Good day, Usually, proofs by contradictions are the easier, and sometimes, even the only ones available. However, there are cases where the easiest proof is not the proof by contradiction. For example, the one below: From the definition of the rational numbers, all of them can be expressed as quotients of two integers. And from […]
I was just thinking about unprovability. I just wanted to know if it is possible to make a concrete boundary between provable problems and unprovable problems in a certain axiomatic system. We know that there is a statement that is true yet unprovable. Then is it possible that a statement is true and unprovable, but […]
Most of the systems mathematicians are interested in are consistent, which means, by Gödel’s incompleteness theorems, that there must be unprovable statements. I’ve seen a simple natural language statement here and elsewhere that’s supposed to illustrate this: “I am not a provable statement.” which leads to a paradox if false and logical disconnect if true […]
How can one prove the statement, “If a function grows fast enough, it cant be proven total in PA, unless PA is inconsistent”? How fast must it grow to be not provably total?