Re:Why not string theory? Because enough is enough
by leni536 on Sunday June 12, 2016 @07:36AM
(#11887638)
> Every mathematical system has things which are true but can't be discovered from first principals.
This is not true, see Gödel's completeness theorem [1], which states "... that a deductive system of first-order predicate calculus is 'complete' in the sense that no additional inference rules are required to prove all the logically valid formulas." It basically means that every "true" statements are provable with a good enough deductive system.
What you possibly think about is Gödel's first incompleteness theorem [2] which is quite a different statement. It states that in "most" formal systems (that are complex enough to be able to describe natural numbers) there is always a statement A that both A and "not A" are not provable. It basically means that these statements can't be "logically valid", which means that there exist models of the formal system where A is true and models where "not A" is true.
This is not true, see Gödel's completeness theorem [1], which states "... that a deductive system of first-order predicate calculus is 'complete' in the sense that no additional inference rules are required to prove all the logically valid formulas." It basically means that every "true" statements are provable with a good enough deductive system.
What you possibly think about is Gödel's first incompleteness theorem [2] which is quite a different statement. It states that in "most" formal systems (that are complex enough to be able to describe natural numbers) there is always a statement A that both A and "not A" are not provable. It basically means that these statements can't be "logically valid", which means that there exist models of the formal system where A is true and models where "not A" is true.
[1] https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_th... [2] https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...