User blog:Alemagno12/Some more set theory questions

Question 3
What is the cardinality of the set of all ordered tuples such that: where the depth of a tuple is defined using these inductive rules:
 * They can't contain something other than ordered tuples
 * They have a countably infinite amount of elements
 * Their depth is at most countably infinite
 * The empty tuple has depth 0
 * Else, the depth of a tuple is the smallest ordinal greater than the depth of all of its elements

Question 4
Define a function F from ordinals to cardinals as follows: Then, let Δ be the smallest cardinal such that Δ = F(ΔOrd), where ΔOrd is the smallest ordinal with cardinality Δ.
 * F(0) = aleph0
 * Else, F(α+1) = the successor cardinal after F(α)
 * Else, F(α) = the smallest weak limit cardinal with cofinality greater than F(β) for all β<α

Has a cardinal equal to Δ been defined before? If so, which one(s)? If not, how big is Δ compared to other cardinals?