Can infinity end?

[Boltzmannbrain]

The main questions of this thread are:

1.  Is it logically possible for the totality of an infinite number of objects to exist?

2. Can infinity end? (such as an infinitely long line or an infinite row of objects)

 

When I think about it logically, the full extent of an infinitely long line or all of an infinite amount of objects should not be able to exist.  I come to this conclusion simply due to the idea that in an infinite row of objects there is no last object.  However, that is just where I am at right now; I may be persuaded otherwise.  

 

The Stanford Encyclopedia of Philosophy has three definitions of infinity here, 

Quote

 

1. In a loose or hyperbolic sense, ‘infinite’ means ‘indefinitely or exceedingly great’, ‘exceeding measurement or calculation’, ‘immense’, or ‘vast’.

2. In a strict but non-mathematical sense that reflects its etymological history, ‘infinite’ means ‘having no limit or end’, ‘boundless’, ‘unlimited’, ‘endless’, ‘immeasurably great in extent (or duration, or some other respect)’. This strict, non-mathematical sense is often applied to God and divine attributes, and to space, time and the universe.

3. There is also a strict, mathematical sense, according to which ‘infinite’ quantities or magnitudes are those that are measurable but that have no finite measure; and ‘infinite’ lines or surfaces or volumes are measurable lines or surfaces or volumes that have no finite measure.

 

Let's use (2) and/or (3) if we can.

 

I don't think that mathematics would agree with my conclusion.  For example, if my conclusion were correct, then there would be some natural numbers that don't exist.  I am pretty sure that cannot be the case in math. 

 

I would like to read what people have to say about all of this.

[Genady]

I see an ambiguity in the question 1 in the concept of "exist." And an ambiguity in the question 2 in the concept of "end." Answers to both questions depend on interpretation of these terms more, or rather than on interpretation of the term "infinity."

[Boltzmannbrain]

1 hour ago, Genady said:

I see an ambiguity in the question 1 in the concept of "exist." And an ambiguity in the question 2 in the concept of "end." Answers to both questions depend on interpretation of these terms more, or rather than on interpretation of the term "infinity."

What are the different meanings of "exist" and "end" are you uncertain about?

[Genady]

Just now, Boltzmannbrain said:

What are the different meanings of "exist" and "end" are you uncertain about?

For example, does a number 3 exist? Does an open interval (0,1) end?

[Boltzmannbrain]

19 minutes ago, Genady said:

For example, does a number 3 exist? Does an open interval (0,1) end?

Here we can use mathematics to answer those questions.  I believe 3 is called an object that exists in mathematics.  And the example (0,1) is what I would like to know about also.  Does it end?  I know it is bounded, but can we say that it ends?

[Genady]

5 minutes ago, Boltzmannbrain said:

Here we can use mathematics to answer those questions.  I believe 3 is called an object that exists in mathematics.  And the example (0,1) is what I would like to know about also.  Does it end?  I know it is bounded, but can we say that it ends?

I think that infinite set is an object that exists in mathematics in the same sense as 3.

I don't think that (0,1) is said to end.

[Boltzmannbrain]

8 minutes ago, Genady said:

I think that infinite set is an object that exists in mathematics in the same sense as 3.

Do all the objects exist that are in an infinite set? 

[Genady]

1 minute ago, Boltzmannbrain said:

Do all the objects exist that are in an infinite set? 

In the mathematical sense we apply now, yes, they do.

[Boltzmannbrain]

16 hours ago, Genady said:

In the mathematical sense we apply now, yes, they do.

I was afraid of that answer.  I don't understand how something can exist but not end.  

This is why I am so interested in this topic.  It doesn't make clear sense.  Something seems wrong, or at least there is a gap of knowledge to be filled.

[Genady]

53 minutes ago, Boltzmannbrain said:

I was afraid of that answer.  I don't understand how something can exist but not end.  

This is why I am so interested in this topic.  It doesn't make clear sense.  Something seems wrong, or at least there is a gap of knowledge to be filled.

I am reasonably clear about what you mean saying "exist", i.e., the mathematical existence. However, I still don't know what "end" means.

For example, number 3 exists. Does it "end"?

