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Sunday, November 20, 2016

Schroder-Bernstein in NBG

@ means less numerous than or equally numerous with
# means less numerous than
~  means equinumerous
/x/ means cardinal of x
e   means element of
€  means subset of
¥  is the intersection symbol

Theorem: (x @ y)(y @ x) --> x ~ y

x # y --> No bijection exists from x to y; ran(x) ¥ y is non-empty.
So x e T, where T is the class of sets each of which is bijective with x.
So x ~ /x/ e T and /x/ e On U 0. Likewise for y.
From this, we see that
x @ y equiv. to /x/ <= /y/
Now /x/ <= /y/ --> /x/ € of /y/
and likewise for /y/ <= /x/
Thus (x@y)(y@x) -> (/x/ € /y/)(/y/ € /x/)  -> /x/ = /y/ -> x ~ y


  1. Nothing original. Just showing that SBT can be kept short.

  2. Officials aware of 'gang stalking' by security agencies

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