The spaceship can keep going, because the universe is expanding faster than the speed of light. The spaceship won't catch up with the expansion of the universe.

So the spaceship isn't creating new space, it is using the space the expansion is creating.

If the dimensional lines (labelled x y and z for height width and depth) can conceptually extend for ever, then even if the universe hasn't expanded that far yet, we can still conceive of the location and so we have the potential to reach it.

Even as a concept, 'that location over there' must exist unless you can explain why it is impossible for anything to get there.

Some physicists tried to argue that space (the dimensional lines) were therefore curved and self-contained within the boundaries of the universe, so you couldn't reach any place beyond the bounds of the universe because it truly didn't exist.

However the 9 year results of NASA's WMAP survey concluded that ....
"The universe is flat, with a 0.4% margin of error, and that Euclidean geometry probably applies".

In other words, the dimensional lines are straight and therefore the Universe is potentially infinite.

He brings the example of a spaceship flying into space and asks what would happen if it went on and on. Is there an end point or does one eventually loop back to the starting point? — spirit-salamander

That is an exact analogy for the flat-earth belief that ships would sail off the edge of the Earth if they went far enough.

No matter where the spaceship flies, it will always be receiving light from billions of years ago/away. The Universe is finite but unbounded.

(@apokrisis please correct me if I’m wrong.)

:up: "Finite and unbounded".

Thanks for the clarification. Brian Greene didn't mention that.

So if you could create a spaceship that is faster than the expansion of the universe, then that spaceship would hit a provisional end of the universe.

In other words, the dimensional lines are straight and therefore the Universe is potentially infinite. — Gary Enfield

Thanks for the reply, it seems to me then that Brian Greene explanations are very misleading.

Such statements by Greene as these are philosophically irritating:

"If space is now infinite, then it always was infinite. Even at the Big Bang. A finite universe can’t expand to become infinite."

https://twitter.com/bgreene/status/839112447923486720?lang=de

He seems to advocate the quilted multiverse model:

"A universe with infinite spatial extent will contain infinitely many mini-universes. An infinite number of these mini-universes will be exactly like our own. Welcome to the mind-blowing nature of infinity – and the sometimes equally mind-blowing nature of the multiverse, which is a common theme among the books in this month’s column. First up is Brian Greene’s The Hidden Reality, which explores nine variations on the multiverse theme. Of these, the type of multiverse that arises as a consequence of infinite space – Greene calls it the “quilted multiverse” because regions of space will repeat like patterns in a quilt – is actually one of the easiest to comprehend." https://physicsworld.com/a/between-the-lines-multiverse-special/

Since Aristotle, the philosophers say that there is only the potentially infinite. — spirit-salamander

That is a gross oversimplification. Philosophers argued for (or at least insisted on) the existence of infinities both before and after Aristotle. The extension of the universe is the most common type of purported infinity, and it was widely believed in the ancient world until the Church made Aristotelian position on the matter something of a theological dogma. For example, both Atomists/Epicureans and Stoics believed in the infinity of space - they only disagreed on whether it was uniformly populated with matter and even other worlds (Atomists/Epicureans) or whether it was void beyond our world (Stoics).

In the West Aristotelian dogma began to crumble during the Enlightenment, and in modern times the infinity of space, at least, was thought to be pretty much self-evident. That only started to change with the development of topology and differential geometry in mathematics and of General Relativity in physics.

Nowadays you would be hard-pressed to find a physicist who denies the possibility of some type of infinity on principle. Even those cosmologists and astrophysicists who propose that the universe is finite in extent do so on contingent empirical grounds, and would readily admit that there is no decisive evidence one way or the other.

Such statements by Greene as these are philosophically irritating:

"If space is now infinite, then it always was infinite. Even at the Big Bang. A finite universe can’t expand to become infinite." — spirit-salamander

Why is this "philosophically irritating"? (He is stating the mainstream position on the matter, BTW.)

That is a gross oversimplification. — SophistiCat

Okay, you're right. I was going by what I assumed was a consensus that may have existed in philosophy since Aristotle. In fact, I think if a survey were done today with academic philosophers, most would "abhor" the infinite mundane.

In the West Aristotelian dogma began to crumble during the Enlightenment, and in modern times the infinity of space, at least, was thought to be pretty much self-evident. — SophistiCat

Giordano Bruno could also be mentioned. Not directly enlightenment, but strongly influenced the Enlightenment.

Nowadays you would be hard-pressed to find a physicist who denies the possibility of some type of infinity on principle. — SophistiCat

My point was about philosophers.

