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Definitions but debates?

Subduction Zone

Veteran Member
There are several different notions of infinity.

Infinity as a limit describes what happens to a function, and simply describes some variable getting larger and larger as some other variable does something.

Infinity as a quantity (known as cardinality) is a different notion of infinity. But, there are different sizes of such infinite quantities. The smallest such infinity is that of the 'number' of counting numbers (positive integers). The 'number' of decimal numbers is a larger infinity than this.

There are also ordinal concepts of infinity. Such can have either infinite ascent, infinite descent, or both.



There is no 'largest number', and no 'largest number before infinity'. And infinite sets need not have a 'start' nor an 'end'.



What happens here depends on the type of infinity. For limits, these equations are all correct.

For cardinals, a sum or product of infinite cardinals is the larger of the two. Exponentiation is a lot trickier. If you want an exposition, I will be happy to give one after I'm back from vacation.

For ordinal infinities, things get stranger.



Boundaries are a concept from geometry (or topology). Most versions of infinity are not in that context, so the notion simply doesn't apply.

Infinite in extent is different than having no boundary. For example, the surface of a sphere has no boundary, but is finite.


Dang beat me to it with the first example. But the OP asked if infinity had a beginning. The counting numbers do have a beginning at 1. They do not have an end.

I am unaware of other examples of infinity that have a beginning but not an end, but I have feeling that they are out there.
 

Heyo

Veteran Member
First, pi is a nice, finite number. It is a bit more than 3 and certainly is less than 4.

You are talking about a *decimal* expansion of the number pi.
Decimal expansions are a bit too complicated for a beginner. My go-to example for infinite series is the geometric series lim( sum( 1/2^n)). It has a nice geometric representation, easy numbers and a reference to the grandmaster of thoughts about infinity, Zeno. And it shows how we can tame some infinities.
 

Subduction Zone

Veteran Member
Decimal expansions are a bit too complicated for a beginner. My go-to example for infinite series is the geometric series lim( sum( 1/2^n)). It has a nice geometric representation, easy numbers and a reference to the grandmaster of thoughts about infinity, Zeno. And it shows how we can tame some infinities.
Nice, another example with a beginning. And this could be said to have an "end". Though the progression itself does not end we do know where the sum ends.
 

YoursTrue

Faith-confidence in what we hope for (Hebrews 11)
Decimal expansions are a bit too complicated for a beginner. My go-to example for infinite series is the geometric series lim( sum( 1/2^n)). It has a nice geometric representation, easy numbers and a reference to the grandmaster of thoughts about infinity, Zeno. And it shows how we can tame some infinities.
Ok infinity is still unending. If it's not, lol, please advise.
 

YoursTrue

Faith-confidence in what we hope for (Hebrews 11)
First, pi is a nice, finite number. It is a bit more than 3 and certainly is less than 4.

You are talking about a *decimal* expansion of the number pi. And, yes, that decimal expansion is infinite. And, yes, we most certainly *can* express that this expansion is infinite. We know that pi is irrational and all irrational numbers have an infinite decimal expansion. But, then, so does 1/3.



If you look at the negative integers, that is an infinite set that has an end (a maximum) but no beginning.
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All that it means to be infinite is that it is not finite. In other words, it cannot be put into a one to one correspondence with some natural number. Some infinite sets have an 'end', (a maximum), some have a beginning (a minimum), some have both, and some have neither.

For example, the collection of all fractions strictly between 0 and 1 has no initial and no final member. if you include both 0 and 1, then it has both while still being infinite.
Sorry I asked. I'm leaving the classroom.
 

Heyo

Veteran Member
Ok infinity is still unending. If it's not, lol, please advise.
Is it always unending, though?
Take the geometric representation of the given example, a square that is tiled with pieces each one half the size of the former.
If I were to completely tile the square with all the infinite pieces, how long would that take me?
Let's assume the biggest tile would take me 1/2 an hour, the second biggest, which is half as big would take me half the time, 1/4 of an hour and so on.
After just one hour I would have laid out infinite many pieces.
 

Polymath257

Think & Care
Staff member
Premium Member
Ok infinity is still unending. If it's not, lol, please advise.

OK, when you talk about a 'beginning' or an 'end', you are assuming what is known as an order (before/after, smaller/larger, etc). Not all sets have such an order. In particular, many infinite sets do not.

For example, the set of points on a sphere. That is an infinite set in which 'start' and 'end' simply do not apply.

Or, take the negative integers (..,-3, -2, -1). This is an infinite set with no 'start', but it *does* have an 'end' (ex: -1).

To be infinite simply means to not be finite. Which means that there is no correspondence with some counting number (0,1,2,3,4,..).

But, there are different sizes of infinity, even when you limit yourself to 'cardinal infinities'. In fact, the number of different sizes of infinity is ALSO infinite.

If you aren't counting (one, two, three,...), but are interested in the order of things (first, second, third, ...), it turns out that there are different infinities of the 'same size'. So, you can simply put a new 'number', usually called omega, after all the finite ordinals.

This looks like

0, 1, 2, 3, ..., omega, omega+1, omega+2, ...,, omega+omega=2*omega, 2*omega+1, ...

In terms of order, all of these are different even though the ones from omega on have the same *cardinal size*.

But, if you go 0,1,2,3,...,omega, you get an infinite ordinal with both a beginning ( ex: 0) and an end (ex: omega).
 

blü 2

Veteran Member
Premium Member
Of course, this is about debates. But -- for me -- it's hard to debate the following: Thoughts are appreciated, though.

