TITLE: What do these old symbols from set theory mean? (Large E, $\cdot$ and $+$ for sets, and $\ \bar{\!\bar X}\,non\!\geqslant\frak n$)
QUESTION [5 upvotes]: So, I'm trying to prove the theorems in this paper by Tarski:
On Well-ordered Subsets of any Set, Fundamenta Mathematicae, vol.32 (1939), pp.176-183
but it is from 1939, and I don't recognize a few of the notations from a modern set-theory perspective. Here are the relevant snippets.
The big-E notation, and set multiplication: (I'm okay with using subtraction to denote set complement.)

More set multiplication, and addition with a singleton:

And the double overbar, fraktur script, and "non" before an inequality: (I assume the double overbar means cardinality, and the fraktur $\frak n$ refers to a cardinal number, although I'm not sure if these cardinals are initial ordinals or the ZF variety using the rank-minimal elements of an equipotence class.)

REPLY [6 votes]: I would be very surprised if the Fraktur cardinals are intended to be well-orderable.  In those days, it was fairly common to distinguish between "cardinals" and "alephs", the latter being the cardinals of well-orderable sets.  Equations, inequalities, and algebraic operations for cardinal numbers were treated simply as abbreviations for statements about sets of those cardinalities.  I don't think people worried about getting actual sets to serve as cardinal numbers.
The double-bar notation for cardinal numbers goes back to Cantor. His idea was that, if you have a set $X$, you obtain its cardinal by two abstractions.  First, you abstract from what particular elements constitute the set; second, you abstract from the order in which the elements are given. (That implies, in contrast to how we think of sets nowadays, that a set is to be regarded as "given" with an ordering of its elements.)  Cantor represented these acts of abstraction by bars.  So a single bar was his notation for the order-type of a set, and double bar meant cardinality.