MHB The set of all sets does not exist.

  • Thread starter Thread starter evinda
  • Start date Start date
  • Tags Tags
    Set Sets
evinda
Gold Member
MHB
Messages
3,741
Reaction score
0
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)
 
Physics news on Phys.org
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)
 
Hi all, I've been a roulette player for more than 10 years (although I took time off here and there) and it's only now that I'm trying to understand the physics of the game. Basically my strategy in roulette is to divide the wheel roughly into two halves (let's call them A and B). My theory is that in roulette there will invariably be variance. In other words, if A comes up 5 times in a row, B will be due to come up soon. However I have been proven wrong many times, and I have seen some...
Thread 'Detail of Diagonalization Lemma'
The following is more or less taken from page 6 of C. Smorynski's "Self-Reference and Modal Logic". (Springer, 1985) (I couldn't get raised brackets to indicate codification (Gödel numbering), so I use a box. The overline is assigning a name. The detail I would like clarification on is in the second step in the last line, where we have an m-overlined, and we substitute the expression for m. Are we saying that the name of a coded term is the same as the coded term? Thanks in advance.

Similar threads

Replies
5
Views
1K
Replies
18
Views
2K
Replies
12
Views
3K
Replies
10
Views
3K
Replies
132
Views
19K
Replies
1
Views
1K
Replies
2
Views
2K
Replies
7
Views
2K
Back
Top