# 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 .

Questions:

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

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