tag:blogger.com,1999:blog-8148573551417578681.post7763572186288074600..comments2021-01-15T10:36:20.149-08:00Comments on Dark Buzz: Logicism did not failRogerhttp://www.blogger.com/profile/03474078324293158376noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-8148573551417578681.post-88688839178725710172019-12-17T14:37:14.574-08:002019-12-17T14:37:14.574-08:00(continued) The consistency of Arithmetic is more ...(continued) The consistency of Arithmetic is more or less self-evident, from the fact that integers exist and we know how to add and multiply them. The consistency of set theory is far from evident, and if someone discovers another contradiction in set theory tomorrow, like Russel did, people won't be too shocked. Hilbert wanted to eliminate the possibility by using methods everyone could agree on, by using operations on whole numbers.<br /><br />When you formulate an axiomatic set theory, proving theorems becomes a calculation on symbols, on a computer. It turns into manipulating whole numbers. What Hilbert wanted was a proof in a theory of arithmetic that this operation will never reach a contradiction. Further, he wanted a proof that every theorem would eventually be provable, with perhaps a proof that takes a long long time.<br /><br />But proving the consistency of set theory is HARDER than proving the consistency of arithmetic, because set theory proves the consistency of Peano Arithmetic! So if you could prove set theory is consistent, you would automatically prove arithmetic is consistent. So the goal of "proving your own consistency" was a WARM UP PROBLEM for the harder problem of proving set theory consistent.<br /><br />The program went well for theories like Pressburger Arithmetic (just addition, no multiplication), and for the theory of the real numbers under addition and multiplication (this is easier than the whole numbers, because there are algorithms to find roots of polynomials). <br /><br />But then Godel showed that even the warm up problem is impossible! Peano Arithmetic can't prove set theory is consistent, because it can't even prove Peano Arithmetic is consistent!<br /><br />The method of proof showed that Peano Arithmetic can always be extended by "Peano Arithmetic is consistent" to get a new theory, we'll call that PA+1. Then PA+1 can be extended to PA+2, and so on.<br /><br />This process of extension can be extended past the integers, to PA+\omega, then to PA+\omega+1, and so on through all computable ordinals.<br /><br />It is not proved, but it is true, that Hilbert's program becomes correct when formulated on ordinals, not on integers. Hilbert did get it wrong. But he wasn't a simpleton. He didn't expect the proof of consistency of Arithmetic to provide evidence that Arithmetic is consistent. That evidence comes from the computational model of the integers.<br /><br />By constructing computable ordinals, using various methods which grow in complexity indefinitely, you can approach mathematical truth in the limit. But it is difficult. Truth does exist, but it requires evolving new ordinal notations.Ron Maimonhttps://www.blogger.com/profile/07091536472786521926noreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-70795200206687530572019-12-17T14:36:57.859-08:002019-12-17T14:36:57.859-08:00You made serious mistakes in this post. First, the...You made serious mistakes in this post. First, the people who don't believe in truth WEREN'T LEFTISTS! They were on the FASCIST RIGHT! Leftists believe in truth.<br /><br />This mistake is particularly maddening because this was exactly the debate between the right wing and left wing in the 40s and 50s, with extreme right wingers like Heidegger taking the position "There is no truth!", "Your being is primary", while left-wingers like Carnap or Godel or Hilbert were busy elucidating what no-nonsense scientific truth meant.<br /><br />Now on to your confusion about "why prove your own consistency"?<br /><br />That's not what Hilbert was after. He was after something more interesting and subtle. The theory of Peano Arithmetic is obviously consistent, because we understand the mental model of the integers, and we can verify that the axioms are true there. Computationally. By defining the operations of plus and times on whole numbers. The axioms of Peano Arithmetic say "there are whole numbers", "there is a plus", "there is a times", and "all the whole numbers are found by stepping up from 0 by one unit indefinitely". Those simple axioms allow you to prove a tremendous amount of stuff, a large fraction of elementary mathematics. But you can't even speak about things like "functions from the real numbers to the real numbers", because those concepts aren't in the language in a simple way. BUT, you can code these more advanced concepts up in a simple way, using a formalism like unicode or coq to represent mathematical symbols. Everything inside a computer is a gigantic whole number. All the transformations mathematica does can be described by Peano Arithmetic.