40 research outputs found
Reverse mathematics and uniformity in proofs without excluded middle
We show that when certain statements are provable in subsystems of
constructive analysis using intuitionistic predicate calculus, related
sequential statements are provable in weak classical subsystems. In particular,
if a sentence of a certain form is provable using E-HA
along with the axiom of choice and an independence of premise principle, the
sequential form of the statement is provable in the classical system RCA. We
obtain this and similar results using applications of modified realizability
and the \textit{Dialectica} interpretation. These results allow us to use
techniques of classical reverse mathematics to demonstrate the unprovability of
several mathematical principles in subsystems of constructive analysis.Comment: Accepted, Notre Dame Journal of Formal Logi
Reverse Mathematics and Algebraic Field Extensions
This paper analyzes theorems about algebraic field extensions using the
techniques of reverse mathematics. In section 2, we show that
is equivalent to the ability to extend -automorphisms of field extensions to
automorphisms of , the algebraic closure of . Section 3 explores
finitary conditions for embeddability. Normal and Galois extensions are
discussed in section 4, and the Galois correspondence theorems for infinite
field extensions are treated in section 5.Comment: 25 page
Banach's theorem in higher order reverse mathematics
In this paper, methods of second order and higher order reverse mathematics
are applied to versions of a theorem of Banach that extends the
Schroeder-Bernstein theorem. Some additional results address statements in
higher order arithmetic formalizing the uncountability of the power set of the
natural numbers. In general, the formalizations of higher order principles here
have a Skolemized form asserting the existence of functionals that solve
problems uniformly. This facilitates proofs of reversals in axiom systems with
restricted choice.Comment: Version: count-20230393arxiv.tex adding publication info in header
and listing modification