Understanding the Intersection of Inductive Sets & the Limits of λ Cardinality

[Look]

By ZFC, the 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 ?

 

[mmwave]

Thank you for your forum post. It seems that you have a good understanding of the definition of the intersection of all inductive sets and how it relates to the axiom of infinity. As for your question about how λ is formally defined as the cardinality of all k in K, let me try to explain it in more detail.

First, let's define what we mean by the cardinality of a set. The cardinality of a set is a measure of its size, or the number of elements it contains. In set theory, we use the concept of bijections to define the cardinality of sets. Two sets have the same cardinality if there exists a one-to-one and onto mapping between them. This means that every element in one set is paired with exactly one element in the other set, and vice versa.

Now, let's look at the set K that you defined as the intersection of all inductive sets. We know that K satisfies the requirements of the axiom of infinity, which means that it contains all the natural numbers and is the smallest set that satisfies the axiom. This means that any other inductive set must also be a subset of K. So, if we can show that K has the same cardinality as the set of natural numbers, we can say that K is the set of all natural numbers.

To show that K has the same cardinality as the set of natural numbers, we can use von Neumann's construction of the natural numbers in terms of sets. In this construction, each natural number is defined as the set of all its predecessors. For example, 1 is defined as the set {Ø}, 2 is defined as the set {Ø,{Ø}}, and so on. In this construction, every natural number is a finite set, and the only set that is not a successor of any other set is the empty set Ø.

Now, let's look at the set K again. We know that all the elements of K are finite sets, and Ø is not a successor of any other set in K. This means that every element in K is a natural number in von Neumann's construction. And since K contains all the natural numbers, it has the same cardinality as the set of natural numbers.

To summarize, λ is formally defined as the cardinality of all elements in K because K contains all the natural numbers, and its elements are defined as finite sets in von Neumann's construction. I hope this helps