Re: { }

>>"Also I find it very difficult to see why { } is not equal to
>>nothingness."

Do you mean "...equal to nothingness"? Well the empty set is a subset
of every set because it logically follows from this argument:

[NOTATION: " !< " means not a subset of; so A !< B, means A not a
subset of B.]

Suppose { } !< of all sets A; then there exists a set B such that { } !
< B. So by definition of not being a subset os some set, there is an
element x in { } such that x is not in B. But by definition { } has no
elements; so this contradicts { } !< B, so our original hypothesis must
be false. So { } < every set A.

The trouble is tat everyday langauage is sometimes ambiguous and if
strict logic is not applied, one finds certain definitions and theorems
"impossible". Just stick to logical arguments and you will be fine.

Leo

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