One equals to two, 1=2

* I have argued that math debates are stupid and whoever starts them should be voted down as a general rule. * I have proven my case by assignment of values between 1 and 2. * I have shown that incoherent mathematics are permitted by the topic. * I have proven that my mathematics are coherent. * I have proven that my opponent cannot (by Godel's) prove his mathematics are coherent. * My opponent has forfeited (round 4). ------------------------------ * For the pope equation I meant three. But, good catch. "How are you the pope?" - I would be the pope on the grounds that the pope and I would be one. One person can't be equal to another, it's a contradiction in the real world. Incoherent mathematics does not have any relevance in the real world. This is why we usually prohibit contradiction, not because there's any real restriction to the system. My opponent doesn't understand the purpose of my point there. I am showing that within an incoherent system proving anything is possible. The statement makes perfect sense, and of course it's contradictory! Have you even paid any attention at all? Contradiction is allowed in incoherent mathematics! Everything is allowed! One can equal two as well as not equal two if when we're talking about "one" and "two" they are abstract concepts not describing anything in reality. You don't seem to be following the argument: Why can't things be contradictory? I can and have proved that with contradiction I can prove my system to be valid. When things are allowed to be contradictory, correct things can be proven correct (incorrect things too) and the system itself can be shown to be valid. However, by Godel's incompleteness theorem I know that your answer (or any readers answers) will be unjustified. The reason for this is coherency. A coherent system cannot prove it's own coherency. Think about it this way, if the system could prove it's own coherency, what if it were actually incoherent? Then the system and the proof of the system are both incoherent. This is Godel's incompleteness theorem: http://en.wikipedia.org... ---------------------------- T1 can prove that if T2 proves the consistency of T1, then T1 is in fact consistent. For the claim that T1 is consistent has form "for all numbers n, n has the decidable property of not being a code for a proof of contradiction in T1". If T1 were in fact inconsistent, then T2 would prove for some n that n is the code of a contradiction in T1. But if T2 also proved that T1 is consistent, i.e. there is no such n, it would itself be inconsistent. We can carry out this reasoning in T1 and conclude that if T2 is consistent, then T1 is consistent. Since by second incompleteness theorem, T1 does not prove its consistency, it can't prove the consistency of T2 either. This easy corollary of the second incompleteness theorem shows that there is no hope of proving e.g. the consistency of first order arithmetic using finitistic means provided we accept that finitistic means are correctly formalized in a theory the consistency of which is provable in PA. It's generally accepted that the theory of primitive recursive arithmetic (PRA) is an accurate formalization of finitistic mathematics, and PRA is provably consistent in PA. Thus PRA can't prove the consistency of PA. This is generally seen to show that Hilbert's program, which is to use "ideal" mathematical principles to prove "real" (finitistic) mathematical statements by showing that the "ideal" principles are consistent by finitistically acceptable principles, can't be carried out. This corollary is actually what makes the second incompleteness theorem epistemically relevant. As Georg Kreisel remarked, it would actually provide no interesting information if a theory T proved its consistency. This is because inconsistent theories prove everything, including their consistency. Thus a consistency proof of T in T would give us no clue as to whether T really is consistent; no doubts about T's consistency would be resolved by such a consistency proof. The interest in consistency proofs lies in the possibility of proving the consistency of a theory T in some theory T' which is in some sense less doubtful than T itself, e.g. weaker than T. For most naturally occurring T and T', such as T = Zermelo-Fraenkel set theory and T' = primitive recursive arithmetic, the consistency of T' is provable in T, and thus T' can't prove the consistency of T by the above corollary of the second incompleteness theorem. --------------- Coherent systems can't prove their own coherency. I advocate, by assigning 1 to equal 2, that my system is incoherent. As a consequence, I can prove the validity of my system and my opponent cannot. -- Either one equals two or one does not equal two. Only one result may be accepted. Therefore, only two results may be accepted. We have two options: "equals two" or "does not equal two" -- Therefore, two of the two options are correct. One equals two and one does not equal two. Therefore, one equals two. By Godel's theorem, I have fulfilled my burden of showing 1 = 2 and he CANNOT. It's actually impossible for him to construct a similar proof unless he is using incoherent mathematics. In incoherent mathematics, 1 = 2, always (one contradiction allows for all contradiction). I have not only proven my statement valid, I have proven that he cannot prove his statement valid. The fact that he seems confused on this point and doesn't so much as try shows the weakness in his case. Further, I'm sick of these stupid math debates. You people should just vote them down when ever you see them. Whoever is first in the debate, started the stupid debate, they should lose just for that. This is not to say I didn't win this debate, I crushed my opponent's argument as well as perhaps informed readers a bit about the foundations of mathematics. Nothing in the topic or introduction required coherency (a relationship to the physical world would have implied such a requirement, none was made). I won hands down. However, math debates are stupid, let people get their win ratio up the old fashion way... by winning. Thank you for the debate.