Theorems

Question 1

Russell class

Answer 1

The collection R such that x is not a member of itself, doesn’t form a set.
Additionally, Proof.

Question 2

Representation theorem for partially ordered sets

Answer 2

If < partially orders X, then there’s a set Y of subsets of X which is such that (X, <=) is order isomorphic to (Y, subseteq).
Additionally, proof.

Question 3

PMI

Answer 3

Suppose Ф is a well defined definite property of sets. The. Ф(0) and for any x in omega (Ф(х) -> Ф(S(x))) therefore for any x in omega Ф(х)

Question 4

WO theorem

Answer 4

Let X be a subset of omega, then either X is empty or there’s an n0 in X, such that for any m in X, either n0=m or n0<m. In other words if it’s not empty X is inductive.