MHB The set of all sets does not exist.

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The discussion revolves around the proof that the set of all sets does not exist, primarily based on Russell's paradox. It begins by assuming the existence of a set of all sets, denoted as V, and defines a subset B containing elements that do not belong to themselves. The contradiction arises when it is shown that if B is a member of itself, it leads to a logical inconsistency, as B cannot both belong to and not belong to itself simultaneously. This contradiction implies that the initial assumption of the existence of V is incorrect, thereby proving that the set of all sets cannot exist. The conversation seeks clarification on the proof's steps and the implications of the contradiction found.
evinda
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Hey! (Wave)

Theorem (Russell's paradox is not a paradox in axiomatic set theory)

The set of all sets does not exist.

Proof

We suppose that the set of all sets exist, let $V$. So, for each set $x$, $x \in V$.
We define the type $\phi: \text{ a set does not belong to itself, so } x \notin x$.

So, from the Axiom schema of specification, the set $\{ x \in V: x \notin x \}$ exists.

Since $V$ is the set of all sets,

$$\{x \in V: x \notin x \} \subset V$$

So, $V'=\{ x: x \notin x \}$ is a set.

Therefore: $V' \in V' \leftrightarrow V' \notin V'$, contradiction.How do we know that $V'$ is a set? Also, could you explain me why, having found this contradiction, we have proven that the set of all sets do not exist? (Thinking)
 
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The proof starts with the assumption: suppose that the set of all sets exists. At the end the proof concludes that $V' \in V' \Leftrightarrow V' \notin V'$ which is a contradiction since a statement can not be equivalent with it's negation. Because of the contradiction the assumption (this is the only thing what was assumed) is incorrect hence the set of all sets does not exist.
 
We want to prove that the set of all sets do not exist.

We suppose that the set of all sets, let $V$, exists.
So, for each set $x, x \in V$.
We define the type $\phi: \text{ a set does not belong to itself , so } x \notin x$.

From the axiom schema of specification, we conclude that there is the set $B=\{ x \in V: x \notin x \}$

$$\forall y(y \in B) \Leftrightarrow y \notin y$$

How can we continue? (Thinking)
 

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