sequences and series - What is the sum of an infinite resistor ladder ...
I am a little confused about how a cyclic group can be infinite. To provide an example, look at $\langle 1\rangle$ under the binary operation of addition. You can never make any negative numbers with
This resolves your problem because it shows that $\frac {1} {\epsilon}$ will be positive infinity or infinite infinity depending on the sign of the original infinitesimal, while division by zero is still undefined. This viewpoint helps account for all indeterminate forms as well, such as $\frac {0} {0}$.
+1 that's a great answer. Especially for the last point: I agree that Zeno's paradox is basically an example of how there can be infinitely many intervals in a finite period of time. I didn't know that there was such a controversy on Zeno's intentions though; looking on wikipedia it seems clear that Zeno invented the paradoxes to support Parmedine's notion that motion is an illusion
If the vector space is finite dimensional, then it is a countable set; but there are infinite-dimensional vector spaces over $\mathbb {Q}$ that are countable as sets.
@foaly: "tensor product distributes over infinite direct products" is a question about the natural map being an isomorphism. Asking for there to be some arbitrary isomorphism is basically never the question you actually care about in practice.
I would like to have some examples of infinite dimensional vector spaces that help me to break my habit of thinking of $\mathbb{R}^n$ when thinking about vector spaces.
Can you partition an infinite set, into an infinite number of infinite sets?