Set Containing Itself: X and {X}

  • Thread starter Thread starter Outlined
  • Start date Start date
  • Tags Tags
    Set
Outlined
Messages
124
Reaction score
0
A set is not allowed to has itself as a member: X \notin X. But I wonder if this is allowed: \{X\} \in X.
 
Physics news on Phys.org
Outlined said:
A set is not allowed to has itself as a member: X \notin X. But I wonder if this is allowed: \{X\} \in X.

Neither are allowed in ZF. But there are theories in which both are allowed -- for example, ZF without the Axiom of Foundation.
 
It's not allowed since the set {{X},X} have no disjoint element of itself.
 
CRGreathouse said:
Neither are allowed in ZF. But there are theories in which both are allowed -- for example, ZF without the Axiom of Foundation.

Is it possible to construct such a set in ZF without Axiom of Foundation?
 
No -- such a construction would amount to a theorem of anti-foundation.

Specific axioms of anti-foundation may provide constructions, however.
 
The confusion here, I think, is in the designator "X", where this letters is trying to stand for both an element as well as a set. {X} in X means the element X as a set is found within the set, which violates the axiom of regularity, otherwise called "axiom of foundation", designed to prevent such expressions.
 

Thread 'Here's a Statistics problem for game of Polo (or Hockey if you like)'

Namaste & G'day Postulate: A strongly-knit team wins on average over a less knit one Fundamentals: - Two teams face off with 4 players each - A polo team consists of players that each have assigned to them a measure of their ability (called a "Handicap" - 10 is highest, -2 lowest) I attempted to measure close-knitness of a team in terms of standard deviation (SD) of handicaps of the players. Failure: It turns out that, more often than, a team with a higher SD wins. In my language, that...
View full post »

Thread 'Roulette wheel physics and probability'

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...
View full post »

Similar threads

MHB Translate the statements into set inclusion
Replies
5
Views
1K
I Countability of binary Strings
Replies
18
Views
2K
MHB The set of all sets does not exist.
Replies
2
Views
5K
I What Does A ⊆ B Mean in Set Theory?
Replies
18
Views
4K
I The set of nonzero values of pmf is at most countable
Replies
3
Views
2K
MHB Why Is the Empty Set Considered Unique?
Replies
10
Views
3K
B Confusion about division by zero in sets
Replies
2
Views
2K
I Are De Morgan's laws for sets necessary in this proof?
Replies
3
Views
2K
I Thinking about equality of infinite sets
Replies
5
Views
2K
MHB How Can I Show That the Symmetric Difference of Sets is Associative?
Replies
3
Views
2K

Hot Threads

Recent Insights

Back
Top