I am developing a ZFC axiomatic system where together with the empty set, there is a singular (and huge) set of constants that are themselves sets and form a complete ordered field (cof) these constants could be called Rcof, +cof, .cof, 
<cof, 0cof, 1cof, etc. 


For sure, in this ZFC axiomatic system, the Real numbers R could be constructed in the standard Dedekin's cuts way. By construction ( 0 = {}, 1 = {0} = {{}}, 2 = {0,1} = {{},{{}}}, etc ), 1 ∈ 2 is obviously true for Real numbers. 


My question is: 1cof ∈ 2cof can be proved, disproved or neither? (That is using the cof definition only). In standard ZFC, (∗) there is, upon isomorphism, just one cof . This mean that must be an isomorphism between R (build in my ZFC system) and Rcof but it seem useless to prove 1cof ∈ 2cof. 


May be (∗) is false my ZFC axiomatic system and I have to live there with two cof. Is this a serious problem in my system?



An axiomatic system with a set of constants that form a complete ordered field
Assuming that ZFC is consistent, then your system will not settle the question of whether 1cof is an element of 2cof or vice versa. One can see this by observing that from any model of ZFC we can make various models of your system, by interpreting your new constant symbols 1cof, 2cof etc. in any of the various ways that we know how to construct a complete ordered field. Some of these constructions will give rise to models of your system where 1cof $\in$ 2cof, and some will not have this. We could also make 2cof $\in$ 1cof, or any other crazy pattern, simply by choosing sets to satisfy the desired type, and then imposing the complete ordered field structure upon that set of choices. 


You didn't say how many constants you had, but your remark that it was "huge" suggests that you possibly intended to have more than continuum many such constants. That is no problem, because if ZFC is consistent, then by the Löwenheim-Skolem theorem there is a model of ZFC whose reals are at least that large, and we can use this model as the starting point for the rearranging process. 


