That's not true, using an infinity is not the same as a singularity occurring in the theory. — MindForged

Yes. So what I am saying is you really want to be able to build "infinities" into your models, and you really want to avoid getting "infinities" back out.

They are great if they can be just assumed in background fashion. They are a horror if that is what the calculation returns as its sum.

But either way, these "infinities" have epistemic status rather than ontic. We realise that as backdrop assumptions, they are strong simplifications. And as calculational outcomes, we are quite within our rights to ignore them and create some kind of work-around.

Space under relativity is treated as a continuum... — MindForged

That is way too simplistic. Relativity treats spacetime as a pseudo-Riemannian differentiable manifold. As a space, the continuity is about the ability to maintain certain general symmetries rather than any physical continuity as such.

Blackholes and wormholes can punch holes in the fabric - those nasty singularities - and yet still the general co-variance can be preserved with the right set of yo-yoing symmetries to take up the slack.

So relativity took away the kind of simple spatial infinity presumed under Euclid/Newton and replaced it with something that still worked. Actual continuity was replaced by the virtual continuity of unifying symmetries ... plus now the stabilising extra of physical measurements of local energy densities. A bunch of discrete local values to be added to the model and no longer able to be taken for granted.

But my point was that we still make assumptions (crucial, necessary ones) regarding the existence of infinity in the world as well (relativity and QM both do so), so the notion of an Actual Infinity isn't off the table. — MindForged

But you are talking about a very classical notion of infinity. And that is clearly off the table so far as modern physics would be concerned.

As I said early on, the best way to characterise things now is that the interest lies in how classicality emerges. So it is the development of finitude from a more radical indeterminism which becomes the story we want to be able to model.

To say the Universe is just "actually infinite" is hollow metaphysics - a way to avoid the interesting questions. What came before the Big Bang? Where does the Cosmos end? You seem to want to shrug your shoulders and say everything extends forever. That is what maths would say. So let's just pretend that is the case.

But questioning these kinds of conventionalised notions of "the infinite" is precisely where current metaphysics needs to start. The answers aren't in. We are only just formulating a clear view of what we need to be asking.

As a space, the continuity is about the ability to maintain certain general symmetries rather than any physical continuity as such. — apokrisis

I didn't make any point regarding physical continuity (if space can even be called physical).

To say the Universe is just "actually infinite" is hollow metaphysics - a way to avoid the interesting questions. What came before the Big Bang? Where does the Cosmos end? You seem to want to shrug your shoulders and say everything extends forever. That is what maths would say. So let's just pretend that is the case. — apokrisis

I don't think the universe is actually infinite in breadth or in the past, I really have no idea. And I'm certainly not saying such questions should be shrugged at. From the very beginning is took issue with the OP's assumption that any sort of actual infinity was impossible in virtue of pure logic (because, supposedly, contradictions crop up). The only points I made about maths were in support of that point. We know no known contradictions are derivable from employing infinity in standard maths. So it doesn't make sense to say actual infinities are impossible because of an inconsistency. They may well be impossible, but as I started by saying that isn't because of anything regarding inconsistency:

You might say that not every aspect of our particular universe can be infinitized, but there's no argument that the concept itself precludes instantiation in the world. — MindForged

I didn't make any point regarding physical continuity (if space can even be called physical). — MindForged

What did you mean by space being "actually infinite" then?

From the very beginning is took issue with the OP's assumption that any sort of actual infinity was impossible in virtue of pure logic (because, supposedly, contradictions crop up). — MindForged

The OP might not have been perfectly expressed but it did seem to be arguing from the famous paradoxes that arise from taking the maths "too seriously" as a physicalist.

Now the usual line from the maths-lover is that the maths got fixed to resolve the problems. And my reply to that is: not so fast buddy. :)

Yes, I do believe that actually infinite quantities exist. For example, I believe that at every point in time an infinite number of events occur. This is an example of actual (or completed) infinity because it's the kind of infinity that does not last forever. Also, I believe that every finite period of time is infinitely divisible which means that it consists of an infinite number of moments. This too is an example of actual infinity.

On paradoxes, for example, Zeno’s paradoxes, there is a very simple solution if you take the view that the actually infinite is impossible:

- Assume time is continuous
- Examine any system over a fixed time period
- Then the system goes through an actually infinite number of states in a finite period
- Actually infinite is impossible so reductio ad absurdum time is discrete
- Time is discrete so Archilles only has to cover a finite number of steps to reach the tortoise — Devans99

The system does not go through an actually infinite number of states in a FINITE period of time. That period of time is INFINITE but we SAY that it is finite because we are only aware of a finite portion of it.

