Naive
Set Theory
Naive
Set Theory
This section will be describing the naive part of set theory. This section will be using the book from Levy (1979/2002) .
Basic Notation
This section will be using first-order predicate calculus with equality. Let us then start with a definition for the basic language that we will be using within this section of set theory.
\(x \text{ and } y\) are any variables, by the sentential connectives \(\neg(not), \rightarrow (if...then..), \vee (or), \wedge (and), \leftrightarrow (\text{if and only if}), \text{ and the qualifiers } \exists (\text{there exists an x})\text{ and } \forall (\text{for all x})\). This is what is called a formula.
Let us now define what a free variable is.
For example, \(x\) is a free variable in each of the following formulas (which are not necessarily taken from set theory):
\[x < 3x,\] \[x^2=y,\] \[x \text{ is a real number},\] \[\sin x > \frac{1}{2}\]\(x\) is not a free variable in the formulas:
\[\forall(x^2 \ge 0),\] \[\neg \exists x(x \in y),\] \[\int_0^y \sin x \text{ } dx \lt \frac{1}{2}\]In the above cases, \(x\) is an auxillary variable which cannot be given a definite value and which can be replaced throughout each formula by another variable, say \(z\), without changing the meaning of the formula. In these cases, \(x\) is used as a bound variable.
Note that \(\forall x (x^2 \ge 0)\) is exactly what \(\forall z (z^2 \ge 0)\) says, while \(\sin x \gt \frac{1}{2}\) isn’t the same as \(\sin z \gt \frac{1}{2}\). The second condition can have values of x and z that may be true for \(\sin x \gt \frac{1}{2}\) while being false for \(\sin z \gt \frac{1}{2}\).
Thus, a formula with free variables says something about the values of its free variables. While the above doesn’t. We shall thus define the above as:
but about the universe which the language describes. This is called a sentence.
Within this section we will then be referring to formulas and sentences as “statements”.