Yet, that's just a waste of time if anything because as we all agree the conclusion is contained within the axioms/premises. — TheMadFool
That is exactly what Gödel's first incompleteness theorem proves. — alcontali
If you construct an abstract, Platonic world using axioms/premises expressed in first-order logic, then there will be perceivable patterns in that world that can neither be falsified nor be proven from its construction axioms. That is exactly what Gödel's first incompleteness theorem proves. — alcontali
This holds for only a finite set of axioms and axiom schemata, an important condition; as new unprovable truths or straightup undecidable propositions can always just then be added as a new axiom to then be provable. — boethius
The pattern already emerges from the existing axioms without explicitly adding the pattern as an axiom. So, what do you gain by adding it? — alcontali
Therefore, I really do not see what you would gain by adding the Riemann hypothesis to the axioms of number theory. — alcontali
If I were to make an educated guess, "unprovable" and "undecidable" mean the same thing. — TheMadFool
I've read some logic and the video gets along well with what standard textbooks say:
1. A sound deductive argument is one in which the conclusion follows necessarily from the premises. — TheMadFool
Undecidable is actually stronger. It means not only unprovable, but also that the negation is unprovable. — alcontali
No, that's validity. Soundness is where the argument is valid and the premises and conclusion are true. — S
Can you have a look at Godel's Incompleteness Theorems vs Justified True Belief — TheMadFool
Hence, mathematics are justified beliefs, but not justified true beliefs. — alcontali
Using a clever hack, Gödel manages to create a theorem that is algebraically "true" in the abstract, Platonic world of number theory, but which by simply asserting its own unprovability, is not provable from the construction logic of that world. Hence, true but not provable. — alcontali
?algebraically "true" — alcontali
If I were to make an educated guess, "unprovable" and "undecidable" mean the same thing. — TheMadFool
This sounds like an endorsement of constructivist logic, such as intuitionistic logic, which requires a positive proof in order to affirm any proposition and accordingly denies the Law of Excluded Middle (LEM, either A or not-A must be true). By contrast, classical logic affirms LEM, which is why it allows double negation elimination (not not-A implies A) and proof by contradiction (reductio ad absurdum).If a proposition P is true then necessarily that a proof must exist for P being true. — TheMadFool
This sounds like an endorsement of constructivist logic, such as intuitionistic logic, which requires a positive proof in order to affirm any proposition and accordingly denies the Law of Excluded Middle (LEM, either A or not-A must be true). By contrast, classical logic affirms LEM, which is why it allows double negation elimination (not not-A implies A) and proof by contradiction (reductio ad absurdum). — aletheist
The works of Kurt Gödel, Alan Turing, and others shook this assumption, with the development of statements that are true but cannot be proven within the system. — Wikipedia
A proof is sufficient evidence or a sufficient argument for the truth of a proposition. — Wikpedia
No, the two Wikipedia quotes are not contradictory. The second one only affirms that a proof is sufficient for the truth of a proposition; it does not state that a proof is necessary for the truth of a proposition.On one hand mathematicians are of the view that truth requires proof and on the other they're claiming, through Godel, that some truths are unprovable. Isn't that a contradiction? — TheMadFool
No, the two Wikipedia quotes are not contradictory. The second one only affirms that a proof is sufficient for the truth of a proposition; it does not state that a proof is necessary for the truth of a proposition. — aletheist
That would be Gödel's proof of his incompleteness theorem, and the correct term is not "unprovable" but undecidable.What was the sufficient proof for the Godel statement "this statement is true but unprovable"? — TheMadFool
In this context, I am not so much denying it as pointing out that it does not apply to all systems of formal logic. Informally, the claim is obviously false, since lots of propositions are true without ever having a formal proof. It seems like you may be confusing truth with justification.Also, it seems that you're denying my claim: truth of a proposition necessarily requires proof. — TheMadFool
Can you provide a proof that the truth of a proposition necessarily requires proof? If not, why do you claim that?That means it's possible for a proposition to be true and without proof. Can you name one such truth? — TheMadFool
On one hand mathematicians are of the view that truth requires proof and on the other they're claiming, through Godel, that some truths are unprovable. Isn't that a contradiction? — TheMadFool
Can you provide a proof that the truth of a proposition necessarily requires proof? If not, why do you claim that? — aletheist
The burden of proof (Latin: onus probandi, shortened from Onus probandi incumbit ei qui dicit, non ei qui negat) is the obligation on a party in a dispute to provide sufficient warrant for their position. — Wikipedia
No, a proof is sufficient but NOT necessary. A true proposition is true regardless of whether humans ever construct a proof for it. One more time: a proof pertains to justification, not truth.A given proof is a sufficient condition to establish truth AND it's necessary too. I'm concerned about the latter aspect viz. necessity. — TheMadFool
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.