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

In summary, the set K, which is the intersection of all inductive sets, satisfies the requirements of the axiom of infinity and has the same cardinality as the set of natural numbers. This means that λ, the cardinality of all elements in K, is formally defined as the set of all natural numbers.
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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 ?
 
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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
 

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

1. What is the intersection of inductive sets?

The intersection of inductive sets refers to the common elements shared by two or more inductive sets. An inductive set is a set that contains its own successor, meaning that it can be extended indefinitely by adding its own elements. The intersection of inductive sets is important in understanding the limits of λ cardinality.

2. What are the limits of λ cardinality?

The limits of λ cardinality refer to the maximum number of elements that can be contained in an inductive set. This limit is determined by the size of the underlying set and the rules of the inductive set. As the size of the underlying set increases, the limits of λ cardinality also increase.

3. How does understanding the intersection of inductive sets help in understanding λ cardinality?

Understanding the intersection of inductive sets is crucial in understanding λ cardinality because it allows us to determine the maximum number of elements that can be contained in an inductive set. By understanding the common elements shared by inductive sets, we can determine the limits of λ cardinality and better understand the size and structure of these sets.

4. What is the role of inductive sets in mathematics?

Inductive sets play a significant role in mathematics as they help define and understand the concept of infinity. They also provide a foundation for mathematical induction, which is a powerful proof technique used in various areas of mathematics. Additionally, inductive sets are used in the construction of mathematical structures such as natural numbers, integers, and real numbers.

5. How are inductive sets and λ cardinality related to Cantor's diagonal argument?

Cantor's diagonal argument is a proof technique used to show that the cardinality of certain sets is larger than others. This argument relies on the concept of inductive sets and their limits of λ cardinality. By understanding the intersection of inductive sets and their limits, we can better understand and appreciate Cantor's diagonal argument and its implications in set theory and mathematics.

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