Godel proved that for any consistent set of axioms of number theory, there are true statements that cannot be proved or disproved using those axioms. This showed that Hilbert's goal of a complete axiomatization of mathematics was impossible. Turing formalized the notion of a computable function as one that can be computed by a Turing machine. He showed that the halting problem - determining if a Turing machine will halt on a given input - is undecidable, demonstrating that there is no mechanical procedure to determine if an arbitrary mathematical statement is provable. This shattered Hilbert's dream of a decision procedure for mathematical proofs.

1. Computability
Anand Janakiraman
UMN, Twin Cities
Hilbert’s dream : Euclid & Formal Reasoning
Euclid successfully axiomized geometry.
Axioms are statements that are assumed to be true
Uncomputability
(see box on formal reasoning )
Theorems are proved/deduced using rules of inference
Hilbert dreamt to determine a complete axiom set assuming axioms are true. There are some functions which cannot be computed by
for mathematics. The Axiom set would be Rules of inference: An example is Modus Ponens any turing machines. The table shows the value computed
by the i th machine on the j th input . Machine 1 produces
the ouput [0110....] No machine can compute the
Finite: without this one could take as one’s axioms the set of all complement of the diagonal [1000....]
true propositions.
Sound: if all provable theorems are true Formal Reasoning was introduced by Euclid in his book
“Elements” which describes Geometry in an axiomatic
Complete: the system is able to prove all true theorems manner.The method of formal reasoning came to be known as
Aristotlean school of thought.
Decidable: if there is a mechanical procedure for determining
whether or not an arbitrary theorem is provable.
Other attempts to axiomize mathematics were “Peano’s
arithmetic” and Elliptical and Hyperbolic geometry which were
done by relaxing Euclid’s fifth postulate
Godel
Godel proved that for any consistent axioms F there is a true
statement of first order number theory that is not provable or
disprovable by F.
( i.e., a true statement that can be made using 0, 1, plus, times, Mechanical computation is limited. Turing machines can
for every, there exists, AND, OR, NOT, parentheses, and compute all that can be computed. The number of turing
variables that refer to natural numbers. ) machines is enumerable, whereas the number of functions
The proof is on the lines of liar's paradox ( "I am lying" ). is not. Thus there are some functions that are not
computable. An example of such a problem is halting
Godel constructs a statement similar to S: problem.
"This theorem is not provable in number theory".
if S is false, then S is provable ( this leads to a contradiction . Is S
provable or not provable) . Thus we are forced to assume S is
true and arithmetic itself cannot prove it
Thus we cannot obtain a system that is complete (since there are Aristotle Euclid
unproven true statements).
It may seem that we could obtain a complete axiomization Halting Problem
by simply taking all true stmts as axioms. But one
requirement is that these axioms should be recognizable
by mechanical method. As Turing subsequently showed Turing showed that the halting problem is uncomputable.
that the true statements about natural numbers cannot be
mechanically recognized.
Turing
Turing showed there no is a mechanical procedure for
determining whether or not an arbitrary theorem is provable.
Mechanical Procedure Hilbert
In order to formalize the notion of mechanical procedure , Turing
introduced a simplified model of computer (the person who
computes ) " assume computation is carried on one-
dimensional paper ie a tape divided into squares.... The
behavior of the computer is determined by the symbols he is
observing and the state of mind at that moment"
Decision Problem
A function is computable if any turing machine computes it.
The Turing Machine is an abstract, mathematical model that
Turing proved that the decision problem is uncomputable from
describes what can and cannot be computed.
the uncomputability of halting problem.
The halting problem (Machine M halts on tape T) can be
expressed as logical formula. If there were a procedure for the
x,y,
provability of arbitrary propositions (the decision problem) , then
y,z,
there would be one for halting problem. The fact that halting
Finite state brain problem is uncomputable means that there is no procedure for
x,y,
Finite alphabet of determining the provability of arbitrary theorem. Thus shattering
symbols
Infinite supply of
Hilbert’s dream.
notebooks
Godel Turing
x