordinal arithmetic is trickier with infinite numbers than it is with finite numbers)

Anyways, this method of "iterating" through the integers has length $\omega 2$. Since $|\omega 2| = 2 \aleph_0 = \aleph_0$, I know there are countably many integers.


Way 2: I will alternate positive / negative

At step 0, I have counted nothing yet
At step 1, I have counted {0}
At step 2, I have counted {0,1}
At step 3, I have counted {0,1,-1}
...
At step 2N, I have counted {0,1,...,N} and {-1,-2,...,-(N-1)}
At step 2N+1, I have counted {0,1,...,N} and {-1,-2,...,-N}
...
------------
At step $\omega$, I have counted every integer


This method of iterating through the integers has length $\omega$. And since $|\omega| = \aleph_0$, we have another proof the integers are countable!


If we intuit that we can count with ordinal numbers, then cardinality is justifiably used for size -- as you see above, if we "count" the same set in two different ways, we can get two different ordinal numbers, but those ordinal numbers have the same cardinality. Conversely, if two ordinal numbers have the same cardinality, then if we can count a set using one ordinal number, then we can always find another way to count it that gives us the other ordinal number.

Put differently, after counting, the ordinal number is what's left if we forget the individual objects and just remember the sequence of steps in the counting process. A cardinal number is what's left after we forget how the steps are ordered.



*: $\omega$ is a limit ordinal. Nothing "new" is counted here -- every particular number I've counted was counted on a previous step. As is typical for transfinite iteration, I've used the limit ordinal to simply collect the results of all previous steps.

 

[lavinia]

You're thinking of the element $+\infty$ of the extended real numbers, not about cardinalities.

The resemblance with $\aleph_0$ is only superficial -- while it is true that the two ordered sets

$$\mathbb{N} + \{ +\infty \} = \{0, 1, 2, 3, \cdots ; +\infty \}$$​

and

$$\mathbb{N} + \{ \aleph_0 \} = \{0, 1, 2, 3, \cdots ; \aleph_0 \}$$​

are order-isomorphic, (+ meaning concatenation, ; meaning the thing to the right doesn't have anything to do with the dots on the left), that's pretty much the only thing that $+\infty$ and $\aleph_0$ share.

The extended real numbers $\pm \infty$ are like ends because they really are meant to be -- e.g. they are the endpoints of the closed, compact interval $[-\infty, +\infty]$. You might find it interesting to read about projective geometry too.

The infinite cardinal numbers, on the other hand, do not measure geometric information. There are "limit cardinals", such as $\aleph_\omega$ which is the limit of the sequence $\aleph_0, \aleph_1, \aleph_2, \cdots$ where the indices range over the natural numbers, but then there is always $\aleph_{\omega + 1}$. These limits don't represent geometric information anyways. (unless you adopt a rather loose meaning of 'geometric')

I am not seeing your point. It seems that for the positive integers, seen as a well ordered set generated by adding 1, infinitity can be thought of as an end point in the ordering defined by the property that each positive integer precedes it. But a Cauchy sequence is mapped onto the integers and its limit point maps naturally - i.e. in an order preserving way - onto infinity and this defines a bijection between aleph null and the Cauchy sequence - I think.