Why is this "philosophically irritating"? (He is stating the mainstream position on the matter, BTW.) — SophistiCat

And therefore possibly philosophically irritating.

Okay, you're right. I was going by what I assumed was a consensus that may have existed in philosophy since Aristotle. In fact, I think if a survey were done today with academic philosophers, most would "abhor" the infinite mundane. — spirit-salamander

Depends on who you ask. I would expect that philosophers of physics, and generally those who have a handle on the mathematical and physical concepts of the last three centuries would, for the most part, be comfortable with the idea of physical infinity in some form, particularly the infinity of space. Classicists and medievalists (such as might use the words "infinite mundane") may well exhibit the prejudices of their subjects.

Giordano Bruno could also be mentioned. Not directly enlightenment, but strongly influenced the Enlightenment. — spirit-salamander

Indeed, the influence goes all the way back to Lucretius, who praised Epicurus and his doctrine of infinite worlds - which, of course, was considered heretical in Bruno's time.

My point was about philosophers. — spirit-salamander

Well, you did ask about physicists. But so far I have found that scientifically and mathematically literate philosophers are largely on the same page with physicists on this. There are, however, both respectable philosophers and physicists who are skeptical about even the least controversial forms of infinity: cosmologist George Ellis, for instance, who has been making inroads into philosophy in his later years. But they seem to be in the minority.

I can't fathom finite.

The Universe is finite but unbounded. — Wayfarer

Or consider a sphere. The mathematics of paths is such that even on earth you could sail forever at a given setting and never arrive again (exactly) on a point you had already crossed once. Or Achilleus's problem of overtaking the tortoise who is an uncountable infinite number of points ahead of him, which, if Achilleus stopped at each of any points in a well-defined set of them, he would never overtake.

The infinities, then, are always there, ready for use when you need them.

So do philosophers have to accept the actual infinite? — spirit-salamander

If you divorce yourself from the myth that is mathematics, it's easy.

Infinity can be applied in different domains. English is infinite, but English isn't French. Math is infinite, but math doesn't contain the letters of the alphabet.

On the scale of the universe, it's hard to disentangle the infinite from temporality. For instance, if the universe were infinite, then how is it that we arose as a species? It would take an infinite number of years to get to this point, yet here we are.

Then again, who knows.

English is infinite? Math doesn't contain letters?
We use infinite in math. That for sure

which would be the myth and why?

Sure. Any human language is infinite. One can write an infinite number of sentences in any language and never run out of things to say.

One can write an infinite number of sentences and never run out of things to say at all.

I said that one can write an infinite number of sentences and never run out of things to say at all.

Etc.

Yeah, I never said that math doesn't use the concept of infinite.

got it. The question would then be, whether there is only potential infinity, like this of English which can never be actualized

which would be the myth and why? — Alexandros

Consider the following...

If you would accept the notion that each object in The Universe occupies unique coordinates and is subject to unique Universal forces, then one might conclude that each object in The Universe is "one of a kind," that is, unique in and of itself. If this is indeed the case, then what exactly does "2" mean?

You have to accept the fact that mathematics sort of avoids this reality and "pretends" that 2 or 3 (or whatever number you choose) exists because it works (until it does not). So, as we go towards 0 and infinity, are we just supposed to say, "Oh well?" Apparently.

If would appear to me that man is quite a ways off from coming anywhere close to "understanding" much of anything, so coming up with concepts like infinity might be similar to tossing a dart across the bar room (backwards and standing on your head) after your twenty-third beer and hoping for a perfect bullseye.

If you would accept the notion that each object in The Universe occupies unique coordinates and is subject to unique Universal forces, then one might conclude that each object in The Universe is "one of a kind," that is, unique in and of itself. If this is indeed the case, then what exactly does "2" mean? — synthesis

There are unique coordinates? I take it you mean two objects don't occupy the same space at the same time. What does "one of a kind" have to do with counting two apples, one red and one green?

You have to accept the fact that mathematics sort of avoids this reality and "pretends" that 2 or 3 (or whatever number you choose) exists because it works (until it does not). — synthesis

No I don't. Nor should you. But we each choose our paths. You might consider joining forces with Metaphysician Undercover. His concern is the supposed equality between 2+2 and 4. :roll:

i don't understand your point, ia it that nor numbers nor infinity exist? They point out to relations which do exist and are independent from physical reality. Infinity can very well be treated in math. Regarding what you say about position and the one of a kind I think you are mixing quantifiable and qualitative. Mathematics doesn't pretend nor avoid reality, it is reality, I hope you understand the difference between the signs we use and the entities, relations and structures it points out.

