“Set of all sets” concept is of fundamental importance because of its intimate relation to the primitive atomic formulas of axiomatic set theory: xϵy and x=y, and as the basis of various “Paradoxes.”
Set of all sets is same as SϵS which is same as (xϵx) which doesn’t exist (undefined).
I suspect this is meant to be addressed by : “Set of all sets that are not members of themselves.” However, since a set can’t be a member of itself, “sets that are not members of themselves" is redundant and is the same as "sets.”
So, “Set of all sets that are not members of themselves” is the same as “Set of all sets.”
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Does “Set S of all sets except S” rectify the situation?”
On Monday I write Sm={Am}, Where Am is all sets except Sm and Sm is a legitimate set.
On Tuesday I write: {At}={Sm,Am} and St={Sm,Am}={At}
So “Set S of all sets except S” is valid at any instant of time and therefore always valid.
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“Set of all Sets” is the set at time t of all sets at time t and doesn’t work.
CONCLUSION: “Set of all Sets” is meaningless and “Set of all S except S” has meaning.
EDIT: There may be a catch. The actual set S of all S except S varies with time . However, if what it actually is at time t doesn't matter in any discussion or derivation, it may not be relevant. Still philosophically vexing.