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#11
10-17-2007, 04:34 PM
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Re: Banach傍arski paradox = Contradiction?
[ QUOTE ]
...but IF you could dissect a tetrahedron, you WOULD be able to prove the axiom of choice was false. This is a contradiction of the Godel result pzhon cited. [/ QUOTE ] Siegmund, I think I may still be missing something. How is this a contraction unless you actually dissect the tetrahedron? 
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#12
10-17-2007, 05:21 PM
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Re: Banach傍arski paradox = Contradiction?
[ QUOTE ]
[ QUOTE ] ...but IF you could dissect a tetrahedron, you WOULD be able to prove the axiom of choice was false. This is a contradiction of the Godel result pzhon cited. [/ QUOTE ] Siegmund, I think I may still be missing something. How is this a contraction unless you actually dissect the tetrahedron? [/ QUOTE ] This is interesting to me because I don't recall ever being exposed to this reasoning. Let me take a stab at explaining. What's been shown is that no proof of the existence of the Dissection is possible using only the standard axioms of set theory - Excluding the AoC. The reason being that if such a proof were possible it would mean that the Logical system of Axioms-Exluding AoC are not consistent with the system of Axioms-Including AoC. This would contradict the Godel result. However, there is another consistent logical system we could adopt just as easily. That is the Standard Axioms + The axiom that AoC is False. It's concievable that in this logical system it can be shown that the Dissection is possible. Only the said proof of the dissection's existence would have to somehow make use of the Axiom that AoC is False. Furthermore, we might adjoin a New axiom to the Standard ones, creating a consistent logical system which makes no mention of the AoC and in which it can be shown that the Dissection is possible. In that case we would get the Free result that the AoC is False in the new system. PairTheBoard
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#13
10-17-2007, 05:30 PM
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Re: Banach傍arski paradox = Contradiction?
[ QUOTE ]
This is interesting to me because I don't recall ever being exposed to this reasoning. Let me take a stab at explaining. What's been shown is that no proof of the existence of the Dissection is possible using only the standard axioms of set theory - Excluding the AoC. The reason being that if such a proof were possible it would mean that the Logical system of Axioms-Exluding AoC are not consistent with the system of Axioms-Including AoC. This would contradict the Godel result. However, there is another consistent logical system we could adopt just as easily. That is the Standard Axioms + The axiom that AoC is False. It's concievable that in this logical system it can be shown that the Dissection is possible. Only the said proof of the dissection's existence would have to somehow make use of the Axiom that AoC is False. Furthermore, we might adjoin a New axiom to the Standard ones, creating a consistent logical system which makes no mention of the AoC and in which it can be shown that the Dissection is possible. In that case we would get the Free result that the AoC is False in the new system. PairTheBoard [/ QUOTE ] Thank for that explanation PTB. All that made sense, but I still don't see a contradiction anywhere....
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#14
10-17-2007, 05:48 PM
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Re: Banach傍arski paradox = Contradiction?
[ QUOTE ]
[ QUOTE ] This is interesting to me because I don't recall ever being exposed to this reasoning. Let me take a stab at explaining. What's been shown is that no proof of the existence of the Dissection is possible using only the standard axioms of set theory - Excluding the AoC. The reason being that if such a proof were possible it would mean that the Logical system of Axioms-Exluding AoC are not consistent with the system of Axioms-Including AoC. This would contradict the Godel result. However, there is another consistent logical system we could adopt just as easily. That is the Standard Axioms + The axiom that AoC is False. It's concievable that in this logical system it can be shown that the Dissection is possible. Only the said proof of the dissection's existence would have to somehow make use of the Axiom that AoC is False. Furthermore, we might adjoin a New axiom to the Standard ones, creating a consistent logical system which makes no mention of the AoC and in which it can be shown that the Dissection is possible. In that case we would get the Free result that the AoC is False in the new system. PairTheBoard [/ QUOTE ] Thank for that explanation PTB. All that made sense, but I still don't see a contradiction anywhere.... [/ QUOTE ] There is no contradiction. It's a proof By Contradiction. PTB
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#15
10-17-2007, 07:24 PM
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Re: Banach傍arski paradox = Contradiction?
The contradiction is between
"If the tetrahedron can be dissected, then the so-called 'axiom' of choice is, in fact, not an axiom, but a provably false statement" (contrapositive of the dissection result) and "Standard analysis with the axiom of choice is logically consistent" (already proven by Godel) but accepting a provably false statement as an axiom leads to logical inconsistency) ... from which we conclude that the tetrahedron cannot be dissected.
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#16
10-17-2007, 08:26 PM
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Re: Banach傍arski paradox = Contradiction?
[ QUOTE ]
"Standard analysis with the axiom of choice is logically consistent" (already proven by Godel) [/ QUOTE ] Just a small nit, but Godel only proved the equiconsistency of ZFC and ZF. The absolute consistency of ZF is, of course, not proven.
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