eh

I think in order for singularities to exist, space must be continuous.

LKS_Blade

if there are an infinate number of points between points A and B, wouldnt there be an infinate number of infinatly small intervals between them? And if something is infinatly small... does it still exist?

ex-xian

The set of prime numbers is an infinite sets. That was proved by Euclid thousands of years ago.

Big Spoon

The prime numbers do indeed form a countably infinite set. A countably infinite set is a set that can be put into a one-one correspondance with a proper subset of itself, or with the integers. The real line is not countable.

AdamWho

I think that the mathematical infinities refered to here are conceptual not actual.

ex-xian

Says who? The proof that there are infinitely many primes rests on assuming that there are a finite number and showing a contradiction. If the finite of the assumption is actual, and the contradiction is actual, why isn't the resulting infinity actual?

The Calculus rests on actual infinities. Calculating an integral as an area under a curve involves adding up rectangles while letting the width of the rectangles approach zero, so that you are summing an infinite number of rectangles to calculate the area.

Keith Russell

I'm not a scientist, and I haven't done a great deal of science reading, but in the little that I have done, always states that if an inquiry yields a result of 'infinity', an error has been made.

K

ex-xian

Not so. Gravitation force is inversely proportional to the distance between the objects. Let two masses, m and n, be seperated by a distance of r. The force between them is

F = G(m*n)/r^2 (times some unit vector for direction, but that can be ignored right now)

where G is the gravitational constant. As r -> infinity, F->0. So if the bodies were seperated by an infinite distance, the force would be 0.

I've only had a half-semester in quantum, so I don't know for sure, but Hawking and Penrose have found solutions to Einstien's field equations that imply the beginning fo the universe as a quantum singularity.

ex-xian

That's probably the result of division by zero. Division by zero is either undefined or infinity. Specifically, 0/0 is a special indeterminate case similiar to infinity/infinity. In some cases, there are mathematical tools to deal with them, in others, the result must be scrapped.

In some cases, infinity is a perfectly legitimate answer, such a the sum of a divergent series.

Clavius

Wittgenstein considered Cantor's theory of transfinite numbers to be useless in a philosophical discussion on infinity. Wittgenstein saw Cantor's work as kind of a "Mathematics gone wild" exercise. It was a cancer on mathematics, which was part of a general sickness.

Isn't this entire discussion thread based on the relationship between a mathematical infinity (ie. limit in calculus) and an actual infinity, if such is possible.

I gather from mathematics, that really this idea of an "infinity" is really irrelevant to begin with. Mathematicians don't work with "Infinity" but with the analysis of limits. To a mathemtician, "infinity divided by infinity" is like "taste divided by smell".(I'm not a mathematician).

at Euler's grave,CLAV