1
$\begingroup$

What is the difference between the two notations below?

{x | x (is an element of) X & P(x)} 

vs.

X = {x | P(x)} 

These both seems to say to me x is in X as long as it abides by property P. The top one is defined as a comprehension and the bottom one is used as a lead into Russle's paradox.

  • 0
    The first describes a certain *subset* of X, quite possibly not all of X. The second says that $X$ is the set of all objects in the universe with property $P$ (unrestricted comprehension, not allowed in ZFC).2011-09-08

2 Answers 2

2

The difference is that $X$ is required to be a set in the first, and in the second it may not be. It's kind of amazing, but this avoids the classical set-theoretic paradoxes. The classic Russell paradox $X=\{x|x\not \in x\}$ is a good example. Now in the first we know we have a set by the power set and separation axioms. In the second we don't.

  • 0
    @André Nicolas: Thanks. That is what I was thinking. For finite sets, we have union, as you say.2011-09-08
0

The first is an object, namely a set. In particular, it can't be true or false. The second is an equation, which is a type of statement. The difference between the two is the same as the difference between "dog" and "my dog has four legs."