[studiot]

2 hours ago, Boltzmannbrain said:

I was afraid of that answer.  I don't understand how something can exist but not end.  

This is why I am so interested in this topic.  It doesn't make clear sense.  Something seems wrong, or at least there is a gap of knowledge to be filled.

OK so I respectfully suggest your vocabulary of concepts is too limited.

 

For instance 'end' is a one dimensional instance of a boundary.

Go to two dimensions and you have for instance the edge of  rectagular piece of paper.

If that paper is now an infinite roll you have two edges but no ends

Carry this line of thinkin g inot higher dimensions.

 

So I have introduced some important new terms for you, boundary, edge, dimension.

But the complexity of the matter does not end there.

 

It is necessary not to confuse boundary with bounded. 

They are quire different concepts with confusingly similar names.

For instance the function f(x) = sin x is bounded, yet x is unbounded. Neither have a boundary.

 

Then again we introduce infinity. Some Astrophysicists like to  argue that the Universe is 'finite yet unbounded', with no end or beginning.

How can that be ?

Well consider na circle.

Does it have a beginning or an end ?

If you travel round it is that journey finite or infini9te ?

But have far do you travel if you make an infinite count of circumnavigations?

 

If you wish your considerations to enter the later 20th/early 21 centuries then you need to consider porous and fractal boundaries (or ends).

 

I don't claim my list of additional terms is exhaustive, just a good start.

 

+1 for  good topic by the way.

[Boltzmannbrain]

4 hours ago, Genady said:

I am reasonably clear about what you mean saying "exist", i.e., the mathematical existence. However, I still don't know what "end" means.

For example, number 3 exists. Does it "end"?

Good point (+1 ), but there is still something unsettling about an infinite "row" or "list", in particular.  The whole row somehow exists, but also doesn't exist from a certain perspective.   Does perception play into mathematics sort of how observation plays into physics?

3 hours ago, studiot said:

OK so I respectfully suggest your vocabulary of concepts is too limited.

 

For instance 'end' is a one dimensional instance of a boundary.

Go to two dimensions and you have for instance the edge of  rectagular piece of paper.

If that paper is now an infinite roll you have two edges but no ends

Carry this line of thinkin g inot higher dimensions.

 

So I have introduced some important new terms for you, boundary, edge, dimension.

But the complexity of the matter does not end there.

 

It is necessary not to confuse boundary with bounded. 

They are quire different concepts with confusingly similar names.

For instance the function f(x) = sin x is bounded, yet x is unbounded. Neither have a boundary.

 

Then again we introduce infinity. Some Astrophysicists like to  argue that the Universe is 'finite yet unbounded', with no end or beginning.

How can that be ?

Well consider na circle.

Does it have a beginning or an end ?

If you travel round it is that journey finite or infini9te ?

But have far do you travel if you make an infinite count of circumnavigations?

 

If you wish your considerations to enter the later 20th/early 21 centuries then you need to consider porous and fractal boundaries (or ends).

 

I don't claim my list of additional terms is exhaustive, just a good start.

 

+1 for  good topic by the way.

+1  Yeah, I am trying to keep all of this in mind.  When I said that there may be a gap in our understanding (or at least in my understanding) of infinity, I was sort of referring to your points in your post.  There might just be a property of infinity that allows it to somehow exist in and out of the ether.  I will look at those properties that you mentioned.  Thanks.  

[Genady]

14 minutes ago, Boltzmannbrain said:

Does perception play into mathematics sort of how observation plays into physics?

Sorry, I've tried but failed to understand the analogy. Thus, don't know what to say.

[Boltzmannbrain]

4 hours ago, Genady said:

Sorry, I've tried but failed to understand the analogy. Thus, don't know what to say.

Forget that analogy. 

I was thinking that maybe it matters in what perspective the infinite objects are being considered.  But we probably don't have to get into that.

[wtf]

On 6/13/2023 at 2:44 PM, Boltzmannbrain said:

2. Can infinity end? (such as an infinitely long line or an infinite row of objects)

 

Sure. Just reorder the natural numbers from their usual order, 0, 1, 2, 3, 4, ... by taking 0 and putting it at the end to get the ordered set:

1, 2, 3, 4, ..., 0

That's an infinite, ordered set of of numbers that's the exact same set as the natural numbers in their usual order, but has both a first and last element. 

