Respectfull.You yourself have said. "Question Everything" in a different context of course, it applies here as well, some of us continue to ponder the issue. It is one of those great questions that we don’t normally talk about because we don’t like to mention questions for which we have no good answers. I believe it's innate within us to do just that. To ponder deeply,question everything, it's how we advance as a species.You have to laugh at this type of stuff or you'll loose you mind
Yes.Can you prove that the title of this thread is true?
Pure mathematics is inexhaustible. So, will there always be problems that can't be solved by existing math rules?Haven't watched the video, but the subtitle is "This is Math's Fatal Flaw", which would have made Gödel laugh. Incompleteness only applies to theories, not math. Consequently, he saw incompleteness as joyful proof that there is no limit to mathematical theorems. Incompleteness implies that math is inexhaustible.
Well yes, but we don't need the incompleteness theorems to see that. A simple counting argument suffices; let's just take decision problems, e.g., those that can be modeled as a function that return "yes/no" answers: \[f:\mathbb{Z} \to \{0, 1\}\] The set of all such problems has the cardinality of the continuum, \( 2^{|\mathbb{Z}|} = |\mathbb{R}| \). So there are an uncountable number of decision problems. However, any mathematical proof must necessarily be a finite set of symbols, and so the set of all mathematical proofs is finite. Therefore, we have more problems than we have mathematical proofs.Pure mathematics is inexhaustible. So, will there always be problems that can't be solved by existing math rules?
Yes. Some problems have unknowable answers but the rules of math are known. But people making YouTube videos need titles that attract audiences so, they contradict their own knowledge of the subject to tell that attracted audience something slightly different than the promise in the title. That's MrSalts Theory of 'STEM' YouTube Titles.Pure mathematics is inexhaustible. So, will there always be problems that can't be solved by existing math rules?
Yes and this has been known for some time.
Everything that is true *CAN* be proven. Do not confuse our ability to prove something with it's ability to be proven. This is a typical 'man's folly' argument. Man attempts to view everything through his own limitations.
Man attempts to view everything through his own limitations.Everything that is true *CAN* be proven. Do not confuse our ability to prove something with it's ability to be proven. This is a typical 'man's folly' argument. Man attempts to view everything through his own limitations.
A proof is ultimately a logical derivation of a true statement within some formal (axiomatic) system. Without the formal system, which defines the language and rules for making such derivations, there is no proof. In other words, a proof is only valid with respect to some given formal system.Everything that is true *CAN* be proven. Do not confuse our ability to prove something with it's ability to be proven. This is a typical 'man's folly' argument. Man attempts to view everything through his own limitations.
Scientific theories are built from empirical hypotheses, not formal statements, and so can neither be proved nor disproved. At best, we can continue to accept or reject the hypotheses based on the accumulation of evidence.Was it not only a few years ago that we "proved" Fermat's Last Theorem
but can we prove quantum theory, or general relativity ?
What exactly do you mean by "hole"?but even maths as shown has many holes,
In many mathematical contexts, division by zero is simply undefined. It doesn't have to be -- we can define 0/0 axiomatically or as shorthand for a limit. The only restriction is that we don't introduce any inconsistencies.what is 0/0 is one I get asked.