Bell's impossibility theorem, equivalence classes: SOS

[Gordon Watson]

:
1. I am an engineer seeking to fully comprehend Set Theory, Logic, Probability; especially as it relates to equivalence/classes, class invariants, etc.,

in the context of an essay that I have posted at http://quantropy.org/12/ [6 pages, 194 Kb, 31 references].

2. The essay relates to John Bell's famous Impossibility Theorem (widely regarded as the most profound discovery of science). The essay is titled: "Bell's theorem refuted in line with Bell's hope and Einstein's ideas".

3. Bell's theorem (BT) is essentially a mathematical impossibility proof -- part of a long line of such "proofs" in quantum mechanics. Bell refuted many earlier "proofs" -- then produced his own --- then hoped for a rebuttal of his own theorem -- as outlined in the first paragraph of the essay.

4. The subject of a possible refutation of BT is controversial in physics; rating high on most crackpot meters. So I could be wrong.

5. BUT, not being crackpot; maybe just wrong: I am to meet with two university mathematicians on 27 December, to discuss the above essay: SO, I want to be sure my maths (as it relates to Set Theory etc. in my essay) is correct in every respect.

6. I also want to introduce the Class Invariants into that essay [see the following post] -- so that my concepts are clear and cover the whole range of Set Theory that is applicable to the ideas in the essay. Such inclusions would serve to clarify the essay, and make it more familiar to mathematicians who may be concerned that they are "not into" the related physics.

7. There is no need to kitty-foot around with me, so CRITICAL comments plus TYPO-identification will be welcomed.

PS: As given in the essay, my direct email is: gorstewat@gmail.com

Thank you.

 

[Gordon Watson]

:
<SNIP>

6. I also want to introduce the Class Invariants into that essay [see below] -- so that my concepts are clear and cover the whole range of Set Theory that is applicable to the ideas in the essay. Such inclusions would serve to clarify the essay, and make it more familiar to mathematicians who may be concerned that they are "not into" the related physics.

7. There is no need to kitty-foot around with me, so CRITICAL comments plus TYPO-identification will be welcomed.

<SNIP>

Is this correct, please?

Question 1:

1. Let [A+} denote an equivalence class (EC), where the notation [.} signifies than an EC is both a class and a set.

2. THEN: If ~ is an equivalence relation on [A+}, and [a]:x is a property of all elements of [A+}, such that whenever x ~ y, [a]:x is true if [a]:y is true, then the property [a]: is well-defined or a class invariant under ~.

3. Given: x ~ y in [A+} and [a]:x = A+ when [a]:y = A+; where [a]:y is defined as response to test [a] on y; etc.

QUESTION: Then [a]: -- "response to test [a] on" -- is well defined and a class invariant of [A+} under ~. ?

Question 2:

In the essay, in the paragraph before equation (4a), we see: ... "a particle may belong to more than one EC."

To be clear here, I am about to make the following modifications:

Eq. (3c) to read: W = H U M = $$\Omega$$ U $$\Omega'.$$

Eq. (3d) to have "= $$\Omega$$" added on the RHS: ..., N} = $$\Omega.$$

Eq. (3e) to have "= $$\Omega'.$$ " added on the RHS: ..., N} = $$\Omega'.$$

Then define my equivalence relations ~ on $$\Omega$$ and $$\Omega'$$ separately; NOT W as a whole as I have done.

QUESTION: Then my probability discussions continue correctly. For they discuss the Probability that an element of an EC defined on one set (say $$\Omega$$) is also an element of an EC defined on another set (here $$\Omega'.$$) ?