Editing Zermelo set theory (section)

== The axiom of separation ==
Zermelo comments that Axiom III of his system is the one responsible for eliminating the antinomies.  It differs from the original definition by Cantor, as follows.

Sets cannot be independently defined by any arbitrary logically definable notion.  They must be constructed in some way from previously constructed sets. For example, they can be constructed by taking powersets, or they can be ''separated'' as subsets of sets already "given".  This, he says, eliminates contradictory ideas like "the set of all sets" or "the set of all ordinal numbers".

He disposes of the [[Russell paradox]] by means of this Theorem:  "Every set <math>M</math> possesses at least one subset <math>M_0</math> that is not an element of <math>M</math> ".  Let <math>M_0</math> be the subset of <math>M</math> which, by AXIOM III, is separated out by the notion "<math>x \notin x</math>".  Then <math>M_0</math> cannot be in <math>M</math>.  For

# If <math>M_0</math> is in <math>M_0</math>, then <math>M_0</math> contains an element ''x'' for which ''x'' is in ''x'' (i.e. <math>M_0</math> itself), which would contradict the definition of <math>M_0</math>.
# If <math>M_0</math> is not in <math>M_0</math>, and assuming <math>M_0</math> is an element of ''M'', then <math>M_0</math> is an element of ''M'' that satisfies the definition "<math>x \notin x</math>", and so is in <math>M_0</math> which is a contradiction.

Therefore, the assumption that <math>M_0</math> is in <math>M</math> is wrong, proving the theorem.  Hence not all objects of the universal domain ''B'' can be elements of one and the same set.  "This disposes of the Russell [[antinomy]] as far as we are concerned".

This left the problem of "the domain ''B''" which seems to refer to something.  This led to the idea of a [[proper class]].