Certainly real wrote:They are just symbols I used to pick a particular semantic I had in mind, but I tried to understand you regarding this, so what I actually did after was ask you the following:

Ok, for the sake of argument, when discussing Infinity, forget numbers. Do the following words have any semantical value: 'Infinitesimal' 'Infinite'

I don't know where you stand on this. Does 'infinite' and/or 'infinitesimal' have any semantical value for you? If so, then any number sequence with '...' is my way of picking out those semantics.

At the time it seemed like you were just changing the subject. Since 888... and 999... have no meaning, it's pointless to ask questions about them. And since you subsequently used this undefined and meaningless notation, it did seem as though you did not read what I wrote.

The infinite definitely has mathematical meaning as the transfinite ordinal and cardinal numbers; that is, Cantor's work and its modern continuations. Infinitesimals don't exist in the real numbers, but they can be implemented in the hyperreal numbers of nonstandard analysis. But these are very different subjects than the way you're using the terms. For example in nonstandard analysis there are lots and lots of infinitesimals, not just one. And they do not in any way shape or form operate as James's fallacious infA concepts.

Certainly real wrote:Again, I am not in disagreement without our inability to empirically observe the Infinite/Infinitesimal. We don't have the tools for this. But I am in disagreement with suggestions that they have no semantical value. I am also in disagreement with suggestions that they have no relevance to our Existence. This is because we cannot accept Existence being finite as it logically implies Existence coming from non-existence.

Well, Existence is a much broader philosophical topic, on which I take no position. I'm only discussing the mathematics of .999... = 1. I don't think it necessarily means anything outside of pure math. It's a statement about the real numbers; and the real numbers arguably have no physical existence at all.

A good analogy is chess. We wouldn't ask if the knight "really" moves that way, or what it means about existence. It doesn't mean anything. It's a rule in a formal game. Likewise .999... = 1 is a logical consequence of the rules of the formal game of mathematics. That's the position I take. I don't try to figure out what it means, because it doesn't really mean anything unless you want it to, as for example using infinite series to solve some differential equation to solve a scientific problem. Even then it's just a calculational device, not a statement about the world or existence or anything else metaphysical.