Undergrad What are the implications of infinity?

[bahamagreen]

A sequence is a mathematical object. Let's take two examples:

$1, 1/2, 1/4, 1/8 \dots$

This sequence has the property that it converges to a limit - in fact, it converges to 0. This is a mathematically well-defined property.

Note that the limit is not part of the sequence; it is a property of the entire sequence. In particular, it is not true that "the sequence eventually reaches 0 at infinity". This may be intuitively how non-mathematicians think of a limit. But, it is mathematically not sound.

However, when modelling a physical process by such a sequence, this allows you to say that eventually the physical process reduces to 0. To be more precise, you could say that eventually the process reduces to the point where it is no longer measurable and is in all practical terms 0.

Your sequence:

$1, 0, 1, 0 \dots$

Has the property that it does not converge. It has no limit.

If you model a physical process by this sequence, then there is no natural end state. There is no mathematical limit that could be assigned to the process once it has terminated.

This means that is you want to include the end state, then you must change the model somehow.

In any case, this says nothing paradoxical about either the mathematics or the physics. Simply that you don't have a fully viable model of the physical situation that includes an end state.

My sequence was 1, -1, 1, -1...
I'm putting it as a function where the count number of the individual numbers in the sequence are not counted as 1, 2, 3... but something like 1/1, 1/2, 1/3... so that although the numbers in the sequence do not have a limit, the count numbers do. In this kind of function where the numbers of the sequence are y and the numbers of the count (indexes) are x, this makes x approach a limit. The numbers of the sequence do not have a limit, but the counts do...

I think this is analogous to a version for Zeno's paradox where instead of counting the decreasing distance measures we count the runner's steps and have him run with increasingly smaller running strides of increasing frequency, and asking which foot hits the ground after the limit. In both the original and in this version of the paradox, Zeno does run through the limit, so asking about what happens past the limit is legitimate... still in the domain of x isn't it?

 

[jbriggs444]

run with increasingly smaller running strides of increasing frequency, and asking which foot hits the ground after the limit.

We should probably take the discussion of supertasks, convergence and infinite cardinalities to another thread.