MHB There is a unique inductive set

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A unique inductive set exists within each inductive set, as demonstrated through the construction of the set B, which comprises all inductive subsets of A. The proof establishes that the union of B is an inductive set and that the intersection of B is contained within any inductive set C. It is shown that A ∩ C is also an inductive set, leading to the conclusion that the intersection of all elements in B must be contained in A ∩ C. The discussion confirms that this intersection is indeed contained in every inductive set, solidifying the uniqueness of the inductive set within the framework of set theory.
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Hi! (Nerd)

Sentence:

There is a unique inductive set that is contained in each inductive set.

Proof:

Let $A$ be an inductive set (we know that there is such a set from the axiom of infinity)

and we define:

$$B=\{ X \subset A: X \text{ is an inductive set}\}$$
($B$ is a set, because if $X \subset A$ where $X$ is an inductive set then $X \in \mathcal{P}(A)$. So, $B$ is a set.)

The set $B \neq \varnothing$ since $A \in B$. So, the set $\bigcup B$ is defined. We will show that $\bigcap B$ has the desired properties.

$\bigcup B$ is an inductive set
(from the Remark: Let $B$ be a nonempty set of inductive sets. Then $\bigcap B$ is an inductive set.)

We will show that $\bigcap B$ is contained in each inductive set.
Let $C$ be an inductive set. Then, from the Remark, the set $A \cap C$ is an inductive set and obviously $\bigcap B \subset A \cap C \subset C$.

It remains to show the uniqueness.
Let $c,d$ sets, so that each of them is contained in each inductive set.
Then $c \subset d$
and $d \subset c$, so $c=d$.Could you explain me how we concluded that $A \cap C$ is an inductive set and how that $\bigcap B \subset A \cap C \subset C$ ? (Thinking)
 
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The set $A \cap C$ is an inductive set, because $\{ A, C \}$ is a non-empty set of inductive sets and so $\bigcap \{ A, C \}=A \cap C$ is an inductive set, right?

But how do we conclude that $\bigcap B \subset A \cap C$? (Thinking)
 
evinda said:
The set $A \cap C$ is an inductive set, because $\{ A, C \}$ is a non-empty set of inductive sets and so $\bigcap \{ A, C \}=A \cap C$ is an inductive set, right?
Correct.

evinda said:
But how do we conclude that $\bigcap B \subset A \cap C$? (Thinking)

Note that $A \cap C \subset A$.
 
Siron said:
Note that $A \cap C \subset A$.

From this, we have that $A \cap C \in B$.
How can we conclude that $\bigcap B \subset A \cap C$ ? (Thinking)
 
evinda said:
From this, we have that $A \cap C \in B$.
How can we conclude that $\bigcap B \subset A \cap C$ ? (Thinking)

By definition $\bigcap B$ is the intersection of all elements in $B$. To formulate it less formally, it's the set that all elements in $B$ have in common. Thus all elements in $B$ contain $\bigcap B$. In particular, since $A \cap C \in B$ it has to contain $\bigcap B$.
 
Siron said:
By definition $\bigcap B$ is the intersection of all elements in $B$. To formulate it less formally, it's the set that all elements in $B$ have in common. Thus all elements in $B$ contain $\bigcap B$. In particular, since $A \cap C \in B$ it has to contain $\bigcap B$.

So, is it like that? (Thinking)

$$\forall x ( x \in \bigcap B \leftrightarrow (\forall b \in B) x \in b)$$

$$\forall x(x \in \bigcap B \leftrightarrow x \in A \cap C)$$
 
Yes.

Suppose $z \in \bigcap B$ then we have $\forall X \in B: z \in X$ by definition. As $A \cap C \in B$ we conclude $z \in A \cap C$, that is, we've proved $\bigcap B \subset A \cap C$
 
Siron said:
Yes.

Suppose $z \in \bigcap B$ then we have $\forall X \in B: z \in X$ by definition. As $A \cap C \in B$ we conclude $z \in A \cap C$, that is, we've proved $\bigcap B \subset A \cap C$

A ok... But I think that we cannot write it like that:

$$\forall x(x \in \bigcap B \leftrightarrow x \in A \cap C)$$

because that would mean that $\bigcap B=A \cap C$.
Or am I wrong? (Thinking)
 
evinda said:
A ok... But I think that we cannot write it like that:

$$\forall x(x \in \bigcap B \leftrightarrow x \in A \cap C)$$

because that would mean that $\bigcap B=A \cap C$.
Or am I wrong? (Thinking)

Indeed. I didn't notice that.
 
  • #10
Having shown that $\bigcap B \subset A \cap C$ have we shown that $\bigcap B$ is contained in each inductive sets, since $A,C$ have been picked randomly and so $A \cap C$ is also an arbitrary set? Or am I wrong? (Thinking)
 
  • #11
evinda said:
Having shown that $\bigcap B \subset A \cap C$ have we shown that $\bigcap B$ is contained in each inductive sets, since $A,C$ have been picked randomly and so $A \cap C$ is also an arbitrary set? Or am I wrong? (Thinking)

You're correct. The inductive set $C$ was picked arbitrary and it was shown that $\bigcap B \subset C$. Hence, $\bigcap B$ is contained in every inductive set.
 
  • #12
Siron said:
You're correct. The inductive set $C$ was picked arbitrary and it was shown that $\bigcap B \subset C$. Hence, $\bigcap B$ is contained in every inductive set.

I see, thanks a lot! (Happy)
 

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