[Intoscience]

I'm not sure but it seems like you are confounding mathematical concepts with physical objects.

For instance you have been given examples of mathematical concepts based on infinite sets. In physical reality however the idea of "infinite number of objects" might not be possible. 

Mathematics can lead to infinities when modelling physical reality. One example being the singularity predicted at the centre of a black hole this brings into question the possibility of physical infinities.

However, space itself maybe infinite, it exists (we are part of it) but it may not end. In which case there would be room to fit an infinite number of objects, in which case we may assume but not confirm that they "all" exist without end. Counting them "all" would take an infinite amount of time though, so in this instance there would be no way to confirm there is an infinite number of them. 

Mind blowing stuff and a subject that is discussed profusely across all science/math forums. 

 

[Genady]

8 hours ago, Boltzmannbrain said:

maybe it matters in what perspective the infinite objects are being considered

I don't know if it matters here, but generally mathematical objects can be considered in a variety of perspectives. For example, a complex number can be considered as built of real and imaginary parts or of phase and magnitude.

[Boltzmannbrain]

9 hours ago, wtf said:

Sure. Just reorder the natural numbers from their usual order, 0, 1, 2, 3, 4, ... by taking 0 and putting it at the end to get the ordered set:

1, 2, 3, 4, ..., 0

That's an infinite, ordered set of of numbers that's the exact same set as the natural numbers in their usual order, but has both a first and last element. 

Interesting +1

But I have to ask, how can 0 come after numbers > 0 in an ordered set?

3 hours ago, Intoscience said:

I'm not sure but it seems like you are confounding mathematical concepts with physical objects.

For instance you have been given examples of mathematical concepts based on infinite sets. In physical reality however the idea of "infinite number of objects" might not be possible. 

Mathematics can lead to infinities when modelling physical reality. One example being the singularity predicted at the centre of a black hole this brings into question the possibility of physical infinities.

However, space itself maybe infinite, it exists (we are part of it) but it may not end. In which case there would be room to fit an infinite number of objects, in which case we may assume but not confirm that they "all" exist without end. Counting them "all" would take an infinite amount of time though, so in this instance there would be no way to confirm there is an infinite number of them. 

Mind blowing stuff and a subject that is discussed profusely across all science/math forums. 

 

Yeah, thinking about how these strange mathematical concepts may cross over into the real world is almost scary.  

 

My math professor told me something that was absolutely mind-blowing to me.  I asked him what he researches as a faculty member with a doctorate in mathematics.  He said that they look at physical phenomena (exotic phenomena I presume) to understand more about math.  

 

So in some limited sense, or maybe not even limited, it seems to me that there is almost no difference between math and physics.  Maybe eventually we will find that they are both the same thing.   

3 hours ago, Genady said:

I don't know if it matters here, but generally mathematical objects can be considered in a variety of perspectives. For example, a complex number can be considered as built of real and imaginary parts or of phase and magnitude.

+1  I find that very interesting and important to remember.

[wtf]

5 hours ago, Boltzmannbrain said:

But I have to ask, how can 0 come after numbers > 0 in an ordered set?

You just define it that way. You make up a relation called the "funny order" on the natural numbers that says, that if n and m are both nonzero, then their new funny order is the usual one.

Except that zero is larger than any other number. 

This new funny order satisfies the axioms of an ordered set: Reflexivity, antisymmetry, and transitivity.

https://en.wikipedia.org/wiki/Partially_ordered_set

It's really no different than taking a bunch of school kids and having them line up by height; and then taking the shortest one and telling then to go to the tallest end.

Another more familiar mathematical model is the closed unit interval [0,1] consisting of all the real numbers between 0 and 1, inclusive. That is an uncountably infinite set that has a smallest and largest value.

Or just think about the points on a circle. That's an uncountably infinite set with no beginning and no end that has a finite length, namely the circumference. 

Edited June 15, 2023 by wtf

[Boltzmannbrain]

20 hours ago, wtf said:

You just define it that way. You make up a relation called the "funny order" on the natural numbers that says, that if n and m are both nonzero, then their new funny order is the usual one.

Except that zero is larger than any other number. 

