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From: Chris Menzel on 20 Apr 2005 23:19 On 20 Apr 2005 18:10:38 -0700, Ross A. Finlayson <raf(a)tiki-lounge.com> said: > Now, I think that any statement that you can say about multiplication, > that the same thing can be said using addition. > > 2 + 2 = 4 > > Now, 2*2 also equals 4, if you remove the word "multiplication" from > the dictionary, in this arithmetic of integrs, I can still say "using > addition, as many times is specified, reptition of addition as many > times as is specified" wherever that word was excised from the > vocabulary, and show the same things about the integers. Roughly, that indefinite "as many times as specified" is why addition alone won't do the job. > That's to say, I don't see anything provable in Peano arithmetic, with > addition and a special word for repeated addition called > multiplication, that isn't provable in Presburger arithmetic. And thus the problem. You just consult your untutored intuitions and your wildly undisciplined imagination instead of actually learning the mathematics. > If that is so,... It isn't. That it isn't is a simple fact of basic mathematics which you simply refuse to learn. I really don't get it. Why would anyone choose just to ramble on aimlessly instead of striving to learn and understand? Really, don't you *want* to know what you're talking about? > and all true statements in Presburger are provable in Presburger, then > all true statements provable in Peano are provable in Presburger, or > Peano, where the Presburger arithmetic's "axioms" are a subset of > Peano arithmetic's "axioms." OK, prove the Presburger version of this statement: (x)[3*(x+1) = (3*x)+3] > Back to Goedel and non-logical axioms, in a way conceptually similar > to talking about Cantor's first and second, you might want to consider > Goedel's first and second incompleteness results, ... And you might want to consider studying them so you actually have a clue as to what they mean. > I'm basically unimpressed with you calling these simple logical > statements "ignorant foolishness." I am not trying to impress you. I am trying to help you. The piece of ignorant foolishness in question was your claim that Goedel's 1st theorem shows that all theories with nonlogical axioms are incomplete. Since you seem not to have put 2 and 2 together: Presburger Arithmetic is a (nontrivial) example of a complete theory. Really. So what you said was false, a product of ignorance. Now you know you were wrong. You are no longer ignorant on this point. Congratulations! Doesn't it feel good to learn something? > What can I say? For starters: "Gee, you're right! Multiplication *isn't* definable in terms of addition. Presburger arithmetic *is* complete! *Not* all infinite sets are the same size! I'd better actually start studying this stuff in order better to understand why!" Yeah, that'd be a great thing for you to say. Chris Menzel
From: Ross A. Finlayson on 21 Apr 2005 01:50 Hi, I'm a computer programmer, by trade. I've studied some, types, and stuff. I've written trivial computer programs in the high and low computer languages, for example, C, C++, Java, assembler, etcetera. So, you might understand why I think it's not much of a stretch to use two variables, towards a simple, even functional where those are imperative languages, and basically efficient counter for reiterating addition. On the Turing tape, in programming terms there are basically only "line numbers" and "goto", like BASIC. The halting tape basically reaches an instruction that points to itself. In a sense, each machine halts, if they ever enter a loop. If they don't, if the instructions don't enter a loop, and each step is finite, on the one-dimensional doubly infinite tape, the non-halting machine, in the sense of not halting into a loop, diverges like the integers: 0, 1, 2, 3, ..., or 0, 1, -1, 2, -2, .... Using static analysis, especially in theories with no undecidable propositions, it is simple and easy to tell whether a Turing tape halts on any input. If the Turing tape program is finite, then it halts in the sense of looping. If it's infinite and loopingly halts, it represents a normal number, or rational number in base other than two. In the infinite, the program loopingly halts. On the Turing tape, in programming terms there are basically only "line numbers" and "goto", like BASIC. The halting tape basically reaches an instruction that points to itself. In a sense, each machine halts, if they ever enter a loop. If they don't, if the instructions don't enter a loop, and each step is finite, on the one-dimensional doubly infinite tape, the non-halting machine, in the sense of not halting into a loop, diverges like the integers: 0, 1, 2, 3, ..., or 0, 1, -1, 2, -2, .... Using static analysis, especially in theories with no undecidable propositions, it is simple and easy to tell whether a Turing tape halts on any input. If the Turing tape program is finite, then it halts in the sense of looping. If it's infinite and loopingly halts, it represents a normal number, or rational number in base other than two. In the infinite, the program loopingly halts. Please feel free to ignore this post, except for the part about using two variables, in a first-order logic, to define multiplication of integers in terms of addition of integers. I appreciate that, Chris, you represent some important academic concerns. By the same token, do not be misleading these people about non-logical axioms and Goedelian incompleteness. You've been following this thread. The universe is infinite. Do you disagree with that in terms of the way that is stated? If you don't, then it's basically empirical evidence, or in the way of physicists a proof, that infinite sets are equivalent. Ross -- "Also, consider this: the unit impulse function times one less twice the unit step function times plus/minus one is the mother of all wavelets."
