If each X Y and Z is a set then F X Y Z is a set each elem

If each X, Y, and Z is a set, then F = { X, Y, Z} is a set, each element of which is a set. We say is a collection, each element of which is a set. Of course, the number of sets in such a collection is not restricted to 3 as in this simple case.

Explain

Solution

Define [x=y] to be equality without the third axiom, i.e. it need only satisfy x=x and x=y Þ y=x.

Define [x =: y, z…] to mean that x equals y and z etc. and anything unequal to y is unequal to x, ditto z, etc.

In symbols: ( [x=y] & [x=z] & …) & ( ( [y’¹y] Þ x¹y’] ) & ( [z’¹z ] Þ [x¹z’] ) & …)

Define xÎy to mean $x’, x’’, y’, y’’, q, q’, q’’ such that [q =: x, q’] & [q’=: y, q, q’’] & [x’=:x, x’’] & [x” =: x, x’] & [y’=:y, y’’] & [y’’ =: y, y’]

There is as set model of [x=y] such that xÎy is membership in a set model of set theory. This theorem is nature of the proof of the so-called stereo equality theorem