|
6
|
|
edited May 29 2011 at 22:22 |
Let the transitive closure graph of a set X be the graph G(X) with V(G)= TC({X}) and (x,y) ∈ E(G) iff x ∈ y. Let H(X) be the reverse graph of G(X) with (x,y) ∈ E(H) iff y ∈ x.
I assume that the following holds:
For every hereditarily finite well-founded set X there is a unique set Y such that G(X) is isomorphic to H(Y).
Questions
Is this assumption correct? [Edit: If the answer is no: for which X?] Has this set Y been given a name (as a function of X)? Something like the reverse set of X? And has it attracted some interest? Can someone give a reference?
Some simple facts:
The finite von Neumann ordinals $\emptyset = 0, 1, 2, ...$ are reverse sets of themselves (self-reverse for short) The finite Zermelo ordinals $\emptyset = 0', 1', 2', ...$, i.e. $\lbrace\rbrace, \lbrace\lbrace\rbrace\rbrace, \lbrace\lbrace\lbrace\rbrace\rbrace\rbrace,...$, are self-reverse. The smallest pairs of not self-reverse sets are $\lbrace 2\rbrace$ vs. $\lbrace 1, 2'\rbrace$ and $\lbrace 0, 2\rbrace$ vs. $\lbrace 0, 1, 2'\rbrace$.
Further question:
- Is there a general correlation between the cardinalities of X, its reverse set Y and TC({X}) (= TC({Y}))?
|
|
|
|
|
5
|
|
edited May 28 2011 at 6:34 |
Let the transitive closure graph of a set X be the graph G(X) with V(G)= TC({X}) and (x,y) ∈ E(G) iff x ∈ y. Let H(X) be the reverse graph of G(X) with (x,y) ∈ E(H) iff y ∈ x.
I assume that the following holds:
For every finite well-founded set X there is a unique set Y such that G(X) is isomorphic to H(Y).
Questions
Is this assumption correct? [Edit: If the answer is no: for which X?] Has this set Y been given a name (as a function of X)? Something like the reverse set of X? And has it attracted some interest? Can someone give a reference?
Some simple facts:
The finite von Neumann ordinals $\emptyset = 0, 1, 2, ...$ are reverse sets of themselves (self-reverse for short) The finite Zermelo ordinals $\emptyset = 0', 1', 2', ...$, i.e. $\lbrace\rbrace, \lbrace\lbrace\rbrace\rbrace, \lbrace\lbrace\lbrace\rbrace\rbrace\rbrace,...$, are self-reverse. The smallest pairs of not self-reverse sets are $\lbrace 2\rbrace$ vs. $\lbrace 1, 2'\rbrace$ and $\lbrace 0, 2\rbrace$ vs. $\lbrace 0, 1, 2'\rbrace$.
Further question:
- Is there a general correlation between the cardinalities of X, its reverse set Y and TC({X}) (= TC({Y}))?
|
|
|
|
|
4
|
|
edited May 28 2011 at 0:51 |
Let the transitive closure graph of a set X be the graph G(X) with V(G)= TC({X}) and (x,y) ∈ E(G) iff x ∈ y. Let H(X) be the reverse graph of G(X) with (x,y) ∈ E(H) iff y ∈ x.
I assume that the following holds:
For every finite well-founded set X there is a unique set Y such that G(X) is isomorphic to H(Y).
Questions
Is this assumption correct? Has this set Y been given a name (as a function of X)? Something like the reverse set of X? And has it attracted some interest? Can someone give a reference?
Some simple facts:
The finite von Neumann ordinals $\emptyset = 0, 1, 2, ...$ are reverse sets of themselves (self-reverse for short) The finite Zermelo ordinals $\emptyset = 0', 1', 2', ...$, i.e. $\lbrace\rbrace, \lbrace\lbrace\rbrace\rbrace, \lbrace\lbrace\lbrace\rbrace\rbrace\rbrace,...$, are self-reverse. The smallest pairs of not self-reverse sets are $\lbrace 2\rbrace$ vs. $\lbrace 1, 1'\rbrace$ 2'\rbrace$ and $\lbrace 0, 2\rbrace$ vs. $\lbrace 0, 1, 2'\rbrace$.
