- #1

Look

- 51

- 0

**minimal set**satisfying the requirements of the axiom of infinity, is the intersection of all inductive sets.

In case that the axiom of infinity is expressed as

∃I (Ø ∈ I ∧ ∀x (x ∈ I ⇒ x ⋃ {x} ∈ I))

the intersection of all inductive sets (let's call it K) is defined as

set K = {x ∈ I : ∀y ((Ø ∈ y ∧ ∀z ((z ∈ y ⇒ z ⋃ {z} ∈ y))) ⇒ x ∈ y)}.

The members of set K can be defined by von Neumann's construction of the natural numbers in terms of sets (https://en.wikipedia.org/wiki/Natural_number#Constructions_based_on_set_theory).

So K satisfying the requirements of the axiom of infinity, where

**all**of its members are

**finite**sets, such that only Ø is not a successor of the rest of K's members.

In that case ∀k ∈ K (k ∪ {k} ∈ K), where |k ∪ {k}| can't be but < |K|, if |K| = λ = weak limit cardinal (such that λ is neither a successor cardinal nor zero).

So, please help me to understand how λ is formally defined as the cardinality of

**all**k in K ?