[sjastro]

Originally Posted by Eratosthenes

Mathematicians work with different conceptual frameworks involving "infinity". Some refer to a "countable" infinity and an "uncountable" infinity.

The correct term is countable and uncountable infinite sets.

A countable set is equivalent to the set of natural numbers N, if one can define a mapping function which is one-one and onto between the sets.
An infinite set that is not equivalent to N is uncountable.
Given that N is the common denominator in defining countable and uncountable sets you are wrong in asserting that mathematicians are using different conceptual frameworks involving infinity.

When infinities appear in derivations or calculations, they are usually a good sign that something is wrong. (Hilbert, the great German Mathematician came up with the Hotel reservation thought experiment). It's useful to get infinities to vanish before they appear in the calculations or derivations, or perhaps cancel each other out. Then they dont need to be dealt with. On the other hand one can integrate to infinity and arrive or approach a limit which is a valid result. Or Sum up a series that approaches a certain finite number, or perhaps infinity itself.

What has this got to do with the subject?
Do you seriously think that cosmologists have calculated the Universe as being infinite?
I suggest you try looking at http://en.wikipedia.org/wiki/Friedmann_equations.

Whether the Universe is infinite or not depends on the value of the curvature k used in the cosmological model under study.

There are some strange Summation series which can be described as nonsense such as {1+2+3+4+5+6......to infinity} = -1/12

Now does the series {1+2+3+4+5+6......to infinity}, a set of positive integers btw, approach a value -1/12 which is not only less than 1, but also negative (less than zero)???? And yet this result is used in several areas of Physics, including String Theory. (String Theory, Joseph Polchinski, Vol. 1, p. 22). Now one can argue that some tricks were used to arrive at this result which contradict mathematical logic and so result is not a valid result. The result is nevertheless use in other areas of mathematics and Physics. Mathematicians use a method called analytic continuation and when applied to divergent series, its possible to rationalise the -1/12 result. Anyway back to the subject at hand...

What absolute hogwash.
Analytic continuation applies to analytic functions which are functions of the complex variable z.
Sorry to disappoint you but {1+2+3+4+....} are not complex numbers, they don't even form a power series on which analytical functions of both complex and real variables are based on.

As far as {1+2+3+4+.....} converging to -1/12 well I think any reasonably smart high school student will tell you otherwise.

I suspect Mr Polchinski is stating something very different.
Why don't you provide the relevant source.

Some infinities are "bigger" than other infinities

So what type of infinite Universe are we talking about here Steven?

Try reading the thread instead of cherry picking statements.