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In writing Principia Mathematica, was one Russell's purposes to formally describe all the possible variations of logical concepts and reasoning that can be used in mathematics?

For example, suppose that a mathematician of the future manages to prove theorems about natural numbers by reasoning on the properties of a subset of the complex numbers that has not yet been defined.

My question is, was one of Russell's purposes to be able to formally capture in advance all abstract concepts upon which the mathematician's conception of the subset will be based, and also his reasoning over its properties, before this subset is conceived by the mathematician's creative power?

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    I am not sure why "conception of a subset", sets played a limited role in Principia, and "creative powers" none at all (intuitionists were concerned with those). The purposes were to elaborate symbolic logic with predicates and quantifiers, and to demonstrate that it is sufficient to reconstruct all of the existing mathematics, i.e. to justify the foundations, build traditional concepts from elementary logical primitives, encode results in the symbolic language, and formalize "intuitive" proofs.
    – Conifold
    Commented Oct 11, 2019 at 0:08
  • See the post What did Whitehead and Russell's “Principia Mathematica” achieve?
    – Mauro ALLEGRANZA
    Commented Oct 11, 2019 at 5:54
  • See PM, page 1 : "THE mathematical logic which occupies Part I of the present work has been constructed under the guidance of three different purposes. In the first place, it aims at effecting the greatest possible analysis of the ideas with which it deals and of the processes by which it conducts demonstrations, and at diminishing to the utmost the number of the undefined ideas and undemonstrated propositions (called respectively primitive ideas and primitive propositions) from which it starts. "
    – Mauro ALLEGRANZA
    Commented Oct 11, 2019 at 5:55

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