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
