Introduction
There are certain problems that computers are unable to
solve, because it is possible to ask a question that is mathematically
unknowable.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
“The trouble is, math is sort of
broken. It's been broken since 1931, when the logician Kurt Gödel published his
famous incompleteness theorems. They showed that in any mathematical system,
there are certain questions that cannot be answered. They're not really difficult
— they're unknowable. Mathematicians learned that their ability to understand
the universe was fundamentally limited.”
https://www.realclearscience.com/articles/2019/01/12/mathematicians_discovered_a_computer_problem_that_no_one_can_ever_solve_110858.html
- - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Gödel's incompleteness theorems are two theorems of mathematical logic that complete and consistent set of axioms for all mathematics is impossible.
The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of the natural numbers. For any such consistent formal system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.
Employing a diagonal argument, Gödel's incompleteness theorems were the first of several closely related theorems on the limitations of formal systems. They were followed by Tarski's undefinability theorem on the formal undefinability of truth, Church's proof that Hilbert's Entscheidungsproblem is unsolvable, and Turing's theorem that there is no algorithm to solve the halting problem.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Gödel's incompleteness theorems are two theorems of mathematical logic that complete and consistent set of axioms for all mathematics is impossible.
The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of the natural numbers. For any such consistent formal system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.
Employing a diagonal argument, Gödel's incompleteness theorems were the first of several closely related theorems on the limitations of formal systems. They were followed by Tarski's undefinability theorem on the formal undefinability of truth, Church's proof that Hilbert's Entscheidungsproblem is unsolvable, and Turing's theorem that there is no algorithm to solve the halting problem.
No comments:
Post a Comment