Is the Density of Points in an Interval Twice as Much with Double Length?

[Stephanus]

[Mentor's Note: Thread moved to Astrophysics since it concerns black hole singularities]

Dear PF Forum,

I am just wondering about this. See if anyone can help me.

Is $\frac{2}{0}$ is twice as much as $\frac{1}{0}$?
Is the above question wrong?

Is the number of points in 2 cm lines twice as much as the number of points in 1 cm lines?
Is the above question wrong?

Thanks.

 

[Simon Bridge]

2/0 and 1/0 are both undefined so the question about their relative sizes is meaningless.

To compare infinite sets the trick is to see if you can assign every member of one set to a member of the other set ... so whole numbers and counting numbers have the same "size" (more accurately: the same way of being infinite) but counting numbers and real numbers are not - with real numbers being "more infinite" or a higher order of infinity.

Look up "Aleph notation" and "Cantor" and you'll be able to answer the question about 1cm and 2cm line segments yourself.

 

[Stephanus]

2/0 and 1/0 are both undefined so the question about their relative sizes is meaningless.

To compare infinite sets the trick is to see if you can assign every member of one set to a member of the other set ... so whole numbers and counting numbers have the same "size" (more accurately: the same way of being infinite) but counting numbers and real numbers are not - with real numbers being "more infinite" or a higher order of infinity.

Look up "Aleph notation" and "Cantor" and you'll be able to answer the question about 1cm and 2cm line segments yourself.

Hi Simon Bridge, glad to see you again.
Actually I want to know the answer of Black Hole singularity.
Is the singularity of 20 solar mass Black Hole is twice as much as the singularity of 10 solar mass Black Hole. But this should belong to other thread.
Thanks for your answer.

 

[Simon Bridge]

Is the singularity of 20 solar mass Black Hole is twice as much as the singularity of 10 solar mass Black Hole. But this should belong to other thread.

(my emph) "twice as much" what?

You cannot have twice as much by itself it has to be twice as much of something.
eg. 4 s twice 2 but it is not twice as much as 2; but 4 cups of water is twice as much water as 2 cups of water.

Remember too that the singularity is not a physical object but a mathematical term that means "undefined".

 

[Stephanus]

(my emph) "twice as much" what?
Remember too that the singularity is not a physical object but a mathematical term that means "undefined".

Twice as much "size"? "density"?
I don't know if size and density is the correct question.
Thanks for your answer. Perhaps I should go back to Cosmology Forum.
It's just that the word "singularity" sounds similar to "infinity"
See you in Cosmology Forum before my thread being deleted by the moderator.

 

[Simon Bridge]

OK - the reason there is a singularity at the center of a non-rotating black hole is because the equations do silly stuff as the radius approaches zero.
i.e. The density approaches infinity and the volume approaches zero.

In terms of volume they are both the same: 0.
The comparison of densities - is the density of one twice the density of the other? That's a question for Cantor - but it is unclear what it would mean.
I'd be inclined to think it is more like comparing 2/0 with 1/0: meaningless due to "undefined".

 

[Mark44]

Is the number of points in 2 cm lines twice as much as the number of points in 1 cm lines?
Is the above question wrong?

There are the same number of numbers in the interval [0, 2] as there are in the interval [0, 1]. Two sets have the same cardinality if there is a one-to-one, onto function that numbers in one of the sets with those in the other set. Although the interval [0, 2] is twice the length of the interval [0, 1], the two sets have the same number of points in them.

 

[Simon Bridge]

Although the interval [0, 2] is twice the length of the interval [0, 1], the two sets have the same number of points in them.

heh heh heh so does that mean the linear point density of the interval [0,2] is half that for the interval [0,1]?
But I was hoping Stephanus would discover about cardinality and application to infinite sets... the problem, though, seems to be the application to the singularity question.