From: Patrick on 21 Apr 2005 09:33 Ross A. Finlayson wrote: > Hi, [.snip.] Instead of carrying on with diversions why don't you just answer Chris' question: "OK, prove the Presburger version of this statement: (x)[3*(x+1) = (3*x)+3]" -- Replace Roman numerals with digits to reply by email
From: Eckard Blumschein on 21 Apr 2005 11:40 On 4/18/2005 8:44 PM, Will Twentyman wrote: >> Well, my expression was not properly choosen. I shoud better have chosen >> "such a quantity" instead. The quality "infinite" is >> an either-or-quality, something that cannot be quantified like pregnant, >> dead, entirely, countable etc. > > My point was that, given sufficient motivation, almost anything can be > quantified. I can certainly quantify "pregnant" by the trimester, > month, etc. I can quantify "dead" by the percent of dead cells/organs, > or the number of hours since death. It is sometimes a matter of the > *will* to quantify than anything else. Of course, any such > quantification is more likely to be accepted if it is reasonable and/or > useful. To be or not to be (Shakespeare) is obviously a matter of abstraction. > >> Different levels of size of infinity are nonsense. > > And yet Cantor found a way to quantify that very thing. You may wish he > hadn't, but he did. He cheated. He did not obey the clear abstract meaning of infinity. Only insane people would say this wife is more than pregnant or more than dead. > >> The notion of infinite numbers is self-contradicting. > > Yet he found a representation for them that is not. Not telling the whole story is also a lie. > >> Cantor felt entitled to create a nonsense-mathematics. This is nothing >> uncommon at all. However, after more than a hundred years it is >> legitimate to ask for at least any tiny benefit. So far I only got aware >> of trouble. > > Then don't use it and don't worry about it. Just make sure your > understanding of mathematics and set theory does have a sound basis. I fear, set theory does not have a sound basis. > >>>>I see most of the paradoxes just superficially remedied. >>> >>>Please name one that is not. I'll admit that some are not remedied in a >>>way that is entirely satisfying, but if the paradox is eliminated, where >>>is the superficiality? >> >> E. g., AC does not provide access to a well-ordered list of the rationals. > > Noting that you meant reals, Yes, of course. > that is precisely what it does. Of course, > it does not give you an explicitly defined function for the > well-ordering, but it doesn't claim to. It just claims there is such a > function. Well, I objected against access to it. >> Already the words countable and uncountable mutually exclude each other. >> I do not see any sense in this formal repetition that contradicts to >> the notion of oo. 130 years of missing benefits seems to confirm my doubt. > > Missing benefit to who? If they are still around, someone must have > seen some benefit. Look around what groups of ideas, illnesses, and organisations are still around. >> Well. It must not be considered a brilliant idea. It is rather stupid. >> Nonetheless, it lacks any justification. > > Yet it was an idea you would not have come up with. It was radical > enough to cause a great deal of controversy. Are you sure it wasn't > brilliant? Yes. >
From: W. Mueckenheim on 21 Apr 2005 12:08
"Ross A. Finlayson" <raf(a)tiki-lounge.com> wrote in message news:<1114002908.531397.20070(a)o13g2000cwo.googlegroups.com>... > No, Wolfgang, the universe has infinite numbers of things. > > Besides the fact that there are infinitely many theoretical theoretical > particles in a non-standard model, of physical particles, of particle > physics, that force vector is not just a scalar, it has infinitely many > components, in a variety of ways that you can look at it. > > When you say, "oh, that's just a method of calculating", that's kind of > ridiculous. To evaluate a solution to that system, you're evaluating > those things in terms of continua, things continuous. Accept that! Even if accepting that, the means to store numbers are limited. The infinitely many components and the virtual exchange particles of forces, vacuum polarization and so on are not suitable to store infinitely many bits. Mathematics is limited. Every reasonable set is limited. Actual infinity in physics is nonsense, as every reasonable physicist will confirm. Regards, WM |