Does $ L $ satisfy a continuous assumption on sets of hereditary sizes?

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

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

$ transitive (t) siff forall r, s (r in s in t to r in t) $, $ “ | x | "$ mean the cardinality of $ x $ which is the smallest ordinal bijective of Von Neumann at $ x $.

In English: $ H _ { alpha} $ is the set of all sets that are hereditarily subhumid 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 $?

or $ L $ is the constructible universe of Godel.