Definition S.N.1Let Tr(z) means O:-z and (/\(x) x:-z => (x u {x}):-z).Remark: We don't define a function Tr(z). We just compress the above formula into Tr(z).Theorem S.N.2If K is a set such that K != O and /\(z:-K) Tr(z) then Tr(n(K)).Remark:In particular if Tr(A) and Tr(B) then Tr(A n B).Proof of Theorem S.N.2It is obvious that O:-n(K). Now take any x:-n(K). We have /\(z:-K) x:-z. Take any z:-K. Since Tr(z) and x:-z, (x u {x}):-z. Since z:-K was arbitrarily chosen, (x u {x}):-n(K). We have proved that O:-n(K) and (/\(x) x:-n(K) => (x u {x}):-n(K)).Theorem S.N.3There is a unique set M such that Tr(M) and (/\(X) Tr(X) => M c X).ProofBy Axiom ZF7 there exists a set m such that Tr(m). Let K = {A:-P(m)|Tr(A)} Let M = n(K). By Theorem S.N.2 Tr(M). It is obvious that M c m. Take any set X such that Tr(X). By Theorem S.N.2 Tr(X n M). Notice that X n M c m. Hence (X n M):-K. Thus M c X n M c X. We have shown that /\(X) Tr(X) => M c X. The proof of uniqueness. Assume that we have a set M1 such that Tr(M1) and (/\(X) Tr(X) => M1 c X). Then M1 c M. By on the other hand, M c M1. Thus M = M1.Definition S.N.4 - Natural Infinite SetWe define that M is a natural infinite set if and only if Tr(M) and (/\(X) Tr(X) => M c X).Remark:By Theorem S.N.3 there is only one natural infinite set.Theorem S.N.5If M is natural infinite set and x:-M then x is an ordinal.ProofLet K = {x:-M | x is an ordinal}. It is easy to verify that O:-K. We will show that /\(x:-K) x u {x} :- K. Take any x:-K such that x is an ordinal. We will show that x u {x} is an ordinal. By Theorem S.CO.6 x is a set of ordinals. Since x is an ordinal, x u {x} is aloso a set of ordinals. By Theorem S.CO.11 and Theorem S.CO.13 it is enough to show that /\(z :- x u {x}) z c x u {x}. Take any z :- x u {x}. If z:-x then z c x c x u {x}. If z = x then x c x u {x}. Hence z c x u {x}. We have shown that x u {x} is an ordinal. So x u {x}:-K. We have shown that /\(x:-K) x u {x} :- K. Since M is a natural infinite set, M = K.Theorem S.N.6If M is an natural infinite set then M is an ordinal.ProofLet K = {x:-M|x c M}. It is obvious that O:-K. We will show that /\(x:-K) x u {x}:-K. Take any x:-K. We have that x c K. Hence x u {x} c K. Thus x u {x}:-K. We have shown that /\(x:-K) x u {x} :- K. Since M is a natural infinite set, M = K. Hence /\(x:-M) x c M. Now by Theorem S.N.5 and Theorem S.CO.15 M is an ordinal.Definition S.N.7 - CountabilityLet x be a set. Let M be a natural infinite set. We define that x is countable if and only if \/(y) y c M and |y|=|x|.Remark:For definition of |x|=|y| go to Cardinality.Definition S.N.8 - FinitenessLet x be a set. Let M be a natural infinite set. We define that x is finite if and only if \/(y) y :- M and |y|=|x|.Remark:For definition of |x|=|y| go to Cardinality.