Infinity

[polymath257]

Either motion is an illusion or we perform infinite tasks all the time.  You win for now, Zeno.  But this isn't over.

Cantor resolved that nicely.

The problem is that 'finite time' and 'infinite number of points in time' are not contradictory: they are discussing different things. One is the measure of an interval of time, the other is counting how many instants there are.

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How many other people think that when equations produce the answer 'infinity' that they are not deep and profound but actually are simply nonsense and point to the fact that we have no idea what's going on.

For example, the singularity which has infinite mass and density is a product of our ignorance and simply cannot exist.

Since infinity is a concept rather than a number it's pretty clear that you've done something horribly wrong with your math if that's what you're getting as an answer.

It seems to me that numbers are concepts as well, so that doesn't seem to be a relevant distinction. There are valid places where an infinite answer is both correct and reasonable (I gave one in another post). Often, that just means you need to look at the reciprocal (which in most cases will be zero).

Mathematicians deal with different aspects of infinity all the time. There are actually several different *types* of infinity (as a limit, as a cardinal, as an ordinal, etc.)
and it is quite possible, for example, to have an infinite cardinality and a finite length (or area, or volume). For some of these *types*, there are also different *sizes* of infinity. So, the infinite size of the set of counting numbers {1,2,3,...} is a smaller infinite cardinality than the infinite size of the set of decimals. There are actually infinitely many sizes of infinite cardinalities. Infinite ordinalities are related but different in specifics.