This new funny order satisfies the axioms of an ordered set: Reflexivity, antisymmetry, and transitivity.

https://en.wikipedia.org/wiki/Partially_ordered_set

It's really no different than taking a bunch of school kids and having them line up by height; and then taking the shortest one and telling then to go to the tallest end.

Another more familiar mathematical model is the closed unit interval [0,1] consisting of all the real numbers between 0 and 1, inclusive. That is an uncountably infinite set that has a smallest and largest value.

Or just think about the points on a circle. That's an uncountably infinite set with no beginning and no end that has a finite length, namely the circumference. 

+1  Thanks for this, but it is very unsettling for me.

What I am interpreting this to mean is that infinity can have a final element (or end) but it also cannot have a final element.

It is also hard to grasp that something that is defined as having no end, can end.

I suppose that infinity ends in one respect and does not end in another.  I am struggling to find the difference between the two "ends".

[wtf]

1 hour ago, Boltzmannbrain said:

What I am interpreting this to mean is that infinity can have a final element (or end) but it also cannot have a final element.

 

You keep referring to "infinity" but that's a vaguely defined word. It's better to talk about infinite sets, which do have a clear definition. A set is infinite if it can be placed into one-to-one correspondence with a proper subset of itself. [Pedantry note, that's the definition of Dedekind infinite, but it will do for present purposes].

By that definition, the natural numbers 0, 1, 2, 3, 4, ... are an infinite set, because they can be placed into one-to-one correspondence with their proper subset the even numbers. 

Now, any set can be ordered in many different ways. Consider a class full of school kids. You can ask them to line up in order of height. You can ask them to line up in order of age. You can ask them to line up alphabetically by last name.

In each case you have the same set, but it's ordered differently. So we see that there are two distinct concepts: The elements of a set, which don't change no matter how you reorder them; and order properties, which can change depending on how you line up the kids, or the elements.

So we can reorder the natural numbers as 1, 2, 3, ..., 0. It's still the same set, but we just ordered it differently. In the usual order 0, 1, 2, 3, ... the ordered set as a first element but no last element. In the reordered set 1, 2, 3, ...,0 there is both a first and last element.

The order properties of a set can vary depending on how we line up the elements.

 

1 hour ago, Boltzmannbrain said:

It is also hard to grasp that something that is defined as having no end, can end.

 

That's not the definition of an infinite set. It's very common in online discussions for people to say that an infinite set is one that has no end. But this is simply false. Dictionary definitions are not helpful in mathematical discussions.

As we've seen, many infinite sets have ends. The funny ordering of the natural numbers 1, 2, 3, ..., 0 has an end. The closed unit interval of the real numbers [0,1] has an end, namely 1. 

Lots of infinite sets have ends. Circles are infinite sets that have no ends at all, yet have a finite length.

So the trick here is for you to unlearn the wrong definition of infinite sets that you've been using. Infinite sets can sometimes have beginnings and ends, other times not.

 

1 hour ago, Boltzmannbrain said:

I suppose that infinity ends in one respect and does not end in another.  I am struggling to find the difference between the two "ends".

Line up the kids by height, line up the kids by weight (and get sued by the parents). Two different orderings on the same set. 

Set membership is one thing. Set orderings are a different thing. You can put many different orderings on a given set. 

Now I am talking about mathematical infinity. I am not talking about physics or the real world (whatever that is, ask a quantum physicist if there even is one). I'm only talking about math. 

But math is a good place to start, because it's the one area of human learning where we have a clear, logical theory of infinity.

[Boltzmannbrain]

On 6/16/2023 at 2:15 PM, wtf said:

You keep referring to "infinity" but that's a vaguely defined word. It's better to talk about infinite sets, which do have a clear definition. A set is infinite if it can be placed into one-to-one correspondence with a proper subset of itself. [Pedantry note, that's the definition of Dedekind infinite, but it will do for present purposes].

By that definition, the natural numbers 0, 1, 2, 3, 4, ... are an infinite set, because they can be placed into one-to-one correspondence with their proper subset the even numbers. 

Now, any set can be ordered in many different ways. Consider a class full of school kids. You can ask them to line up in order of height. You can ask them to line up in order of age. You can ask them to line up alphabetically by last name.

