Thread: can every statement be proven

there are many hypothesis that have not been proven yet (as the Riemann hypothesis etc.) I am just wondering if, theoretically, there would be a hypothesis that can not be proven at all.
I mean, I know that the Godel incompleteness theorem states that you can not prove the axioms of mathematics. Apart from that, has there been any efforts made to "prove" that certain hypothesis can simply not be proven? I guess the math involved would be far to complicated for me. even the fermat hypothesis has been proven but in such a complicated way that you would imagine that there must be even more complicated ways and detours to prove many other things.

This statement can not be proven.

Originally Posted by DivideByZero

This statement can not be proven.

Ghost of Godel salutes you.

The continuum hypothesis is one such unprovable hypothesis. This states that there is no set whose cardinality is strictly between that of the integers and that of the real numbers.

The set of all integers is countable and has cardinality denoted by (aleph nought). All countable sets have this cardinality. The set of all real numbers is uncountable and its cardinality is denoted . (It’s supposed to be lowercase gothic “c” – but I don’t know how to do gothic in TeX. :|) It can be shown that – in other words, the cardinality of the real numbers is equal to the cardinality of the power set of a countable set. The continuum hypothesis states that there is no set with cardinality such that .

Kurt Gödel showed in 1939 that the hypothesis cannot be disproved using the axioms of Zermelo–Fraenkel set theory. He didn’t show that it couldn’t be proved, only that it couldn’t be disproved. The job was completed in 1963 when Paul Cohen showed that Zermelo–Fraenkel axioms are indeed inadequate to prove or disprove the hypothesis.

Thanks for that, Jane.

Was it Greg Chaitin who demonstrated that there are, in fact, more unprovable theorems than provable ones? I'm pretty certain they worked this out as a direct extension from Godel, and didn't name/number these theorems, but it was a theoretical proof.

Thanks for the link. I’m afraid I’ve not heard of Gregory Chaitin before.

\mathfrak{c} =

Am I missing something here? This question seems to have a very simple answer that doesn't have anything to do with Goedel or self reference.

Originally Posted by evariste.galois

I am just wondering if, theoretically, there would be a hypothesis that can not be proven at all.

Yes: False hypotheses cannot be proven.

Originally Posted by serpicojr

\mathfrak{c} =

Whoa, this place has LaTeX now?

Originally Posted by evariste.galois

there are many hypothesis that have not been proven yet (as the Riemann hypothesis etc.) I am just wondering if, theoretically, there would be a hypothesis that can not be proven at all.
I mean, I know that the Godel incompleteness theorem states that you can not prove the axioms of mathematics. Apart from that, has there been any efforts made to "prove" that certain hypothesis can simply not be proven? I guess the math involved would be far to complicated for me. even the fermat hypothesis has been proven but in such a complicated way that you would imagine that there must be even more complicated ways and detours to prove many other things.

No, when they introduced neutrons into science there was no statement, that the government could not prove, could not corrupt, alter or ridicule.

The scientists the government put in charge either knowingly committed treason against America. For which they could be shot. Or they unknowingly committed treason against America and could be threatened with treason. Either way there is no more solid basis for truth in America.

Sincerely,


William McCormick

Originally Posted by serpicojr

\mathfrak{c} =

I just saw this.

Thanks!

Originally Posted by William McCormick

Originally Posted by evariste.galois

there are many hypothesis that have not been proven yet (as the Riemann hypothesis etc.) I am just wondering if, theoretically, there would be a hypothesis that can not be proven at all.
I mean, I know that the Godel incompleteness theorem states that you can not prove the axioms of mathematics. Apart from that, has there been any efforts made to "prove" that certain hypothesis can simply not be proven? I guess the math involved would be far to complicated for me. even the fermat hypothesis has been proven but in such a complicated way that you would imagine that there must be even more complicated ways and detours to prove many other things.

No, when they introduced neutrons into science there was no statement, that the government could not prove, could not corrupt, alter or ridicule.

The scientists the government put in charge either knowingly committed treason against America. For which they could be shot. Or they unknowingly committed treason against America and could be threatened with treason. Either way there is no more solid basis for truth in America.

Sincerely,


William McCormick

Must be a full moon out somewhere.

Originally Posted by William McCormick

Originally Posted by evariste.galois

there are many hypothesis that have not been proven yet (as the Riemann hypothesis etc.) I am just wondering if, theoretically, there would be a hypothesis that can not be proven at all.
I mean, I know that the Godel incompleteness theorem states that you can not prove the axioms of mathematics. Apart from that, has there been any efforts made to "prove" that certain hypothesis can simply not be proven? I guess the math involved would be far to complicated for me. even the fermat hypothesis has been proven but in such a complicated way that you would imagine that there must be even more complicated ways and detours to prove many other things.

No, when they introduced neutrons into science there was no statement, that the government could not prove, could not corrupt, alter or ridicule.

The scientists the government put in charge either knowingly committed treason against America. For which they could be shot. Or they unknowingly committed treason against America and could be threatened with treason. Either way there is no more solid basis for truth in America.

Sincerely,


William McCormick

Ahhh a perfect example of an unprovable hypothesis!

Originally Posted by quantumdude

Yes: False hypotheses cannot be proven.

Why not? You can prove something that is false to be false just as you can prove something that is true to be true.

Actually, you can rarely (if ever) prove any theory to be true. You can only attempt to disprove it and fail repeatedly. This does not constitute a proof, but it provides an ever-growning set of evidence that the theory is at least close to the truth. 8)

Originally Posted by Faldo_Elrith

Originally Posted by quantumdude

Yes: False hypotheses cannot be proven.

Why not? You can prove something that is false to be false just as you can prove something that is true to be true.

A proof is nothing other than a logically valid argument with true premises, so the conclusion of a proof must be true. When someone says, "Prove statement P" it is always understood that what is meant here is "Prove that P is true". Now consider a false statement Q. If someone says, "Prove that Q is false" he is really asking you to prove the true statement P="Q is false".

So false hypotheses can be proven false.

Originally Posted by MagiMaster

Actually, you can rarely (if ever) prove any theory to be true. You can only attempt to disprove it and fail repeatedly. This does not constitute a proof, but it provides an ever-growning set of evidence that the theory is at least close to the truth. 8)

I believe evariste.galois is referring to mathematical hypotheses here, not scientific theories. Mathematical truths are deductive (or a priori) while what’s accepted as “true” in science is generally inductive (or a posteriori). They are not the same.

:facepalm: You're right of course. I've been reading too much on the physics board.

Originally Posted by Faldo_Elrith

So false hypotheses can be proven false.

Sure, just like the paradox in the second post can be proven to be a paradox. But that isn't what is normally meant by proof in mathematics. By definition, one can only prove true statements. When false statements are exposed as false, one doesn't write a proof. Instead one provides a counterexample.

Originally Posted by JaneBennet

Originally Posted by MagiMaster

Actually, you can rarely (if ever) prove any theory to be true. You can only attempt to disprove it and fail repeatedly. This does not constitute a proof, but it provides an ever-growning set of evidence that the theory is at least close to the truth. 8)

I believe evariste.galois is referring to mathematical hypotheses here, not scientific theories. Mathematical truths are deductive (or a priori) while what’s accepted as “true” in science is generally inductive (or a posteriori). They are not the same.

I see a greater flaw in math today then I do in science. And, I see an 85 percent error in science. This is not to stun or solicit replies. It is my actual and heartfelt belief. No need to reply.



Sincerely,


William McCormick