Negation of the axiom of choice and Evil

Beside the particular case of the axiom of choice

CC(2 through m), countable choice for sets of n elements

n=2 through m, there is the particular case where the whole

axiom is negated, no choice at all.

In “All things are numbers” in Logic Colloquium 2001, and in “About

the strength of Evil” in ASL Winter Meeting 2004 2005, I wrote

that the numbers of attributes of Evil are infinite sums and products

of integers.

If there is no choice at all, which is a standing valid case,

as I am changing my mind to include this case beside

CC(2 through m),the only infinite sums or products well defined

(existing) is the infinite sum 1+1+1+… which is equal to

aleph zero.

So, Evil is really restricted.

The case where the whole axiom of choice is negated is not a

parametric case.

We apply it to ethics.

Another way for clarification is the following :

Let us consider the infinite cardinal product XiAi Ai being sets of

attributes of Evil of size the same or different integers, without the

ai being urelements.

In some model of the negation of the axiom of choice, the product is

void.

So, the number of combined attributes of Evil is not like with the axiom

of choice an infinite product of integers (which is the cardinality of

the continuum), but 1+1+1+…(which is aleph zero).

As for the numbers of attributes of Good, I wrote in “All

things are numbers” in Logic Colloquium 2001 that they

are Dedekind cardinals.

A criticism that I am making to myself is that, in such a case,

attributes must be indistinguishable as urelements are.

Attributes would go like : Good, Good, Good, …

Adib Ben Jebara.