Quote:
Originally Posted by sjastro
......Hence the intuitive idea that infinity is larger than anything we can imagine is not terribly useful.
Steven
|
Mathematicians work with different conceptual frameworks involving "infinity". Some refer to a "countable" infinity and an "uncountable" 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. mathematicians deploy some clever techniques that avoid infinities. 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.
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...
Some infinities are "bigger" than other infinities
So what type of infinite Universe are we talking about here Steven?