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If we have two objects that are infinitely far apart and one object moves one metre nearer the other are they still infinitely far apart?

Yes, but a different infinity.

So, infinity B = infinity A – 1m?

As a mathematical concept that may well make sense to a mathematician, ..

I would be the last to argue with that, but as a physical reality it has a lot in common with unicorns.

I agree with Pete, “There's no such thing as "infinitely far apart."” I admit that it is a term I have used, but only in an effort to establish that any attempt to measure (non-mathematical) infinity is meaningless.

By defintion you cannot measure infinity. You can invent as many infinities as you like, so you could indeed define A = 1/x and B = ((1/x) - 1). Both definitions are valid and for all nonzero values of x, they are distinct and physically realisable, so you have defined two different infinities.

Absolutely, but that wasn't what I wrote or meant! My x's were tending to zero. Aha! It seems I missed out the crucial bit, now re-edited!

No. That expression is nonsense.

Nope, it doesn't.

The point of my question was to establish that infinity can never be reached and so the question is moot. If we say the universe is infinite and no two objects can be infinitely far apart how do we resolve this? No where in the universe can any two objects be infinitely far apart even though it resides in an infinite space. There appears to a circularity in the concept of infinity.

Easy. The universe is bigger than the distance between any two objects in it. As it must be, if it contains them.

That doesn't even come close to answering Jeffrey's question if you can have different infinities.

By definition, any infinity will be larger than any finite number or quantity.

If you can identify two objects, let alone actually measure the distance between them (a) they can't be infintely far apart and (b) any universe, domain or manifold that contains them must have at least one dimension greater than the distance between them.

The number of rational vectors within a sphere is infinite, even if the radius of the sphere is finite. The number of irrational vectors is also infinite. It is left a an exercise to the reader to determine which is the larger, but either could be called a spherical infinity because the symmetry of vector distributions is spherical. That is why balloons inflate the way they do (at least in physics textbooks).

What we can possibly say is that infinity cannot be based on a particular geometry as that is by definition bounded by surfaces. So a spherical infinity should not exist, nor a cubic or triangular infinity. This then leads on to the geometry of spacetime. As spacetime has geometry then it should not be infinite as infinity is not geometrical.

Which highlights that infinity can only consist of 1 dimensional and not 3 dimensional geometry. The 3 dimensional space is then an expansion of this 1 dimensional space.

Quote from: JeffreyWhat we can possibly say is that infinity cannot be based on a particular geometry as that is by definition bounded by surfaces. So a spherical infinity should not exist, nor a cubic or triangular infinity. This then leads on to the geometry of spacetime. As spacetime has geometry then it should not be infinite as infinity is not geometrical.I like that line of thinking; it took me quite a long time to reach that point (lots of dog walks), and even longer to get beyond it to something resembling Barbour's "Platonia". QuoteWhich highlights that infinity can only consist of 1 dimensional and not 3 dimensional geometry. The 3 dimensional space is then an expansion of this 1 dimensional space.I tend to think in terms of one dimension for infinity, but why do you opt for one, rather than infinite dimensions?

It isn't quite that straightforward.

A straight line is unbounded so therefore a directional vector is also unbounded.

I don't think we have a term for what I mean.