0
$\begingroup$

Can a pure class be an atom? i.e., can there be a pure class that is a member of another class?

This question is from Maschovakis' Notes on set theory (2nd edition).

He says that "We assume at the outset that there is a domain or universe $\cal W$ of objects, some of which are sets, and certain definite conditions and operations on $\cal W$. We call the objects in $\cal W$ which are not sets atoms, but we do not require that any atoms exist. Definite conditions and operations are neither sets nor atoms."

"We will profess that for every unary, definite condition $P$ there exists a class $A=\{x|P(x)\}$ such that for every object $x$, $$x\in A\iff P(x)$$."

In exercise 3.23, he asks to prove: For every class $A$, $$\text {$A$ is a set $\iff$ for some class $B$, $A\in B$ $\iff$ for some set $X$, $A\subset X$. } $$ where inclusion among classes is defined as they were classes.

The author provides solution to this exercise, which for $(2)\implies (3)$ is as follows: We will assume that $A$ is a class. If $A$ belongs to some class $B$, then it must be an object (atom or set), since only objects are put into classes by definition of class; and since $A$ is a class by the hypothesis, it is not an atom, and so it must be a set. Since $A\subset A$, $A$ is a subset of a set.

What I can't understand is that how could we conclude that a class can't be an atom.

2 Answers 2

3

By "class", we mean "collection of objects". An "atom", by definition, is an object that is not a collection. A "set" is a class which is also an object. None of these are deductions - this is just what the words mean. Note that an "atom" is not what your first sentence suggests - it's not just a thing that can be in a class, it's a thing that isn't a collection. Sets can be in classes, but aren't atoms.

They key is to pay attention to what the sentence means: "since $A$ is a class, it is not an atom, and so it must be a set". In other words, $A$ is a collection and also an object (because it's in a class), so it must be the variety of object that is allowed to be a collection; i.e., a set.

To answer the question in your first line: no. There cannot be pure classes that are elements of other classes - by definition. A class is a collection of objects, and pure classes are not objects. The question you might want to ask is "why must there be pure classes at all?" The answer to that is Russell's Paradox: consider the collection of all sets that don't contain themselves. This collection can't contain itself, but must contain itself; a contradiction. This collection, therefore, must be a proper class - a collection which isn't a set, so it can't be in itself one way or another.

1

This is just the definition of the term "class". By definition, a "class" is a collection of objects, and every object is a set or an atom. (This is not very clearly stated in the material you have quoted, but is the intended definition.)