Nope, and we have already been over your “self-referential” nonsense before.
Again don’t try positing your simply lack of reasoning onto others.
Well, as simple statements of equality (or the lack thereof) and not a “comparison”, they are not “fundamentally different” even though the variables involved can be. However simply equating something with it self is just trivial. While not equating something with it self just demolishes any possible consistency. It is in equating or not equating something with something else that such statements can become practical and can maintain some consistency.
Again…
So your “serial observation” is still just superfluous nonsense.
Nope, under your “Singular comparisons” you just get a trivial statement that something equals it self or the self-inconsistent statement that it does not. As both sides of those statements must be the same (by your own requirement) they must always be symmetrical.
Nope, under your “Non singular comparisons” you just get statements of equality or inequality between something and something else. Although such statements can be practical and maintain consistency there is nothing inherently “symmetric” about them. Particularly in statements of inequality, which as a result of that inequality must be asymmetrical (not the same on both sides).
Again…
So your “Serial-only observation simply can't get” anything that is, well, serial.
The Man, you simply can't get the difference between building-blocks and their use by collections.
Self-reference (whether it is True or False) is the building-block of certain ids, because it is the simultaneity of no more than one value.
Non self-reference (whether it is True or False) is the building-block of uncertain ids because it is the simultaneity of more than one value.
Since the considered framework is at least the linkage of Self-reference AND Non Self-reference, we get these different states:
a) A collection (which is actually based on both building-blocks) has an asymmetrical character if we are focused on the certain id's aspect (the simultaneity of no more than one value) of the linkage (we are able to define direction that is based on some order among certain id's).
b) A collection (which is actually based on both building-blocks) has symmetrical character if we are focused on the uncertain id's aspect (the simultaneity of more than one value) of the linkage (we are unable to define direction that is based on some order among uncertain id's, which is exactly the uncertainty of id's superposition).
c) A collection (which is actually based on both building-blocks) has several symmetrical degrees between Symmetry (superposition of ids) and Asymmetry (non superposition of ids)
d) Any given form is both global AND local state of the considered framework, for example:
Jsfisher and you clearly demonstrate that the number of given things cannot fully capture the non-trivial meaning of what Number is, simply because the traditional meaning of Number is based only on clear distinction of the involved.
By using Non-locality/Locality linkage as the qualitative foundation of Number, clear distinction is simply one of the options, for example:
Quantity 2 (and it does not matter if it is a whole number or two places of some 0.xx fraction) can't be used unless there is at least connector/connected linkage, where the connector has non-local quality and the connected has local quality.
From this qualitative foundation, Uncertainty and Redundancy are the fabric of the mathematical space that enables:
1) Strong symmetric observation of the linkage, which is resulted by superposition of identities (uncertain ids, for example: (AB)).
2) Weak symmetric observation, which is resulted by non-distinct replacement among clear ids (redundant ids, for example: (A,A), (B,B), (AB,AB) (in the last case AB superposition is ignored and taken as 'AB' notation for clear id of superposition representation)).
3) Asymmetric observation, which is resulted by clear ids (for example: (A,B))
By the way, the ( 1) , 2) , 3) ) explanation above uses the Asymmetric observation (3), but again, no one of the options above has any privilege and we as participators (and not only observers) of this mathematical universe actually design it for our purpose.
The coherence of this mathematical universe is guaranteed by its Non-local/Local linkage qualitative foundation, where Non-locality and Locality complement each other into a one fabric.
Nope, under your “Singular comparisons” you just get a trivial statement that something equals it self or the self-inconsistent statement that it does not. As both sides of those statements must be the same (by your own requirement) they must always be symmetrical.
Yep, your "bla bla ..." reasoning simply can't get Symmetry and Asymmetry as observed in the following diagram: