Give me ONE digit position of any real that isn't computable up to that digit position!
in [Math]
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From: Transfer Principle on 8 Jun 2010 14:08 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.
From: Transfer Principle on 8 Jun 2010 14:21 On Jun 7, 7:03 pm, Tony Orlow <t...(a)lightlink.com> wrote: > On Jun 6, 1:54 am, Transfer Principle <lwal...(a)lausd.net> wrote: > > I agree (except for the word "drivel"). > Would you like an example, if only in idealization, about a real-world > every-day occurrence? Got one, with cosmic consequences... OK, I'd like to see an example of a real-world occurrence, especially one which can be modeled using T-riffics or H-riffics -- though perhaps in the main TO thread, not this Herc thread. Hey, I notice that both TO and Nam Nguyen have entered this ever-growing thread. To me, it's always interesting to read a thread with several different viewpoints appear.
From: MoeBlee on 8 Jun 2010 14:35 On Jun 8, 1:08 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > Please > include what I was responding to when I said "would do that"? What > EXACTLY is the "that" that I was speaking of? CORRECTION (and to similar comments elsewhere in my post): I overlooked that you did include the passage. AS you presented, this is the exchange: [If] JSH were to state that the sky is blue, the _standard theorists_ would be the ones to start coming up with obscure counterexamples such as the Doppler effect at velocities approaching c, alien languages in which "blue" means "red," and so forth. - Transfer Principle The standard theorists" would do that? How do you know? WHICH "standard theorists"? And would you please say exactly what you mean by "a standard theorist"? - MoeBlee (Repeating some of what I just mentioned:) (1) STILL what do you mean by "a standard theorist"? (2) I guess Fred Jeffries is a "standard theorist"? (3) Whatever Jeffries has said, I would put the point this way: If Joe Blow says "the sky is blue", of course one can truthfully (though not necessarily pertinently) say "No, it isn't when 'sky' refers to this stuff at my feet and 'blue' refers to the color of my hair". So, of course, "2+2=5" is even finitistically true when the numerals '2' and '5' designate the numbers two and four. But this is not even the kind of thing I thought was at stake in your "Doppler" example. I thought you were referring to mere contrarianism for its own sake, for merely finding some example that is aside the point when we are meant to consider the more common-place generalization. The sky is gray or black or whatever at different times. But that's not what's at issue when someone says "The sky is blue" in a more modestly informal sense, and when we take that sense, we don't need to quibble that the sky is sometimes gray or black or whatever. And also, we can always reinterpret words to make sentences come out true or false relative to such interpretations. So, if a crank says "1+1=2", no, I don't know anyone who would say, with a straight face, "WRONG, because in my system '1' stands for the set of real numbers, '+' stands for intersection, '=' stands for 'subset' and '2' stands for pi", unless it were simply to make the point about mathematical logic or unless one really wanted to point out that some other interpretation was being used. So, when I say "1+1=2" is a finitistic fact, I don't mean that there is not some interpretation in which "1+1=2" is false, but rather that "1+1=2" is finitistically true when we interpret those symbols in their ordinary way. So, no, a reasonably informed person should not hassle cranks for merely asserting finitistic facts. That there may be some contrarians- for-contrariness-sake out there who may be wont to hassle anybody for virtually anything, of course, I can't deny. MoeBlee
From: herbzet on 8 Jun 2010 14:41 Transfer Principle wrote: > Also, the mathematician Willard van Orman Quine > came up with a perfectly respectable theory which > proves the negation of Cantor's Theorem. Good point, which you made in a previous post to me, and which I've been thinking about. If the question is whether or not it is true of sets that they are all lesser in cardinality than their respective powersets, then I suppose that what the word 'set' denotes, or could be taken as denoting, would be a different object in ZFC and NFU. That would be a more interesting discussion, minus the polemics over mathematical truth. I guess any of ZF(C), NF(U), etc, etc, would do as a foundation for mathematics, so long as the bridges don't fall down. > Thus, according to Bender's logic, Quine must have been > an anti-Cantor "kook" as well. Non-sequitur, btw. -- hz
From: Transfer Principle on 8 Jun 2010 15:57