"Does Infinity have a beginning? It really just depends on the infinity you describe, whether it will have a beginning or not. Most infinities do have beginnings, simply because in order to tangibly grasp the concept of whatever infinity we are talking about (just based on the limitations of the human mind) we generally need a starting point." What Is The Largest Number Before Infinity? - The Biggest (the-biggest.net)
First of all, all infinities are conceptual ─ there are no infinities in the real world that we can go and look at.

So you can postulate a series that starts at 0, or 1, or -1, or -[inf], and goes to [inf], or you can simply say a series is unbounded (the literal meaning of 'infinite'). And after that you can get into Cantorian orders of infinity, and so on.

But like the Euclidean point or line or plane, [inf] has no real referent, like 0, 1, 2, π, e, or any other number. They're all abstractions, concepts, hence depend on the observer. For example you can count 1, 2, 3 in your head; but in the real world before you can count things you have to make two choices ─ first, what it is you want to count, and second, in what field, range, setting, you want to count them. How many [cows] in the [barn]? for example. Without you there to make those judgments, impose those conditions, there's nothing to count.

As for eternity, that problem wouldn't exist if time is an effect, bi-product, property of (say) mass-energy, hence only exists because mass-energy (or whatever) does.
 

YoursTrue

Faith-confidence in what we hope for (Hebrews 11)
OK, when you talk about a 'beginning' or an 'end', you are assuming what is known as an order (before/after, smaller/larger, etc). Not all sets have such an order. In particular, many infinite sets do not.

For example, the set of points on a sphere. That is an infinite set in which 'start' and 'end' simply do not apply.

Or, take the negative integers (..,-3, -2, -1). This is an infinite set with no 'start', but it *does* have an 'end' (ex: -1).

To be infinite simply means to not be finite. Which means that there is no correspondence with some counting number (0,1,2,3,4,..).

But, there are different sizes of infinity, even when you limit yourself to 'cardinal infinities'. In fact, the number of different sizes of infinity is ALSO infinite.

If you aren't counting (one, two, three,...), but are interested in the order of things (first, second, third, ...), it turns out that there are different infinities of the 'same size'. So, you can simply put a new 'number', usually called omega, after all the finite ordinals.

This looks like

0, 1, 2, 3, ..., omega, omega+1, omega+2, ...,, omega+omega=2*omega, 2*omega+1, ...

In terms of order, all of these are different even though the ones from omega on have the same *cardinal size*.

But, if you go 0,1,2,3,...,omega, you get an infinite ordinal with both a beginning ( ex: 0) and an end (ex: omega).
How about something that has no begining? While it is daunting for the human mind (I doubt fishes or bonobos think about it much) to comprehend something with no beginning, infinite generally means something with no end, triangle or cube. :) Or spheres.
 

Polymath257

Think & Care
Staff member
Premium Member
How about something that has no begining? While it is daunting for the human mind (I doubt fishes or bonobos think about it much) to comprehend something with no beginning, infinite generally means something with no end, triangle or cube. :) Or spheres.

No problem. The collection of all integers (positive and negative whole numbers, like -3, or 5) has no beginning or end.

I'm not sure why it is 'daunting', though. Seems perfectly simple to me.

As for 'infinite' typically meaning there is no 'end', a LOT more precision is required in this subject. As I pointed out, there are different notions of infinity and they can give different answers.

For example, think of a straight line of length 3 meters. is it finite or infinite? Well, it has a finite length (3 meters), but has infinitely many points (between any two points, there is another). So saying it is 'without end' needs clarification as to which process is being undergone. Is it length or subdivision? The answers can be different.

Lumping the different notions together will cause a LOT of confusion. But if you are precise enough, those difficulties go away pretty quickly.

Infinity really isn't the mystery it was at one time. We understand many aspects quite well.
 

Revoltingest

Pragmatic Libertarian
Premium Member
Some musings....
The set of counting numbers is infinite.
It has a beginning at one.
There are other boundaries.
Points on a line are bound by the line.
Points on a plane are bound by the plane
 

YoursTrue

Faith-confidence in what we hope for (Hebrews 11)
No problem. The collection of all integers (positive and negative whole numbers, like -3, or 5) has no beginning or end.

I'm not sure why it is 'daunting', though. Seems perfectly simple to me.

As for 'infinite' typically meaning there is no 'end', a LOT more precision is required in this subject. As I pointed out, there are different notions of infinity and they can give different answers.

For example, think of a straight line of length 3 meters. is it finite or infinite? Well, it has a finite length (3 meters), but has infinitely many points (between any two points, there is another). So saying it is 'without end' needs clarification as to which process is being undergone. Is it length or subdivision? The answers can be different.

Lumping the different notions together will cause a LOT of confusion. But if you are precise enough, those difficulties go away pretty quickly.

Infinity really isn't the mystery it was at one time. We understand many aspects quite well.
Oh boy, your concept of a multi-aspected line is yes, daunting. And a bit on the comical side of a line at the same time. :) Have a real nice day, with the infinite points of time you count or don't count from dawn to dusk.
 

Windwalker

Veteran Member
Premium Member
Of course, this is about debates. But -- for me -- it's hard to debate the following: Thoughts are appreciated, though.

"Does Infinity have a beginning? It really just depends on the infinity you describe, whether it will have a beginning or not. Most infinities do have beginnings, simply because in order to tangibly grasp the concept of whatever infinity we are talking about (just based on the limitations of the human mind) we generally need a starting point." What Is The Largest Number Before Infinity? - The Biggest (the-biggest.net)
Does a mobius strip have a beginning?
 
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