On the Embedding of the First Nonconstructive Ordinal in the Rogers Semilattices

The embedding of the first nonconstructive ordinal in the Rogers semilattices of families of arithmetic sets is considered. It is proved that, for any infinite family of arithmetic sets, the first nonconstructive ordinal can be embedded over any minimal element of its Rogers semilattice. It is also shown that if the family is principal or finite, then the first nonconstructive ordinal is embedded over any nongreatest element of its Rogers semilattice.