What do these old symbols from set theory mean? (Large E, $\cdot$ and $+$ for sets, and $\ \bar{\!\bar X}\,non\!\geqslant\frak n$)

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.)

big-E

More set multiplication, and addition with a singleton:

addition and multiplication

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.)

set "non"

Solutions Collecting From Web of "What do these old symbols from set theory mean? (Large E, $\cdot$ and $+$ for sets, and $\ \bar{\!\bar X}\,non\!\geqslant\frak n$)"

This answer collects our educated guesses about this notation.

The operations $+$ and $\cdot$ between sets mean union and intersection, respectively. The symbols $\sum$ and $\prod$ extend these to indexed unions and intersections in the obvious way. From what I can tell this notation was pretty much standard in the first half of the 20th century.

The large $E$ operator seems to stand for set comprehension (in which case it probably comes from the word ensemble) and translates to modern notation as $E_y[\varphi(y)]=\{y;\varphi(y)\}$. There is evidence for this interpretation on page 181 of the linked paper.

The double bar $\bar{\bar{X}}$ undoubtedly means the cardinality of $X$, although it isn’t clear how this is interpreted if $X$ is not well-orderable. It is possible, since the symbol always appears in relation to another cardinality, that the relations mean the existence of certain functions (probably injections) and the symbol $\bar{\bar{X}}$ by itself has no meaning.

It is likely that initial ordinals are meant when talking about cardinal numbers (written in fraktur). This is supported by the appearance of $\aleph_0,\aleph_1$ and others in the paper. Also, I expect it was much too early for the minimal rank representative workaround (which is basically Scott’s trick) to have been known.

The symbol $\mathrm{non}\leq$ stands for $\not\leq$ and similarly for other relations. Note that $\mathrm{non}\in$ also appears in the paper.

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.

The reference is too old. I only know:

$\overline{\overline{X}}$ denotes the cardinality, i.e., $|X|$.