As I understand N's argument it runs something like this: matter being finite and time infinite, then everything that has happened will eventually happen in the future in exactly the same way. This doctrine first occurs in the Gay Science (sections 285 and 341) and N. thought it could be proved scientifically. He subsequently restated and modified it, so it is difficult to understand his "proofs."
Most modern commentators agree that there are major flaws in the ground for the eternal return of the same, and concentrate instead on its functions to measure the spiritual health of the individual and his acceptance of life in N's overall thinking. A discussion of the ER and its place in N's thought can be found in Schacht, Nietzsche (1983), Part IV. If you are looking for objections to one statement of the theory, see Danto, Nietzsche as Philosopher (1965), pps. 204-209.
I hope this helps or at least stimulates your thinking about the topic.
John

First, I would recommend that if you do construct a thesis about eternal recurrence, you have to put the theory in the right perspective. So put eternal recurrence in the form of a hypothetical, like it was put forward in Gay Science. Suppose you learned out of the blue that the world reoccurred. Would this be a good thing or a bad thing? Nietzsche is presupposing quite a few things, like nihilism and naturalism and all that fun stuff Nietzsche is known for.

But first and foremost, this is a cosmological

A nice quick challenge might be to come up with an argument that there is no proof that time is any more infinite than matter.

VideCorSpoon: im going to have to analyze what you wrote because I'm only in highschool and you seem really educated in this matter, so thanks for opening my mind thank you kindly though

david: I was thinking of using the argument of saying that you cannot prove time is infinite while matter is not. I could probably throw that in my thesis, but, I think it would be really dificult to write one whole paragraph on that particular argument. That is something else I will have to look into. I thank you as well.

The Eternal Recurrence is both a parody and a counterexample to the traditional metaphysics and an existential imperative. I want to emphasize the latter more than the former, if only because the other posters seemed to have missed this point.

The doctrine runs like this:

Suppose that there is a fininite number of possibilities which can be instantiated in the world, but an infinite amount of time in which these possibilities can be instantiated. What follows? What follows is that everything happening now has already happened, and will happen over and over and over again an infinite number of times.

As a metaphysical doctrine, it's not very good (I'll get to this after). As an existential imperative, though, it's quite good, because it begs this question: Are you willing to live your life over and over again for all eternity in exactly the same way?



Nietzsche gives us a litmus test: Is there even a single moment of joy in your life which makes all the boredom, the bull, the suffering etc. worth living over and over again just to get to that one moment of joy?

Criticisms:

It's not very good as a metaphysical doctrine...see the Neo-Platonists' classic argument. Suppose that there are an infinite number of days preceding this one. Ok, to arrive to the present day, each particular point preceding this one must be traversed, which is impossible, as per the definition of "infinite." Therefore, we can never arrive to the present day...wait. We have arrived to the present day! Hm...I guess there aren't an infinite number of days before this one.

Another objection you could use is that it completely ignores the next life. People have lived miserable lives (at least according to worldly ways of thinking) in expectation of the joy of the next.

I'm not sure what a third objection you could use is.