I haven't divided anything, that's the point.

I start with something I will give a symbold, a. a is a number defined such that:

0 = { }

a = {0}

Then I say that:

1 = {a, a, a}

And that:

2 = {a, a, a, 1, 1, 1}

And that:

3 = {a, a, a, 1, 1, 1, 2, 2, 2}

There's no division involved.

Exactly, there is no division involved. That is what I tried to explain to origin but he prefers to ridicule rather than try to understand the subtleties under discussion.

Anyhow, as to your exercise per se, I already explained that merely making symbols for things is not actually identifying/proving them as numbers/strings in a fractional state, now is it? Especially if that state is not YET (as you agreed above) identified by actual division operation/process. Yes? That was my point. Mere trivial constructions/de-constructions based on symbolic 'definitions' of multi-unitary 'composite unitary' symbol/set is not actually proving the identity/behaviour of the fractional number/string in the first instance, is it?

Really? You still can't figure this one out?

Let me ask you a question that gets back to the root of what division is: If I have no pies, and I share them out equally among no people, how much pie does each person get? That's right, none! An answer which is in and of itself consistent with the rules of multiplication and division.

I think where it starts to ge a little messy (and I'm sure Rpenner will likely chip in with a few examples here) is when you start considering functions like 1/x where the limit as you approach zero is ±∞ rather than zero with x=0 representing a discontinuity. And so, they choose to leave it undefined rather than define it as zero.

The issue is not the division, as we agreed above it is not yet involved; it is the AXIOMS regarding zero treatment according to those axioms; which axioms treat like/like constructions (such as 3/3, 8/8, 9/9 etc) as UNITARY as per the definitions. When that like/like construction comes to the example of 0/0, then we have a problem. IF that 0 is a number on the number line, then it has a VALUE on the number line as a POINT on that line. Yes? If we have any LIKE POINT/LIKE POINT 'value' then they are equal' and so the expression of 0/0 should be UNITARY also, just like all the others.

BUT as you now bring in the REALITY into the maths (which I have been trying to do all this time), we find that the 0/0 is a NON-action at best, and an axiomatic quandary at worst. Which is why mathematicians who do NOT resort to Reality to inform the axiom will have no choice in the matter but to call it "axiomatically undefined", since they have no other way to 'handle' that 0 as a number/value under the like/like contexts. Naturally I have the change to the axioms needed for 0 to be consistently treated without such "undefined" situations arising at all. I will be publishing that too soon.

The other aspect of 0 as a placeholder is already evident in the NOTATION convention which leas to all the problems of expressing symbolically a 'something' which is not YET a 'result' of any 'division' that has been 'completed'; let alone any division which has yet to BE started when we use the 1/3 symbol for what we WANT to do but have not YET done UNTIL we 'generate' the conventional notation FRACTIONAL string of 0.333... which we call 'a number/point' on the number line BUT have not yet done anything more than 'equate' 1/3=0.333... AS A NOTATIONAL STATEMENT rather than a true 'complete' mathematical process on EITHER/BOTH sides of that purely notational '=' sign.

These are the subtle aspects which get lost in the cross-purpose exchanges where things YET to be 'done' are ASSUMED 'already done'. So I will leave it at that and leave you with the last word on our exchanges on these matters, Trippy, everyone.

Anyhow, thanks again for all your trouble and courteous contributions to these discussions, Trippy, everyone. I have all I need to finish my complete and consistent, reality-inter-connected 'contextual maths and physics' theory/publication. I will say everything I have to say on this and other matters in that. Until then, cheers to all genuine seekers of scientific and mathematical reality-contextual understandings!

I probably won't be posting again for many weeks. So thanks again and good luck and good thinking to you all.

PS: By the way, Trippy, everyone; just for your information I have 'demolished' the 'Hilbert Hotel' device for 'explaining' infinity and its so-called 'properties' in the context currently being applied/manipulated by the maths. Reality has come to the rescue again to make sense of it all! Oh, and I also have identified what the 'infinitesimal' really is in reality (so QQ will be pleased to have the answer to his Zeno Paradoxes and his 'Reducing Sphere' exercises!). It will all be in my book under the Physics/Maths 'loose ends' section.