Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: jeffreyH on 30/04/2015 18:26:14
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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?
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Yes, but a different infinity.
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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?
There's no such thing as "infinitely far apart." You've made it sound as if infinite is a distance whereas it's really a concept. It means "that which increases without bound". So while you can meaningfully say that lim 1/x = 0 as x approaches infinity has meaning you can't say that A and B are infinitely far apart with it having any meaning.
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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.
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So, infinity B = infinity A – 1m?
No. That expression is nonsense.
As a mathematical concept that may well make sense to a mathematician, ..
Nope, it doesn't.
I would be the last to argue with that, but as a physical reality it has a lot in common with unicorns.
My sentiments exactly!
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.
I quite agree. It should be remembered that infinity is not a number, it's a concept.
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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 if you let x → 0.
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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.
That is incorrect. As I said above infinity is a concept and not a number. I'm certain that you know that it's a number but I'm doubtful if you fully understand precisely what the concept is. It only has to do when discussing limits. Thus
lim (x -> inf) 1/x = 0
lim (x -> inf) (1/x - 1) = 0
Here the "inf" means "increases with out limit. So "increases without limit" means identically the same thing in each example. You should know that my friend!
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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!
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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!
Thanks, pal. I appreciate you catching my bonehead moves and pointing them out to me. Very much appreciated. :)
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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.
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Easy. The universe is bigger than the distance between any two objects in it. As it must be, if it contains them.
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No. That expression is nonsense.
My point precisely.
Nope, it doesn't.
For once I’m on the same page as the mathematicians – well some of them. [:)]
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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.
You will always have this problem as long as you think of infinity as a space. It's not a number, it's not a space and it's not a time - it's infinity.
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Yes, but a different 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.
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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.
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That doesn't even come close to answering Jeffrey's question if you can have different infinities.
Different infinities is something I can have difficulty getting my head round when they get complex. However, at the simple level eg the set of all whole numbers is infinite, but it is contained within the set of all real numbers which is also infinite. I saw a debate between mathematicians who argue the set of real numbers must be larger so is a bigger infinity vs others who say infinite is infinite.
As you say, it doesn't help Jeffery!
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By definition, any infinity will be larger than any finite number or quantity.
Of course, but the statement that "any infinity will be larger than any finite number", while it is perfectly true, says nothing about the relationship between different sizes of infinity; nor does it establish that a value that is smaller than another value is necessarily finite, if you accept that there can be different sizes of infinity.
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.
Once again, we agree completely, but it still does not address the OP, because if you accept that there are different sizes of infinity, none of which can be measured, you have to use some other means of establishing sizes. Comparison seems to be the standard method, and undoubtedly this is meaningful in mathematics. Outside maths, though, it has no meaning, as you are just comparing one unknown non-quantity with another.
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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.
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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).
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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).
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.
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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.
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".
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.
I tend to think in terms of one dimension for infinity, but why do you opt for one, rather than infinite dimensions?
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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.
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".
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.
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. A closed geometry such as a sphere isn't unbounded. Therefore Alan is correct. One dimensional is probably a bad term for what I mean. I don't think we have a term for what I mean.
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It is also interesting to note the fact that as celestial bodies tend towards a more or less spherical shape then all the vectors away from them tend to be straight line vectors. Once angular momentum is introduced this curves the spacetime so may make it finite by definition.
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It isn't quite that straightforward.
It is never likely to be straightforward when trying to compare finite with infinite, unless you remember that the rules that apply to one may not apply to the other.
A straight line is unbounded so therefore a directional vector is also unbounded.
Absolutely. The trouble comes if you assume that “unbounded” = “infinite”. Pete, for one, has argued that etymologically they are the same, because infinite simply means without end. However, that ignores the fact that beginning and end are directional terms and can be interchangeable.
I don't think we have a term for what I mean.
Once you get away from mathematical infinities there is a sad, but understandable lack of terms.
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Angular momentum is a vector. It has nothing to do with the curvature of anything.
All vectors in any space can be represented by the shortest path from an origin, e.g. 40.7127° N, 74.0059° W is the vector for New York, as represented by a sector of a great circle that passes through the equator south of London. There will be an equivalent formulation for any vector in any curved space.
A straight line is unbounded so therefore a directional vector is also unbounded.
Beware of imprecise terminology. "Directional vectors", conventionally i,j,k in Euclidean 3-space, are indeed bounded - they have unit length by definition, and other vectors are specified as multiples of those elements. But a direction, as defined by the ratio of elements i,j,k, is of course unbounded.
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A great circle can be considered as unbounded but only a continuous rotation can make the path infinite. That is angular momentum. This circle has a bounded symmetry. The start and endpoints of the circumference can be set at any point. Though the symmetry is broken if we tilt the plane it sits on and rotate the new orientation..