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can every statement be proven

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evariste.galois

Posted: Wed Jul 09, 2008 1:38 am Post subject: can every statement be proven

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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.

DivideByZero

Posted: Wed Jul 09, 2008 4:33 am Post subject:

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This statement can not be proven.

sunshinewarrior

Posted: Wed Jul 09, 2008 4:38 am Post subject:

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DivideByZero wrote:

This statement can not be proven.

Ghost of Godel salutes you.

JaneBennet

Posted: Wed Jul 09, 2008 5:08 am Post subject:

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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 $\aleph_0$ (aleph nought). All countable sets have this cardinality. The set of all real numbers is uncountable and its cardinality is denoted $\textit{$\textbf{c}$}$ . (Itâ€™s supposed to be lowercase gothic â€œcâ€ â€“ but I donâ€™t know how to do gothic in TeX. Neutral

) It can be shown that $\textit{$\textbf{c}$} = 2^{\aleph_0}$ â€“ 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 $\mathcal{C}$ such that $\aleph_0<\mathcal{C}<2^{\aleph_0} = \textit{$\textbf{c}$}$ .

Kurt GÃ¶del showed in 1939 that the hypothesis cannot be disproved using the axioms of Zermeloâ€“Fraenkel set theory. He didnâ€™t show that it couldnâ€™t be proved, only that it couldnâ€™t be disproved. The job was completed in 1963 when Paul Cohen showed that Zermeloâ€“Fraenkel axioms are indeed inadequate to prove or disprove the hypothesis.
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Last edited by JaneBennet on Wed Jul 09, 2008 8:10 am; edited 1 time in total

sunshinewarrior

Posted: Wed Jul 09, 2008 6:00 am Post subject:

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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.

JaneBennet

Posted: Wed Jul 09, 2008 8:11 am Post subject:

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Thanks for the link. Smile

Iâ€™m afraid Iâ€™ve not heard of Gregory Chaitin before. Embarassed

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serpicojr

Posted: Wed Jul 09, 2008 1:46 pm Post subject:

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\mathfrak{c} = $\mathfrak{c}$

quantumdude

Posted: Thu Jul 10, 2008 8:24 am Post subject: Re: can every statement be proven

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Joined: 10 May 2005
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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.

evariste.galois wrote:

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

Yes: False hypotheses cannot be proven.

quantumdude

Posted: Thu Jul 10, 2008 8:25 am Post subject:

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serpicojr wrote:

\mathfrak{c} = $\mathfrak{c}$

Whoa, this place has LaTeX now?

$\int uper$

William McCormick

Posted: Thu Jul 10, 2008 8:01 pm Post subject: Re: can every statement be proven

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evariste.galois wrote:

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
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JaneBennet

Posted: Fri Jul 11, 2008 5:05 pm Post subject:

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serpicojr wrote:

\mathfrak{c} = $\mathfrak{c}$

I just saw this.

Thanks! Very Happy

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quantumdude

Posted: Sat Jul 12, 2008 3:02 pm Post subject: Re: can every statement be proven

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William McCormick wrote:

evariste.galois wrote:

Must be a full moon out somewhere. Laughing

DivideByZero

Posted: Sat Jul 12, 2008 3:25 pm Post subject: Re: can every statement be proven

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William McCormick wrote:

evariste.galois wrote:

Ahhh a perfect example of an unprovable hypothesis!

Laughing

Faldo_Elrith

Posted: Sat Jul 12, 2008 7:04 pm Post subject: Re: can every statement be proven

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Joined: 02 Jul 2008
Posts: 74

quantumdude wrote:

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.

MagiMaster

Posted: Sun Jul 13, 2008 11:29 pm Post subject:

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Joined: 16 Jul 2006
Posts: 288

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. Cool

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