Doron,
All the words you use are like those melting watches in Dali's painting.
Fine, but to communicate with others you need to define and settle the usage.
When you use a word in a way that seriously departs from recognized usage, you need to admit what you are doing and clearly state your unique usage.
You don't need to tell others their usage is wrong.
You are using the word "Cardinality" in a way that is very unfamiliar to English speakers. It's a word associated with numerality and "how many?"
You are using it to primarily speak of what's present on the "stage."
This is such a departure from the ordinary meaning of the term that people can't but be confused and misunderstand.
You are The Master of what you write. But if you are writing for others, you have to be able to write for your reader.
The potential for confusion runs very deep here, because numbers are terms of numerality. We commonly use numbers to designate quantities or order in a series.
Your Organic Numbers designate the presence of entities prior to quantity.
At that point you need to be very careful in how you pare your words, for your reader's sake.
Because its very easy to misunderstand your parallel aspects of organic number for .. well, numbers, denoting quantity.
When it comes to getting your idea across, Humpy's attitude is conterproductive.
I am using the "how many?" question and so does Standard Math.
Standard Math puts arbitrary limitations on this question, for example:
This framework is based on arbitrary limitations:
|{{a,b,c,…}}|=|{{}}|=|{{{}}}|=|{{{{}}}}|= …= 1 and it can't deal with total-existence, and can't deal with Complexity, because the considered is limited to the first level of existence of the measured objects.
My framework does not use these arbitrary limitations, for example:
|{{}}|= 0+1 = 1 <
∞ (= the cardinality of the stage).
|{{{}}}|=|{{},{}}|= 0+1+1=0+1+0+1=2 <
∞ (= the cardinality of the stage).
|{{},{{}}}|=|{{{{}}}}|=0+1+0+1+1=0+1+1+1=3 <
∞ (= the cardinality of the stage).
|{{a,b,c,…}}| = |{N}| = |N| + 1 <
∞ (= the cardinality of the stage).
It has to be stressed that it is possible to use any limitations that we wish on the existence of things, but then it must be clearly understood that our framework is a partial case of the general existence of things.
The problem with Standard Math is that it claims that its limitations are actually the general case, and this is exactly the problem that I expose at the foundations of the Standard approach of the mathematical science.
As a result it is clear that Standard Math is nothing but a partial case of the mathematical science, and I show this fact right at the foundations of this science.
Apathia there are no half-ways or short-cuts to paradigm-shifts and what I show in this thread is not less than a paradigm-shift.
In order to get a paradigm-shift one simply has no choice but to go beyond his current paradigm of already agreed definitions, terms, axioms etc… and explore unfamiliar frontiers. It is defiantly not an easy task but, as I said, there are no half-ways or short-cuts here, and a lot of guts are needed in order to do that, because one finds himself unarmed and without his familiar knowledge, in a foreign space.
