# 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 .

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.