True Sentence in Calculus of Classes: Individual Domain and Cardinality I was reading Woodger's translations of Tarski's papers from 1923 to 1938, in specific, the one regarding the Concept of Truth in Formalized Languages from 1931, section III relating to Calculus of Classes.

I've had a hard time understanding difinitions 28, 29, 30 and 31. The
  first two state:
(i) Def. 28: ${\epsilon }_{k}=\overline{(\bigcap_{k+1}^{}{\iota
}_{k,k+1}}).\bigcap_{k+1}^{}(\bigcap_{k+2}^{}{\iota
}_{k+1,k+2}+\overline{{\iota }_{k+1,k}}+{\iota }_{k,k+1})$
and
(ii) Def. 29: $\alpha = \bigcap_{1}(\bigcap_{2}\iota_{1,2}+\bigcup_{2}\iota_{2,1}.\epsilon_{2})$

These are explained in the following paragraph, in the following way:

(iii) "...${\epsilon }_{k}$ states that the class denoted by the variable
  ${v}_{k}$ consists of only one element; the sentence ${\alpha}$, (...)
  states that every non-null class includes a one-element..."

I was translating the equations and managed to produce something like this:

(1) Def. 28 (what I could translate): ${{{\epsilon }_{k}\equiv \left\{{\exists
{v}_{k+1}|{v}_{i}\not\subseteq{v}_{k+1}} \right\}\wedge \left\{\forall
{v}_{k+1}:({v}_{k+1}\not\subseteq{v}_{k})\vee \left\{\forall
{v}_{k+2}:({v}_{k+1}\subseteq{v}_{k+2})\right\}\right\}}}$
(2) Def. 29 (what I could translate): $\alpha \equiv \forall v_{1}:\left \{\exists
v_{2}|\epsilon _{2}\wedge(v_{2}\subseteq v_{1})\right \}\vee \left
\{\forall v_{2} : v_{1}\subseteq v_{2} \right \}$

And I'm not very sure if this translations are correct. On the other side, I was thinking I should arrive to something more like:

(3) Def. 28 (about what I'd expect): $\boldsymbol{{\epsilon
 }_{k}\Leftrightarrow1=\overline{\overline{{v}_{k}}}}$
(4) Def. 29 (about what I'd expect): $\boldsymbol{\alpha \Leftrightarrow \left \{
 \forall \upsilon_{2}: \upsilon_{2}\not\equiv \left \{\varnothing\right
 \}\leftrightarrow  \left \{ \exists \omega \in \upsilon_{2} |  1=
 \overline{\overline{\omega}} \right \}\right \}}$

But now I'm thinking I'm really lost, dont know if the translations are fit. Can someone please help me confirm if (1) and (2) are correct interpretations of the definitions of Tarski (i) and (ii)? Are (3) and (4) correct interpretations of Tarski's paragraph (iii)? Can someone point out how is it possible to connect the meaning with the definitions?
Thanks!!
 A: We have to note that [see page 193] :

$\bigcap_2 i_{1,2}$

is satisfied only by the null-class. 
In modern notation, it is equivalent to: $\forall x_2 \ (x_1 \subseteq x_2)$.
Thus, rewriting Def.29 as (that fits with your translation):

$\forall x_1 \ [ \lnot \forall x_2 (x_1 \subseteq x_2) \to \exists x_2 \ (x_2 \subseteq x_1 \land \epsilon_2)]$

we have that, assuming that $\epsilon_2$ states that "the class denoted by the variable $x_2$ consists of only one element", the sub-formula following the first universal quantifier means:

"if the class denoted by the variable $x_1$ is not the null-class, then there is a class included in it that is a one-element class".

Thus, the complete formula means:


"every non-null class includes a one-element class as a part." 



Now for the "easy part" : Def.28.
The formula is made of to conjuncts, where the left one is: $\lnot \forall x_{k+1} \ (x_k \subseteq x_{k+1})$, which means: "the class denoted by $x_k$ in a non-null class".
The right conjunct is:

$\forall x_{k+1} \ [ \lnot \forall x_{k+2} (x_{k+1} \subseteq x_{k+2}) \to (x_{k+1} \subseteq x_k \to x_k \subseteq x_{k+1})]$

which means:


"every non-null class included into the class denoted by $x_k$ is identical with it"


and this holds only if the class denoted by $x_k$ has no "proper" subsets.
