Meaning of Logic

To understand what does, we need to understand what it is, Russell project vision was not to find a relationship between Mathematical Logic and Logic, while Gödel with inconsistent theorem show otherwise, make to question what it is.

Dedekind–Peano axioms (19 Century) come from question rice by Pierce work to formalize arithmetic (which based on the work of Hermann Grassmann work was noted in his late sixties, 1870´s). Cantor approach to Mathematics the used of contraposition (nonconstructive proofs), a mathematical system consistence can be evaluated in his totality, where constructivists (that it is necessary to find (or construct) a mathematical object to prove that it exists), (two extensions from this school of thought intuitionism founded by Brouwe (intuitionism an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality), finitism of Hilbert and Bernays (Finitism accepts the existence only of finite mathematical objects), constructive recursive mathematics of Shanin and Markov, and Bishop (mathematical analysis done according, some principles of constructive mathematics.).

The difference between Cantor by assuming its non-existence and then deriving a contradiction from that assumption, this proof by contradiction is not constructively valid.

The underline of all this mathematical logic is consistence, where logic per say is deductive (at most of the case), a bridge to far, where consistence is not enough, in philosophical logic interpretation is necessary, case of point Halt Theorem.

A general algorithm to solve the halting problem for all possible program-input pairs cannot procedure compute_g(i):

    if f(i,i) == 0 then

        return 0

    else

        loop forever exist