Further question:
- Is there a general correlation between the cardinalities of X, its reverse set Y and TC({X}) (= TC({Y}))?
|
|
|
|
3
|
|
edited May 28 2011 at 0:29 |
Let the transitive closure graph of a set X be the graph G(X) with V(G)= TC({X}) and (x,y) ∈ E(G) iff x ∈ y. Let H(X) be the reverse graph of G(X) with (x,y) ∈ E(H) iff y ∈ x.
I assume that the following holds:
For every finite well-founded set X there is a unique set Y such that G(X) is isomorphic to H(Y).
Questions
Is this assumption correct? Has this set Y been given a name (as a function of X)? Something like the reverse set of X? And has it attracted some interest? Can someone give a reference?
Some simple facts:
The finite von Neumann ordinals $\emptyset = 0, 1, 2, ...$ are reverse sets of themselves (self-reverse for short) The finite Zermelo ordinals $\emptyset = 0', 1', 2', ...$, i.e. $\lbrace\rbrace, \lbrace\lbrace\rbrace\rbrace, \lbrace\lbrace\lbrace\rbrace\rbrace\rbrace,...$, are self-reverse. The smallest pairs of not self-reverse sets are $\lbrace 2\rbrace$ vs. $\lbrace 1, 1'\rbrace$ and $\lbrace 0, 2\rbrace$ vs. $\lbrace 0, 1, 2'\rbrace$.
Further question:
- Is there a general correlation between the cardinalities of X, its reverse set Y and TC({X}) (= TC({Y}))?
|
|
|
|
|
2
|
|
edited May 28 2011 at 0:20 |
Let the transitive closure graph of a set X be the graph G(X) with V(G)= TC({X}) and (x,y) ∈ E(G) iff x ∈ y. Let H(X) be the reverse graph of G(X) with (x,y) ∈ E(H) iff y ∈ x.
I assume that the following holds:
For every set X there is a unique set Y such that G(X) is isomorphic to H(Y).
Questions
Is this assumption correct? Has this set Y been given a name (as a function of X)? Something like the reverse set of X? And has it attracted some interest? Can someone give a reference?
Some simple facts:
The finite von Neumann ordinals $\emptyset = 0, 1, 2, ...$ are reverse sets of themselves (self-reverse for short) The finite Zermelo ordinals $\emptyset = 0', 1', 2', ...$, i.e. $\lbrace\rbrace, \lbrace\lbrace\rbrace\rbrace, \lbrace\lbrace\lbrace\rbrace\rbrace\rbrace,...$, are self-reverse. The smallest pairs of not self-reverse sets are $\lbrace 2\rbrace$ vs. $\lbrace 1, 1'\rbrace$ and $\lbrace 0, 1, 2'\rbrace$ 2\rbrace$ vs. $\lbrace 0, 2\rbrace$1, 2'\rbrace$.
Further question:
- Is there a general correlation between the cardinalities of X, its reverse set Y and TC({X}) (= TC({Y}))?
|
|
|
|
|
1
|
|
asked May 28 2011 at 0:07 |
Musings in set theory: Reverse sets?
Let the transitive closure graph of a set X be the graph G(X) with V(G)= TC({X}) and (x,y) ∈ E(G) iff x ∈ y. Let H(X) be the reverse graph of G(X) with (x,y) ∈ E(H) iff y ∈ x.
I assume that the following holds:
For every set X there is a unique set Y such that G(X) is isomorphic to H(Y).
Questions
Is this assumption correct? Has this set Y been given a name (as a function of X)? Something like the reverse set of X? And has it attracted some interest? Can someone give a reference?
Some simple facts:
The finite von Neumann ordinals $\emptyset = 0, 1, 2, ...$ are reverse sets of themselves (self-reverse for short) The finite Zermelo ordinals $\emptyset = 0', 1', 2', ...$, i.e. $\lbrace\rbrace, \lbrace\lbrace\rbrace\rbrace, \lbrace\lbrace\lbrace\rbrace\rbrace\rbrace,...$, are self-reverse. The smallest pairs of not self-reverse sets are $\lbrace 2\rbrace$ vs. $\lbrace 1, 1'\rbrace$ and $\lbrace 0, 1, 2'\rbrace$ vs. $\lbrace 0, 2\rbrace$.
Further question:
- Is there a general correlation between the cardinalities of X, its reverse set Y and TC({X}) (= TC({Y}))?
|
|
|
