Give me ONE digit position of any real that isn't computable up to that digit position!
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From: herbzet on 8 Jun 2010 16:26 Transfer Principle wrote: > On Jun 7, 4:08 pm, Ostap Bender <ostap_bender_1...(a)hotmail.com> wrote: > > On Jun 7, 3:57 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > > > > Thanks for illustrating my point. Indeed, only adults with severe > > > > mental handicaps would continue discussing this little proof and > > > > trying to understand it using "box" analogies. What difference is it > > > > if it's a box or a cup or a piece of paper? Who needs props when the > > > > idea is so simple for anybody with IQ above 90? > > > So you avoid this proposition because you "don't need" boxes? > > No, what I am saying is that different people have different mental > > abilities. While people with IQs above, say, 95 find Cantor's proof to > > be easy and trivial, others, like yourself, need various props like > > "boxes" to help themselves visualize the diagonalization idea. > > I don't believe that those who reject Cantor's Theorem > must therefore have IQ's under 90 or 95. > > Once again I don't dispute that Cantor's Theorem has an > easy, possibly even trivial, proof in ZFC. But that's > just the thing -- _in_ZFC_. But this doesn't mean that > anyone who opposes ZFC must therefore have a low IQ. > > Bender states that he first learned of Cantor's Theorem > in a 7th grade math club. Of course, no one is going to > start discussing NFU or theories other than ZFC to a > group of 7th graders -- ZFC is automatically assumed, > as it should be. > > But now in adulthood, one can be made aware of some of > the theories other than ZFC. George Greene mentioned a > set theory with a universal set, and Barb Knox also > mentioned some ideas in her post. Ths, blind acceptance > of ZFC is no longer necessary. Bought a book recently -- "Classical and Non-Classical Logics" by Eric Schechter (got it a huge discount). It is *very* elementary in some parts, and it is mostly about propositional logics -- it hardly touches quantified logic. http://www.math.vanderbilt.edu/~schectex/logics/ The author argues that non-classical as well as classical logic should be taught at the elementary level, a breadth- first rather than depth-first approach. I like the book very much -- I've been thinking of writing a "review" of it, but I'm not that verbal. It turns out that I'm interested in logics that are close to relevance logic -- although the book also neglects the S1 modal logic of strict implication, which is in the neighborhood of relevance logic too. In a similar spirit, it might be well to mention alternative set theories to 7th grade math club members. -- hz
From: Aatu Koskensilta on 8 Jun 2010 21:51 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Another example would be suppose we define AI formal system as one in > which some of its theorems would be syntactically isomorphic to a > formal system in which in turn there's a formula than can't be model > theoretically truth definable. The clause "a formula than can't be > model theoretically truth definable" would require a close inspection > of our current knowledge about the foundation of reasoning via FOL. Well, the notion of a formula being "syntactically isomorphic to a formal system" also needs some elucidation. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Tim Little on 8 Jun 2010 22:05 On 2010-06-08, Transfer Principle <lwalke3(a)lausd.net> wrote: > But the term: > >> > standard theorists > > I invented, and so it's forbidden. I can talk about standard > _theories_, since theories like ZFC, ZF, PA, FOL are standard. But > as soon as I talk about "theorists" the phrase is forbidden. MoeBlee is more charitable than I am about your phrasing. MoeBlee asks you what you mean by the term "standard theorist", implicitly giving you the benefit of the doubt that perhaps you meant something that could be interpreted so as to make your statements true. I say you are simply wrong, that standard theorists (or by any other name) say no such things as you are imputing to them, and that your saying they would is strong evidence that you have extremely poor reading comprehension skills and/or next to no ability to form mental models of other people. Quite probably both. > Herc and Jeffries both write statements that are refuted by ZFC, yet > are treated differently when they do so. Why? The reason is that Herc and Jeffries have very different treatments of other posts themselves, very different demonstrated levels of mathematical competence in their postings, and very obviously different types of claims in the context of their statements. The