In each case you have the same set, but it's ordered differently. So we see that there are two distinct concepts: The elements of a set, which don't change no matter how you reorder them; and order properties, which can change depending on how you line up the kids, or the elements.

So we can reorder the natural numbers as 1, 2, 3, ..., 0. It's still the same set, but we just ordered it differently. In the usual order 0, 1, 2, 3, ... the ordered set as a first element but no last element. In the reordered set 1, 2, 3, ...,0 there is both a first and last element.

The order properties of a set can vary depending on how we line up the elements.

 

That's not the definition of an infinite set. It's very common in online discussions for people to say that an infinite set is one that has no end. But this is simply false. Dictionary definitions are not helpful in mathematical discussions.

As we've seen, many infinite sets have ends. The funny ordering of the natural numbers 1, 2, 3, ..., 0 has an end. The closed unit interval of the real numbers [0,1] has an end, namely 1. 

Lots of infinite sets have ends. Circles are infinite sets that have no ends at all, yet have a finite length.

So the trick here is for you to unlearn the wrong definition of infinite sets that you've been using. Infinite sets can sometimes have beginnings and ends, other times not.

 

Line up the kids by height, line up the kids by weight (and get sued by the parents). Two different orderings on the same set. 

Set membership is one thing. Set orderings are a different thing. You can put many different orderings on a given set. 

Now I am talking about mathematical infinity. I am not talking about physics or the real world (whatever that is, ask a quantum physicist if there even is one). I'm only talking about math. 

But math is a good place to start, because it's the one area of human learning where we have a clear, logical theory of infinity.

Thanks for this +1

I definitely see what you are saying.  I was not specific enough in the OP. 

But I think you answered my questions for the other "types" of infinity.   

[Alysdexic]

Since childhood I'v been working on eliminating infinities; in so doing, I independently invented a Riemannian infinity plot whilst other children were piddling their minds away going out and getting tans, or whatever they did, and later wrote a proof that it take finite vis (energhy) to accelerate mass to celerity.

In any case, I'd argue everything is finite and the infinite does not and can not exist (and if it did exist, could not be known), and tangentially but importantly, infinite and eternal are often confused. The root of eternity makes it mean “however long something is,” or “lifetime,” not “after time” (“eternity” comes from the Latin words “aeternus,” which came from “aeviternus,” which came from “aevum” [age or lifetime], which came from the from Proto-Indo-European root “*aiw-” [lifetime]).

Thus, everything is eternal; that is, it lasts however long it lasts. Nothing is infinite; that is, something must be quantified with upper and nether bounds. The univers has finite mass and extent; both of those values define de Broglie bounds at both ends.

[Genady]

2 hours ago, Alysdexic said:

Since childhood I'v been working on eliminating infinities

Please, don't eliminate infinities. I like them. A lot.

 

2 hours ago, Alysdexic said:

I'd argue everything is finite and the infinite does not and can not exist

I'd argue to the contrary.

 

2 hours ago, Alysdexic said:

and if it did exist

So, it can exist then.

 

2 hours ago, Alysdexic said:

infinite and eternal are often confused

But not by us.

[exchemist]

On 6/19/2023 at 8:43 PM, Alysdexic said:

Since childhood I'v been working on eliminating infinities; in so doing, I independently invented a Riemannian infinity plot whilst other children were piddling their minds away going out and getting tans, or whatever they did, and later wrote a proof that it take finite vis (energhy) to accelerate mass to celerity.

In any case, I'd argue everything is finite and the infinite does not and can not exist (and if it did exist, could not be known), and tangentially but importantly, infinite and eternal are often confused. The root of eternity makes it mean “however long something is,” or “lifetime,” not “after time” (“eternity” comes from the Latin words “aeternus,” which came from “aeviternus,” which came from “aevum” [age or lifetime], which came from the from Proto-Indo-European root “*aiw-” [lifetime]).

Thus, everything is eternal; that is, it lasts however long it lasts. Nothing is infinite; that is, something must be quantified with upper and nether bounds. The univers has finite mass and extent; both of those values define de Broglie bounds at both ends.

I’m familiar with de Broglie’s relation. What’s a de Broglie bound?