If it is, it’s a potential infinity rather than an actual Infinity (you do understand the distinction?).

The division of space takes time, first we must cut one inch, then 1/2 an inch, then 1/4... No matter how many cuts we make we never get to actual infinity, just some small number. — Devans99

It's not possible for a human being to count an infinite number of things in a finite number of steps.

However, that DOES NOT mean that it is impossible for an infinite number of things to happen between two points in time.

However, that DOES NOT mean that it is impossible for an infinite number of things to happen between two points in time. — Magnus Anderson

You are assuming time is continuous.

- Assume we have a system
- Watch it evolve over a finite time period
- Will we observe it pass though an actually infinite number of states?

My gut feeling is no so time is probably discrete.

You say that you believe time is continuous but you don’t give an argument why.

I say time is discrete because otherwise we get logical contradictions.

Ok here is the proof that Actual Infinity does not exist:

We have the concept of ‘Reductio ad absurdum’ which Wikipedia defines as:

‘In logic, reductio ad absurdum is a form of argument which attempts either to disprove a statement by showing it inevitably leads to a ridiculous, absurd, or impractical conclusion, or to prove one by showing that if it were true, the result would be absurd or impossible.’

Returning to the Actually Infinite, my proof that it does not exist is that THE ARGUMENT STARTS WITH AN ABSURDITY.

For example infinite time implies anything can happen will happen an infinite number of times which is absurd.

You are assuming time is continuous.

- Assume we have a system
- Watch it evolve over a finite time period
- Will we observe it pass though an actually infinite number of states?

My gut feeling is no so time is probably discrete.

You say that you believe time is continuous but you don’t give an argument why.

I say time is discrete because otherwise we get logical contradictions. — Devans99

If time is infinitely divisible, it follows that it is impossible to experience what happens at every single point in time. So yes, you cannot observe a system passing through an actually infinite number of states. But just because you cannot be directly aware of something does not mean that that something does not exist. We are only ever aware of a small subset of what is "out there".

I want to understand why you think that the belief that time is continuous (= infinitely divisible) leads to logical contradictions.

But just because you cannot be directly aware of something does not mean that that something does not exist. — Magnus Anderson

You are making things complicated. Zeno’s paradoxes disappear if we assume time is discrete for example (IE then Archiles only has to cover a finite number of steps to catch the tortoise).

Don’t you get it, logical contradictions like Zeno’s, Hilberts Hotel etc... exist because we have an absurdity (Actual Infinity) at the core of our reasoning

You are making things complicated. Zeno’s paradoxes disappear if we assume time is discrete for example (IE then Archiles only has to cover a finite number of steps to catch the tortoise).

Don’t you get it, logical contradictions like Zeno’s, Hilberts Hotel etc... exist because we have an absurdity (Actual Infinity) at the core of our reasoning — Devans99

Not quite. Zeno's "paradoxes" exist because people do not understand the concept of actual infinity. That's the problem. For even if you accept that time and space are infinitely divisible and that in order to move from point A to point B you must cross every point in between the two points, Zeno's conclusions still do not follow.

What’s wrong with the simplist argument?

The Actually Infinite exists. Reductio ad absurdum. No it doesn’t.

Give me a counter example from nature of the Actually Infinite (one you can actually prove exists please) if you can...

For example infinite time implies anything [that] can happen will happen an infinite number of times which is absurd. — Devans99

The concept of infinite time does not imply that things will repeat an infinite number of times. You can have an infinite number of completely different moments.

Yes it does:

- Assume infinite time
- so anything than can possibly happen will eventually happen
- If it happens once it will eventually happen again
- So it will eventually happen an infinite number of times

Not quite. You can have an infinite number of one and the same thing. Or you can have an infinite number of completely different things. Or you can have a mix of the two. The concept of infinite time is not defined in such a way that it means that everything that can happen will happen an infinite number of times. That's simply NOT how the concept of infinite time is defined.