On Jun 8, 11:08 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 8, 12:20 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > Back in March, I had the following discussion with MoeBlee: > > Me: > > > > [If] JSH were to state that the sky is blue, the _standard theorists_ > > > > would be the ones to start coming up with obscure counterexamples > > > > such as the Doppler effect at velocities approaching c, alien > > > > languages in which "blue" means "red," and so forth. > > MoeBlee (9th of March, 8:37AM MoeBlee's local time): > > "The standard theorists" would do that? How do you know? WHICH > > "standard theorists"? And would you please say exactly what you mean > > by "a standard theorist"? > > MoeBlee was skeptical that an adherent of standard theory > > would come up with obscure counterexamples to generally > > accepted facts. > As far as I can tell, you still seem to think as if there are two > political parties, the "Standards" and the "Rebels" so that people in > these threads neatly and naturally fall into one or the other camp. And so MoeBlee caught me using the forbidden phrase. One thing that I need to do is to avoid using the grouping terms. I quoted a post from back in March in which I used the grouping terms. Note that I may use a grouping term if another poster has used it. In this case, "adherents" refers to Herc's notion that ZFC is in some ways like a "religion." 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. So what should I do now that I was caught using the forbidden grouping phrase? I need to rewrite the question without the offending grouping term and refer only to the two individuals, namely Herc and Jeffries. So the statement becomes: Herc and Jeffries both write statements that are refuted by ZFC, yet are treated differently when they do so. Why? > I have a cluster of notions and interests and questions regarding all > kinds of theories and various methods of logic even. I've never taken > any pledge of allegiance to ZFC or even to first order logic or > whatever. You've never quoted anything by me that determines I'm a > "standard theorist" and still without your saying what the hell > constitutes a "standard theorist". And I still won't, since that's a forbidden term that I shouldn't have used in the first place. > > > > It is precisely for this reason that I am open to > > > > reading about alternate theories, as long as those > > > > theories don't contradict empirical evidence. Thus, > > > > even I won't defend a theory which seeks to prove > > > > that 2+2 = 5, since one can prove in the real world > > > > that 2+2 = 4, not 5. > Again, please include the context of my comment that has anything to > do with "obscure counterexamples to accepted facts". What does that > even MEAN? If something is a fact, in what sense does it have a > counterexample? And what kind of facts? Empirical facts? Finitistic > mathematical facts? Below is the post of MoeBlee, dated the 9th of March, at 5:37PM Greenwich time. In order to mention the grouping terms, I replace them with [group] in this post. > On Mar 9, 12:21 am, Transfer Principle <lwal...(a)lausd.net> wrote: > > if a known so-called [group] let's say > > JSH, were to state that the sky is blue, the [group] > > would be the ones to start coming up with obscure counterexamples > > such as the Doppler effect at velocities approaching c, alien > > languages in which "blue" means "red," and so forth. > The [group] would do that? How do you know? WHICH > [group]? And would you please say exactly what you mean > by [group]? > > Case in point -- in a thread in which the [group] > > demanded that a [group] accept Cantor's theorem as beyond dispute, > > I mentioned that there are some statements, such as 2+2=4, which, > > unlike Cantor's theorem, I do accept as unequivocally true. Then > > a [group] immediately brought up 2+2 == 1 (mod 3). > (1) I'd like to see the full context of that. (2) So because one > [group] said such and such in one instance, then you > conclude that [group] (whatever you mean by that) > generally say such and such? > MoeBlee And since Google wouldn't show me the post that I had in mind to answe MoeBlee's question (1), I told him that I would wait until I saw a post of the type that I was describing that day. And today, I see the post by Jeffries in which he discusses 6+4+8 = 26. > I never criticized anyone for not "conforming" to ZFC. (Of course, IF > one presents something as if within ZFC that is not within ZFC then it > is correct to point out that, contrary to their representation, their > argument or supposed "reductio" or whatever, is not a ZFC argument or > not a reductio in ZFC). I've critized people for incorrectly and > ignorantly shooting their fat mouth off about ZFC. Have whatever > theory you like. But Herc does not present a theory. Rather, he > presents a bunch of confused and dogmatic rambling. He doesn't even > know what ZFC IS. Object to ZFC as much as you like. But if one's > objections are incorrectly premised or confused or ignorant as to what > ZFC is, then I'll point that out if I wish. That is not denying anyone > the prerogative still to hold philosophical objections to ZFC or to > aspire to a different theory let alone to actually presenting a > theory. This reminds me of another type of grouping that I may mention since another poster (Chandler???) mentioned it earlier. Instead of those who "adhere" to ZFC, there are those who _understand_ ZFC vs. those who don't. MoeBlee implies this above as well. Herc objects to ZFC because he doesn't _understand_ ZFC or how the diagonal argument works. 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. 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. |