The notion of an actual infinite makes zero sense if, as per my assumption, actual means what it seems to mean to wit, completed in one sense or another for it flies against the definition of infinity as being necessarily that which can't be completed.

Maths, set theoretical infinities, kind courtesy of Georg Cantor, is an altogther different story as maths is essentially an axiomatic system, anything goes so long as you don't contradict yourself within one.

No I don't. Nor should you. But we each choose our paths. You might consider joining forces with Metaphysician Undercover. His concern is the supposed equality between 2+2 and 4. :roll: — jgill

There is a fundamental problem with the concept of numbers. The numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc. But then we want "2" and "3", each to represent a distinct unity as well. So we have to allow that "1" represents a different type of unity than "2" does, or else we'd have the contradiction of "2" representing both one and also two of the same type of unity.

What does "one of a kind" have to do with counting two apples, one red and one green? — jgill

Each object in The Universe is unique because it occupies its own space. What more do you need to know?

You have to accept the fact that mathematics sort of avoids this reality and "pretends" that 2 or 3 (or whatever number you choose) exists because it works (until it does not).
— synthesis

No I don't. Nor should you. But we each choose our paths. — jgill

You are correct, you don't. Although it is easy to see the illusions that people cling to for whatever the reasons, sometimes it harder to acknowledge the ones to create the very foundations of our society. Mathematics is a system made up over time that works sometimes for some things, but has no real existence outside of this space.

If you cannot tell me what happens as zero and infinity is approached, what kind of system is that? And (again) the entire system is based on the idea that identical objects exist (when it is clear that this is not the case).

There is a fundamental problem with the concept of numbers. The numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc. But then we want "2" and "3", each to represent a distinct unity as well. So we have to allow that "1" represents a different type of unity than "2" does, or else we'd have the contradiction of "2" representing both one and also two of the same type of unity. — Metaphysician Undercover

I don't believe you even need to go there. The idea that more than one (of anything) exists is simply untrue.

The convention to use multiples is just the lazy person's way of arranging things together instead of dealing with each thing as a unique individual. Maybe it's the ultimate form of identity group-think?

i don't understand your point, is it that nor numbers nor infinity exist? They point out to relations which do exist and are independent from physical reality. — Alexandros

Really, numbers exist independent from physical reality? If you happen to be visiting a Universe that had no objects, what does "10" mean?

Infinity can very well be treated in math. Regarding what you say about position and the one of a kind I think you are mixing quantifiable and qualitative. Mathematics doesn't pretend nor avoid reality, it is reality, I hope you understand the difference between the signs we use and the entities, relations and structures it points out. — Alexandros

"We?" Who are we?

Mathematics is to reality as words are to feeling. The closest you can get to reality is through direct experience.

1. Set theory does not have, in this context, formal terms 'actual infinity' or 'completed infinity'. So for formal concerns we don't need to vindicate those notions that are not even used.

2. Set theory, in this context, does not use 'infinity' as a noun, but instead uses the adjective 'infinite'. This is important since, in this context, set theory does not point to a set named 'infinity' but rather mentions that various sets have the attribute of being infinite.

3. The non-formal notion of actual infinity does not need to resort to the term 'completed'.

4. Moreover, where does one find even a non-formal definition of 'infinity' that includes 'that which can't be completed'? If 'cannot be completed' is not part of the actual definition, then it is question begging to use this to argue that 'completed infinity' is a contradiction in terms.

6. Where does one find a definition of 'actual' that includes 'completed'?

7. Cantor's work was not axiomatized by him. It was only later mathematicians who axiomatized infinitistic mathematics.

8. The set theoretic definition of 'is infinite' is given this way.

x is finite iff there is a natural number n such that there is a 1-1 correspondence between x and n.

x if infinite iff x is not finite.

9.

maths is essentially an axiomatic system, anything goes so long as you don't contradict yourself within one. — TheMadFool

Yes, in a broad sense, and from a certain point of view, that is correct:

"It is not our business to set up prohibitions, but rather to arrive at conventions." - Rudolph Carnap

as we go towards 0 and infinity, are we just supposed to say, "Oh well?" Apparently. — synthesis

What exactly do you mean by "go towards 0 and infinity"? And who apparently says "oh well" in this context?

There is a fundamental problem with the concept of numbers. The numeral "1" represents a basic unity. an individual. The "2" represents two of those individuals together, and "3" represents three, etc. But then we want "2" and "3", each to represent a distinct unity as well. — Metaphysician Undercover

Whose concept is that? Where can I actualy read anyone explaining the concept of numbers that way?

we is people who understand mathematics. Your analogy is completely wrong, you should have learnt it at high school, I see I waste my time here