MathDB

Let

V_0 = \varnothing

be the empty set and recursively define

V_{n+1}

to be the set of all

2^{|V_n|}

subsets of

V_n

for each

n=0,1,\dots

. For example V_2 = \left\{ \varnothing, \left\{ \varnothing \right\} \right\} \text{and} V_3 = \left\{ \varnothing, \left\{ \varnothing \right\}, \left\{ \left\{ \varnothing \right\} \right\}, \left\{ \varnothing, \left\{ \varnothing \right\} \right\} \right\}. A set

x \in V_5

is called transitive if each element of

x

is a subset of

x

. How many such transitive sets are there?
Proposed by Evan Chen

Problem Statement