- 10 Oct '11 21:00I wanted to pry this concept loose from some other discussions that seem to be going on. Really, I just want to hear from those who have some maths background that can apply it to philosophical questions (there are a number of you here!).

I read a book called*Aleph*some years ago, but don’t know if I still have it (will make a search of my bookshelves). I have forgotten pretty much all the maths I once knew (which just went through differential calculus)—I mean that for real: I have a great talent for forgetting things that I have not used in some time.

With that, I’d like to open this as a kind of free-form thread on the discussion of the meaning of “infinity”…

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Note: I realize that some of this might belong on the Science Forum, but the Spirituality Forum also became the de facto philosophy forum—from the beginning, whether some religionists like it or not—and so I’d like to keep it here if we can. - 10 Oct '11 21:06

This might spur some thinking:*Originally posted by vistesd***I wanted to pry this concept loose from some other discussions that seem to be going on. Really, I just want to hear from those who have some maths background that can apply it to philosophical questions (there are a number of you here!).**

I read a book called*Aleph*some years ago, but don’t know if I still have it (will make a search of my booksh ...[text shortened]... e beginning, whether some religionists like it or not—and so I’d like to keep it here if we can.

http://en.wikipedia.org/wiki/Infinity_%28philosophy%29 - 10 Oct '11 21:33

However it can be true depending on how the perimeters are set up, which allows*Originally posted by Agerg***In light of a recent argument I have had I put it here for posterity that, contrary to popular opinion, it isn't necessarily true that (hypothetically speaking) given infinite time, all things that can be potentially be done will be done.**

for an argument to be had on the subject ;-)

Infinities are usually used in philosophical/theological arguments (in my experience)

in an attempt to confuse everyone and then claim victory while everyone is too

confused to disagree. - 10 Oct '11 21:34

Please, lay that out for me. Even if you have before (or cut & paste). That certainly has philosophical implications...*Originally posted by Agerg***In light of a recent argument I have had I put it here for posterity that, contrary to popular opinion, it isn't necessarily true that (hypothetically speaking) given infinite time, all things that can be potentially be done will be done.** - 10 Oct '11 21:36
*Originally posted by googlefudge***However it can be true depending on how the perimeters are set up, which allows**

for an argument to be had on the subject ;-)

Infinities are usually used in philosophical/theological arguments (in my experience)

in an attempt to confuse everyone and then claim victory while everyone is too

confused to disagree.**Infinities are usually used in philosophical/theological arguments (in my experience) in an attempt to confuse everyone and then claim victory while everyone is too confused to disagree.**

Hence my interest. There ought to be a level of understanding where even the layperson need not be confused by such arguments (of course, that was not intended as an ethical "ought" ). - 10 Oct '11 21:41
*Originally posted by vistesd***[b]Infinities are usually used in philosophical/theological arguments (in my experience) in an attempt to confuse everyone and then claim victory while everyone is too confused to disagree.**

Hence my interest. There ought to be a level of understanding where even the layperson need not be confused by such arguments (of course, that was not intended as an ethical "ought" ).[/b]**There ought to be a level of understanding where even the layperson need not be confused by such arguments**

There is but you won't find it here, you won't find it in science, you won't even find it in religion. You will only find it through introspection. - 10 Oct '11 21:51 / 5 edits

Suppose we point to some infinitely large set X and say that for any x in X, doing x is possible. Then whatever is the size of infinity for this set (Aleph-0, Aleph-1 or whatever), if we take the power set of this*, the size of this new set will be a bigger infinity than the one you started with. Now there is no reason why one should confidently say it is true that not all of these things can be done also...but then you can take the power set of this set too, and so on and so forth.*Originally posted by vistesd***Please, lay that out for me. Even if you have before (or cut & paste). That certainly has philosophical implications...**

But then allowing that for each power set one is able to do the things which lie inside then for any given size of infinity as it relates to time you can always, excuse my phraseology, find some set of things possible to do which is "bigger" than the set of time points you have to do them in

* You probably know this but using a finite set for simplicity if we have X = {a,b} then the power set P(X) = {{a},{b},{a,b}, {}}, and the powerset of P(X) = P(P(X)) is given by { {{a}}, {{a},{b}}, {{a},{a,b}},{{a},{}},...(12 more elements) } - 10 Oct '11 21:55 / 1 edit

Agerg will probably answer this better, but sometimes different explanations in concert help make things easier to understand.*Originally posted by vistesd***Please, lay that out for me. Even if you have before (or cut & paste). That certainly has philosophical implications...**

Of course it is possible that it will just confuse things further but hey ho ;-)

It has to do with there being classes of infinity.

The lowest/smallest of which being what's called the countably infinite.

Countable infinity implies as the name suggest that you could (hypothetically) count it.

So if you had an infinite number of apples you could number each apple with an integer and as the integer numbers go on forever

you can keep doing so for ever counting the apples.

Indeed the base example of a countable infinity are the integer numbers.

Now you might be thinking well anything you have must be countable, you just assign a number to each object and there

are infinite numbers so you will never run out.

Unfortunately this is where I introduce the real number line.

So lets say I want to count the real numbers from 1 to 2.... that shouldn't be too hard should it.....

However there are an infinite number of real numbers between 1 and 2. (inclusive)

but were still ok as there are an infinite number of integers to muse to count them...

except that there are more real numbers between 1 and 2 than there are integers.

The way I show that is like this.

The first number is 1 so I can label that with the integer 1.

lets say the next is 1.1 and label that with the integer 2....

