1 Russell’s paradox with Sets

Russell’s paradox starts by designing a Set S that contains/specifies all Sets that are not elements of itself. (Note everything in our current model is a set; even the elements of sets are sets) \[ S = \{ x | x \notin x \} \tag{1}\] Next we ask can S be an element of S?
[Assume S is element of itself] \[S \in S \tag{2}\]
[This would imply by definition (1)] \[S \notin S \tag{3}\] [By contradiction (2) and (3)]
\[S \notin S \tag{4}\]

[but by the definition of (1) ]
\[S \in S \tag{5}\]
…thus the whole process repeats infinitely
It is a paradox because there is no end to this proof ( we cannot conclude whether it is true or false )

The resolution is to bound the universe so that S must be within a universe.

2 Russell’s paradoxal set S as an AST

3 Analogy and Abstracting Russell’s paradox with bears

In set theory, the propositional function is elementhood. Elementhood takes two sets (lets call the universe that contain our sets of interest, \(SET\)) and return a truth value. Also remember everything is a set.
We can think of our standard set theory as a pair that contains the universe and it’s associated function. \[(SET,\in)\]

\[ \in :: (SET,SET) \rightarrow bool \]

Let’s design another theory \[(BEAR, like)\] with \(BEAR\) being defined as the universe of bears.
\[ like :: (BEAR,BEAR) \rightarrow bool \]

Just like our initial set model, lets pretend everything in our universe is a bear and instead of set containing sets, we have bears liking bears.

And with theory let’s design a paradox using groups of bears and “like”.

• Arbitrary bear y that likes bear B iff P(y,likes) is true
• Arbitrary set x are elements of set S iff P(x,\(\in\)) is true
  • where P(x,action) stands for negation of action(x,x)

Note the similarities between our Set theory and Bear theory.

In other words,
any bear y must like bear B if and only if bear y does not like itself

4 AST with Bears

[Let us define bear B as this AST]

[bear B likes itself]

[therefore bear B does not like itself]

[But this means bear B does not not like itself, which is a double negative, meaning bear B likes itself]

[and so on and so forth…]