obsrvr524 wrote: \(0 + 0 + 0 + \dots\) - that - is an infinite SERIES. It has a final sum which is the "sum of an infinite series".

Some people might say, "infinite sum" to mean the sum of an infinite series. It's shorter. But it is misleading and technically wrong.

That might be true but it doesn't seem to be a relevant correction (it's more of an effort to help me better express myself.) The point is that what we're dealing with here is an expression involving an infinite addition of terms that exists outside of time rather than inside of time. What I am asking is "What does it mean for such an expression to never stop?" To say that such an expression stops or that it never stops is a figurative use of the word "stop" since the word "stop" can be literally applied only to things that exist in time. Since such an expression can be represented as a sequence of terms (not merely as a set of terms), the question can be reframed as "What does it mean for a sequence to have no end?" That's the question I asked you, right? And it is better to ask Silhouette the same question if it's possible to do so. So that's what I did.

Magnus wrote:Let's say that the word "infinity" represents the number of natural numbers.

Observer wrote:[wrong word but ok]

Actually, the truth value of that statement can be barely contested (: That's because I provided a PROVISIONAL definition -- basically my own -- to be used for the duration of that particular argument I presented. I did NOT claim that's the official definition. (The official definition, I believe, is more general than that.)

No. When you added the "a", you started a new sequence.

We didn't add \(a\) to a different sequence (one that was moreover empty.) Rather, we added it to the existing sequence. Thus, we didn't really start a new one.

"a" does not represent the next higher number

\(a\) is not a number at all, it's a letter. It's a letter that we added to a sequence of natural numbers. The index of the place it occupies in the sequence, however, is a number and that number is basically the number of natural numbers. (The only thing I am not entirely sure about is whether that number is the number of natural numbers OR the number of natural numbers plus one. The reason being that the index of the first place in the sequence is \(1\) rather than \(0\). In fact, I am far more inclined to believe its actual index is infinity plus one.)

By the way, assuming that this can more easily change your mind, James had no problem using infinity as an index. (There was a discussion between him and Carleas on whether or not the set of even natural numbers is in one-to-one correspondence with the set of natural numbers -- or something like that.)