Except that can't be right because there are numbers between 1 and 1.1

so lets say its 1.01...

Except that there are numbers between 1 and 1.01....

In fact there are an infinite number of numbers between any real number and any other real number.

Any number I try to nominate as my second number I can find an infinite number of real numbers between the two.

It isn't possible to match up a countable infinity with the real number line because there will always be an infinite number

of real numbers between any two you pick.

So in fact you can see that the real number line is infinitely bigger than the integer number line, despite the fact that the

integer number line is infinite.

And that in fact the real numbers between any to integers outnumber the total number of integers by infinity...

What has this to do with doing everything if you exist forever....

Well it depends on what infinities you are using.

Time is usually considered to be granular and is thus countably infinite.

And thus if god can only do a finite number of things per time unit, the things he can do in an infinite amount of time are still countably infinite.

However if god can do a non-countably infinite number of things then he not only need not have done them all, but he can't possibly have done them all.

That said I find the idea that god can do a non-countably infinite number of distinct things to be absurd.... but that is a different argument.

EDIT: Agerg I would be interested in any comments/corrections you have on my description of the problem (I went with as non-technical as possible

as I suspected you would go down the technical route) - 10 Oct '11 22:10

I seem to recall that the initial controversy over power sets was that you end up with an infinity larger than the infinity of the original set? (Likely I have not said that well; but I am open to correction.) This*Originally posted by Agerg***Suppose we point to some infinitely large set X and say that for any x in X, doing x is possible. Then whatever is the size of infinity for this set (Aleph-0, Aleph-1 or whatever), if we take the power set of this*, the size of this new set will be a bigger infinity than the one you started with. Now there is no reason why one should confidently say it is true ...[text shortened]... of P(X) = P(P(X)) is given by { {{a}}, {{a},{b}}, {{a},{a,b}},{{a},{}},...(12 more elements) }***seems*to be a paradox.

With that, I understand what you’re saying. But—can there be a power set of all sets? Or is the number of power sets infinite (as I suspect is the case)? What can “relative infinities” possibly*mean*outside the conventions of mathematics—that may have no other meaning than to make the maths work (i.e., they may have a use without a meaning)?

Please don’t misunderstand! None of this is in the way of argumentative questions. They are simply questions of principle. And I am not sure that even a “complete” mathematics can describe all the principles (conditions) for such a mathematics to cohere (Gödel?). But, of course, I am interested in the implications for philosophy.

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BTW, what, if anything, is the significance of the power set including the null set { , }. What if there is—ontologically speaking—no null set? - 10 Oct '11 22:16

I need to think through this (and I saw Agerg's response first) but--*Originally posted by googlefudge***Agerg will probably answer this better, but sometimes different explanations in concert help make things easier to understand.**

Of course it is possible that it will just confuse things further but hey ho ;-)

It has to do with there being classes of infinity.

The lowest/smallest of which being what's called the countably infinite.

Countable infi ...[text shortened]... t with as non-technical as possible

as I suspected you would go down the technical route)

Is this not a matter of defining "infinity"? Some might take "infinity" to be an absolute, not subject to "levels". I am happy with different infinities, but I am not sure that that means anything other than different (competing) definitions of infinity. Can we start with a definition of "infinity" that acoomodates both the maths and ontology?

[I get the "countable" and "non-countable" distinction, btw.] - 10 Oct '11 22:23 / 1 editI suspect that the notion of “relative infinities” is a linguistic concession. Nevertheless, is it possible that the mathematics can describe what is ontologically impossible? [Again, that is
*not*an argumentative question!]

I want to say that infinity implies that no infinity is more or less than another, and that*that*support Agerg’s argument just as well. And that the idea of power sets of infinite sets is just a (formal) way of ackowledging this. Because “infinity” applies to both time and the possibility of outcomes. It is not that there is a “bigger” infinity, but that infinity is—infinity. That—say—a god or daemon has infinite duration does not imply that such a god or daemon could thereby accomplish all the infinite possibilities for outcomes…

??? - 10 Oct '11 22:24

We are simply talking numbers here. Let's go to see about infinite space, as this read will give you lots of relativistic understanding about how infinity can actually have and end, and yet still be infinite.*Originally posted by vistesd***I seem to recall that the initial controversy over power sets was that you end up with an infinity larger than the infinity of the original set? (Likely I have not said that well; but I am open to correction.) This***seems*to be a paradox.

With that, I understand what you’re saying. But—can there be a power set of all sets? Or is the number of ...[text shortened]... he power set including the null set { , }. What if there is—ontologically speaking—no null set?

http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/cosmo.html

yes, html.

Enjoy!

-m. - 10 Oct '11 22:24 / 1 edit

The way you argue that the size of one set (the reals) is bigger than the other (the integers) by mentioning that for any two numbers there are infinitely many reals between them isn't enough unfortunately...look up Cantor's diagonal argument instead ;]*Originally posted by googlefudge***Agerg will probably answer this better, but sometimes different explanations in concert help make things easier to understand.**

Of course it is possible that it will just confuse things further but hey ho ;-)

It has to do with there being classes of infinity.

The lowest/smallest of which being what's called the countably infinite.

Countable infi t with as non-technical as possible

as I suspected you would go down the technical route)

There are infinitely many rationals between any two integers but the size of the rationals is equal to the size of the integers!

I agree though that if we start out with a finite set (though the theist generally doesn't!) then the set of things that can be done is still finite.