For example infinite time implies anything can happen will happen an infinite number of times which is absurd. — Devans99

It is? I think it's just a consequence of infinity, which is a very odd 'number' indeed. In scrabbling to make sense of infinity, some silly things happen. So, in a genuinely infinite amount of time, anything that can happen, will happen an infinite number of times. This isn't absurd, it's logical. But we just used "infinity" to mean two different quantities in one sentence. Didn't we? Maybe INFINITY is absurd? That certainly seems possible.... :chin:

The "set of moments after the present moment" is unbounded but no one gets up in arms about defining such a collection of moments as a set. — MindForged

There is no such "set". The moments after the present moment have not yet come into existence so you cannot collect them into a set, nor can they be members of "a set" in any way or fashion, as they are non-existent. You are claiming to have a set of things which do not exist, but that's impossible so it's pure fiction, nonsense.

Again, what is the non-question begging argument for this? — MindForged

It's clearly not a matter of begging the question. It's a matter of understanding the definition of "set", and understanding the definition of "infinite", and realizing that it is impossible to have an infinite set. These two are incompatible, by definition, so talking about infinite sets is contradictory nonsense. Of course we all know that because of the many paradoxes which are known to arise from the assumption of infinite sets, but some like you, choose to ignore this obvious fact.

They aren't "collected" in a mechanistic process, i.e. going out and declaring "You go in this set" and such. Just sharing a property is enough, and it happens to be perfectly compatible with there being infinite collections. — MindForged

I think you are mistaken here. That something has a particular property is a judgement. The thing is a particular the property is a universal. Therefore if "sharing a property" is what is required to be a member of a set, then a judgement is required in order that things be of the same set. So the declaration "you go in this set" is exactly what is required in order that a thing be a member of a particular set.

You seem to either believe that sets just naturally exist without ever being created by human minds, or else that things automatically jump up and join any set which they are supposed to be a member of, without being counted into that set. So either the green grass is naturally a member of the set of green things without that set ever being created by a human mind, or else the green grass jumps into the set, of its own power, as soon as "the set of green things" is named by a human being. Both of these, I tell you are nonsense.

All the paradoxes of infinity (https://en.m.wikipedia.org/wiki/Paradoxes_of_infinity) result from us having a logical facalcy (Actual Infinity) at the core of our reasoning.

The Actually Infinite has no place in the material world.

The Actually Infinite has no place in the material world. — Devans99

You have to understand that actual infinity is just a concept, nothing more than that, and that you have to understand it.

I don't think that you understand it.

Exactly my point Actual Infinity is a concept but it does not exist in reality.

My point is that you do not understand what the word means. If you know what actual infinity means, there is no way in hell you can take Zeno's "paradoxes" seriously.

2nd proof that Actual Infinity does not exist:

- Actual infinity plus one equals Actual infinity
- but X+1 <> X for all X
- So Actual Infinity is absurd

2nd proof that Actual Infinity does not exist:

- Actual infinity plus one equals Actual infinity
- but X+1 <> X for all X
- So Actual Infinity is absurd — Devans99

Fabulous argument.

Here's a variant of it:

- More than Two Things plus one equals More than Two Things
- but X + 1 <> X for all X
- So More than Two Things is absurd

More than Two Things plus one equals More than Two Things — Magnus Anderson

No it does not. X+1<>X

So if you have a number greater than 2 and you add 1 to it you don't get a number greater than 2?

Interesting.

When you add something to a number you get a different, larger number

Yes, when you have two apples and you add another apple, you get three apples. You don't get three oranges or three bananas. But you nonetheless get something that belongs to the same class that three oranges and three bananas belong to. And that class is the class represented by the word "three". In the same way, when you add 1 to a number that belongs to the class of numbers greater than two, you get a number that belongs to the class of numbers greater than two. You can hardly deny this. So yes, More than Two Things + 1 = More than Two Things.

But the concept of something that you add something to and it remains the same is nonsensical

It's nonsensical to you, because you don't understand it. When I say More than Two things + 1 = More than Two Things what I am saying is that if you take a number greater than two and add one to it you will get a number that is greater than two.

Yes, but a different number so I still assert:

- Actual infinity plus one equals Actual infinity
- but X+1 <> X for all X
- So Actual Infinity is absurd

Yes, you will get a different number, that's true, but that's not the point. Noone is saying you'll get the same number. More than Two things + 1 = More than Two Things does not mean you will get the same number. Infinity + 1 = Infinity does not mean you will get the same number. It means you will get a number that belongs to the same class of numbers. When you have an infinite set of elements and you add another element to it, you'll get a different set of elements, one that is larger, but one that is nonetheless infinite.