Does $ L $ satisfy a continuous hypothesis about the sets of hereditary sizes?


Define: $$ H _ { alpha} = {x | forall and in TC ( {x }) (| y | leq | alpha |) } $$

Where: $ TC (x) = {y | forall t (transitive (t) wedge x subseteq t to y in t) }, $

$ transitive (t) iff forall r, s (r in s in t to r in t) $

In English: $ H _ { alpha} $ is the set of all the sets that are hereditarily subnumerable to $ alpha $.

Define recursively: $$ daleth_0 = omega_0 $$

$$ daleth _ {i + 1} = | H _ { daleth_i} | $$

$$ daleth_j = bigcup_ {i <j} ( daleth_i), text {if} not exists k (k + 1 = j) $$ .

Questions:

1 is $ daleth_1 = aleph_1 $ satisfied in $ L $?

2 is $ daleth_i = aleph_i $ satisfied in $ L $?