Intereting Posts

Set of non-units in non-commutative ring
Is axiom of choice required for there to be an infinite linearly independent set in a (non-finite-dimensional) vector space?
What's the nth integral of $\frac1{x}$?
How to study for analysis?
Combinatorial prime problem
Solving a Diophantine equation of the form $x^2 = ay^2 + byz + cz^2$ with the constants $a, b, c$ given and $x,y,z$ positive integers
Gradient, tangents, planes, and steepest direction
General solution of second-order linear ODE
Which harmonic numbers have prime numerators?
Finding invertible elements in $\mathbb{Z}/m\mathbb{Z}$
For every infinite $S$, $|S|=|S\times S|$ implies the Axiom of choice
How to say that the “geometry is the same” at every point of a metric space?
Weak categoricity in first order logic
Proof verification: $\langle 2, x \rangle$ is a prime, not principal ideal
Let $E$ be a Banach space, prove that the sum of two closed subspaces is closed if one is finite dimensional

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.

- Why don't we use base 6 or 11?
- Polarization: etymology question
- What's an example of an infinitesimal?
- How did Bernoulli approximate $e$?
- Why is Lebesgue so often spelled “Lebesque”?
- What is the origin of “how the Japanese multiply” / line multiplication?

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.)

- Why the terminology “monoid”?
- Once and for all - “Rational numbers” - because of ratio, or because they make sense?
- Conventions for function notation
- Why sum of two little “o” notation is equal little “o” notation from sum?
- Notation for “vectorized” function
- What are some examples of notation that really improved mathematics?
- Why is $\cos(x)^2$ written as $\cos^2(x)$?
- Who discovered this number-guessing paradox?
- Why are the order-of-operations conventions good?
- Weierstrass M-Test

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., $|X|$.

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- Showing that a graph has a cycle length less than something
- Find irreducible but not prime element in $\mathbb{Z}$
- Evaluating $\int_0^\infty \frac{dx}{1+x^4}$.
- Finding the spanning subgraphs of a complete bipartite graph
- If $\mid a_{jj}\mid \gt \sum_{i \neq j} \mid a_{ij} \mid$ then vectors $a_1,\dots ,a_n \in \mathbb{R}^n$ are linearly indendent.
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- Characteristic function of Normal random variable squared
- how to show a series is not uniformly converges?
- Can a function be applied to itself?
- Explanation on arg min
- About the order of the $L^1$ norm of the Dirichlet kernel.