<br /><br />But now we have set theory. In set theory, there are statements like "consider all function from the real numbers to the set of all functions on the real numbers". These enormous entities are much, much bigger than the set of integers and plus and times, and WE CAN'T verify that the axioms are true by any calculation. We just have to piddle around with intuition.<br /><br />Worse than that, when people did think they understood things intuitively, Russel found a paradox. "What about the set of all sets that don't contain themselves?" That was a construction in Frege's set theory, and it made the whole thing inconsistent.<br /><br />So Hilbert wanted to make sure that never happened again. He wanted to prove the consistency of SET THEORY. Not of Arithmetic. He wanted to prove the consistency of set theory using the ideas of arithmetic.Ron Maimonhttps://www.blogger.com/profile/07091536472786521926noreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-83726752662532770612015-09-29T10:30:19.814-07:002015-09-29T10:30:19.814-07:00Thanks. You can download a free copy of the first ...Thanks. You can download a free copy of the first paper <a href="http://opus.ipfw.edu/philos_facpubs/298/" rel="nofollow">here</a> or <a href="http://philpapers.org/rec/BULTSO-4" rel="nofollow">here</a>.<br /><br />Yes, there are some consistency proofs of Peano arithmetic and other systems, but they are not very satisfying.Rogerhttps://www.blogger.com/profile/03474078324293158376noreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-1639681257095337702015-09-29T07:37:40.468-07:002015-09-29T07:37:40.468-07:00Your views are strongly supportive of a reversed n...Your views are strongly supportive of a reversed narrative about Cantor's set theory. Set theory came after most useful mathematics and the use of the term "finite" is quite abused. Hilbert had trouble explaining what he meant by it. Edward Nelson said “[f]initism is the last refuge of the Platonist.” The argument Godel made was the same type of diagonalization that Cantor performed and Turing after Godel. It has nothing to do with productive mathematics. By the way, people who think Godel and Turing spawned the computer revolution are plainly mistaken. It's a myth of mathematics departments.<br /><br />"No compelling evidence has yet been presented that G1 affects, or future refinements of it will affect, mainstream mathematics."<br />http://link.springer.com/article/10.1007/s11787-014-0107-3<br /><br />Self-referencing formulas and impredicative sets are not about mathematics proper. I can't stress how badly people don't understand this! By the way, people who are trying to connect this to the continuum hypothesis are crackpots. They similar to the extent they are both about nonsense.<br /><br />Chaitin finds that systems have a specific complexity bound beyond which provability falls to zero. Ron Maimon: "Gödel's theorem is a limitation on understanding the eventual behavior of a computer program, in the limit of infinite running time." (http://physics.stackexchange.com/a/14944)<br /><br />Roger: "Supposedly Hilbert thought that an axiomatization of math should first prove the consistency of its axioms. If so, that was a stupid belief, because inconsistent axioms allow proof of anything."<br /><br />Precisely! Poincare knew that assuming induction to prove it is circular anyways. I won't bother mentioning the weak retort about infinitely axiomitized systems because it's a fanciful and stupid reply to the absurdity of incompleteness.<br /><br />"Generalizing a bit, this suggests that logicism may be safe from Gödel's first incompleteness theorem and even the second, as well. If we do not and cannot know the consistency of a putative logicist system, then we are probably no better off with regard to mathematical knowledge than it is (assuming it is otherwise powerful enough). And, if we can know of a system that it is consistent, then, for the benefit of logicism, we should regard the system as too weak to be the relevant one for logicism"<br />http://www.jstor.org/stable/2214847Anonymousnoreply@blogger.com