FOM: Some Problems in Reverse Mathematics

[Harvey Friedman]

Consider the following principles, which are frequently used throughout
mathematics.

A. Let n >= 1 and M1,...,Mn be a sequence of countable structures in a finite
relational type, where for all 1 <= i <= n-1, Mi is embeddable into Mi+1.
Then for all 1 <= i <= j <= n, Mi is embeddable into Mj.

B. Let n >= 1 and M1,...,Mn be a sequence of countable structures in a
countable
relational type, where for all 1 <= i <= n-1, Mi is embeddable into Mi+1.
Then there are structures M1',...,Mn', where for all 1 <= i <= n-1, Mi' is
a substructure of Mi+1', that is isomorphic to Mi.

C. Let M1,M2,... be an infinite sequence of countable structures in a
finite relational type, where each Mi is embeddable into Mi+1. Then there
are structures M1',M2',... such that for all i >= 1, Mi' is a substructure
of Mi+1' that is isomorphic to Mi.

Finite and infinite sequences of subsets of omega are coded as subsets of
omega in the usual way, using cross sections.

What is the reverse mathematics status of A,B,C over RCA_0?