ability to distinguish these may or may not be a skill you are able to learn. I hope that one day you can. > But I don't like this idea that the only people (on sci.math, at > least) who attack ZFC are those who don't _understand_ it, and > everyone who does understand ZFC defends Cantor. Why should liking an idea affect whether or not it is actually true? The empirical evidence is that the vast majority of those who have attacked ZFC on sci.math actually *don't* understand it. Whether you like that or not is of little relevance to anyone but you. > What I'd love to see is a poster who _fully_ understands ZFC, > perhaps as much or even more than MoeBlee understands the theory -- > yet still doesn't believe that ZFC is the best theory. There are plenty of people in this newsgroup who understand ZFC, and are perfectly willing to accept that it may not be the best theory. Isn't it nice that your wish is already granted? However, I suspect that you actually meant the stronger statement that they should consider ZFC to be "unacceptable" for some reason. Bad luck there - all of them here at least appear to consider ZFC to be an acceptable theory. I have heard that there are people who understand ZFC and consider it to be unacceptable. However, they are not flooding newsgroups with a multitude of diatribes against it. The vast majority of posts arguing against ZFC or its theorems in sci.math are actually from incompetent cranks. - Tim
From: Marshall on 9 Jun 2010 09:23 On Jun 8, 6:37 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Marshall <marshall.spi...(a)gmail.com> writes: > > That's right. Uninteresting. The TM is already too simple to be of > > much interest to me. What is done with it could be done with any model > > of computation; TMs are adequate for what we do with them but they are > > horrifically awful by almost any metric I can think of. > > They are conceptual simple, a more-or-less straightforward idealised > model of the way human computers go about computing stuff, as Turing > clearly explains. This is important when discussing, say, the > Church-Turing thesis. Other models of computation are better suited for > other purposes. Just as I said. Marshall
From: MoeBlee on 9 Jun 2010 15:21
On Jun 9, 1:38 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jun 8, 1:19 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Jun 8, 2:57 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > So how should I define "standard theorist"? Let me > define it simply by saying that a "standard > theorist" is an adherent of the ZFC "religion" as > defined by Herc. I don't know Herc's "definition". But I surely don't adhere to ZFC as a religion. > I believe that ZFC is a "religion" to which posters > "adhere," if and only if Herc's "rambling" is > "dogmatic" You believe a lot nutty things. The above is another of them. > -- indeed, I'll use "adherent" to refer > to both MoeBlee and Herc, "adhering" to their > respective ideas about ZFC. What ideas do I have about ZFC that you think I adhere to (other than that such obvious things as that ZFC is a formal theory and certain finitistic facts such as that such and such a sentence has a ZFC proof)? > > > What I'd love to see is a poster who _fully_ > > > understands ZFC, perhaps as much or even more than > > > MoeBlee understands the theory -- yet still doesn't > > > believe that ZFC is the best theory. > > How about MoeBlee himself? I make no claim that ZFC is the best theory. Moreover, "best" theory is probably best evaluated relative to "best for WHAT end?". Moreover, it's not clear to me that I even have more than a quite small stake in deciding what is the best theory. > > You won't find a post of mine in which I > > declare ZFC is the best theory. > > I defer to Little, who understood my intent as well as > gave a reasonable response: What sense is there in deferring to Little about MY beliefs? > "However, I suspect > that you actually meant the stronger statement that they should > consider ZFC to be "unacceptable" for some reason. Bad luck there - > all of them here at least appear to consider ZFC to be an acceptable > theory. > "I have heard that there are people who understand ZFC and consider > it > to be unacceptable. However, they are not flooding newsgroups with a > multitude of diatribes against it. The vast majority of posts > arguing > against ZFC or its theorems in sci.math are actually from incompetent > cranks." So? Anyway, 'acceptable' is like 'best' in the sense that if one asks me "Is ZFC an acceptable theory" then I'd have to ask "What do you mean by 'acceptable' in this regard?" Theories are not things I go around dividing into "acceptable or unacceptable". MoeBlee |