Free modules with isomorphic duals

Abstract: Let M,N be free modules over a Noetherian commutative ring R and let F be a field whose cardinality card(F) does not exceed the continuum. We prove the following : 1) The assertion that [Any two F-vector spaces with isomorphic duals are isomorphic] is equivallent to the ICF (Injective continuum function) hypothesis and it is a non-decidable statement in ZFC. 2) If the dual of M is a projective R-module and rank(M) is infinite then the ring R is Artinian. 3) If R is Artinian and card(R) does not exceed the continuum then the the dual of M is free. 4) If M,N have isomorphic duals then they are themselves isomorphic, when R is a non-Artinian ring that is either Hilbert or countable. 5) If R is a non-local domain then R is a half-slender ring. 6) If R is Artinian ring and its cardinality card(R) does not exceed the continuum then the assertion that [any two free R-modules with isomorphic duals are isomorphic] is non-decidable in ZFC. 7) If R is a domain and rank(M) is infinite then the Goldie dimension of the dual of M is equal to its cardinality . 8) If R is a complex affine algebra whose corresponding affine variety has no isolated points then [any two projective R-modules with isomorphic duals are themselves isomorphic]. 9) Let V be an F-vector space of infinite dimension. The dimension of its dual is equal to the cardinality of the powerset of the dimension of V. We also prove that if the powersets of two given sets have equal cardinalities then there is a bijection from the one powerset to the other that preserves the symmetric difference of sets.