An uncountable countable set
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From: mueckenh on 27 Aug 2006 10:34 Dik T. Winter schrieb: > In article <1156500654.035798.315570(a)m79g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > > > You apparently do not know what the word "computable" does mean. To > > > > > be precise: all algebraic numbers are computable in the mathematical > > > > > sense. > > > > > > > > To be precise: Each algebraic number is computable. But not all. > > > > Because then the list of all algebraic numbers was computable. > > > > > > But it is. > > > > So you can compute all solutions of a polynomial equation even of > > higher than fourth degree in finite time? I doubt that. > > I did not state that. I said that the numbers were computable, where > I use the mathematical sense of computable. And that means you can compute a number even if you need infinite time. Therefore in the mathematical sense each algebraic number is computable. (In reality it is not.) But even in mathematical sense not *all* algebraic numbers are computable. Should you be able to quote a serious scientist who asserted that the list of all algebraic numbers was computable? (You would prove this scientist unserious.) Regards, WM
From: mueckenh on 27 Aug 2006 10:36 Dik T. Winter schrieb: > In article <virgil-B3927D.12230325082006(a)news.usenetmonster.com> Virgil <virgil(a)comcast.net> writes: > > In article <1156501289.435365.119480(a)75g2000cwc.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > ... > > > The problem is not indexing but "indexing without covering". > > > That is easy to prove impossible for any finite natural number in unary > > > representation. And, alas, there are only finite natural numbers. > > > > I have no idea what "indexing without covering" means, and until it has > > a clear definition, I will continue to state that indexing all of them > > in any way indexes all of them. > > What I have been able to find was that with indexing of digit WM means > that every digit position can be indexed by a natural number. With > covering WM means that all positions to a certain one can be indexed > by a natural number. What WM is asserting is that when a number can > be totally indexed, it also can be totally covered. Quantifier dislexia > disguised. > Quantifier dyslexia is the assumption that there is a number that counts all natural numbers, i.e. which is larger than all natural numbers. My arguing holds for the unary numbers of my list. If you think not, then give an example of a finite number n which indexes the digit position n but does not cover all positions m =< n. Here is my example: The number 3 = 0.111 in my unary expression indexes the digit position 3 of any unary number and it covers the digit positions 1, 2, and 3 of any unary number. The same can be proven for any finite number n. In 0.111... there are allegedly only finite digit positions. Hence this proof is valid for every position of this number. Every position can be indexed, and every position can be covered. This is dictated by logic: Position n can be indexed ==> every position m < n can be covered. Not every position m can be covered ==> not every position n > m can be indexed. Regards, WM
From: Tony Orlow on 27 Aug 2006 10:45 MoeBlee wrote: > Tony Orlow wrote: >> Logic determines truth. Induction is more than just a form of proof. > > Logic determines VALIDITY. And induciton is indeed more than just a > form of proof. But your concept of induction is uninformed, > superficial, and confused. > > MoeBlee > That second sentence refers to inductive logic, as opposed to deductive logic, of which "inductive" proof is a form. Inductive logic derives rules from facts, whereas deductive logic derives facts from facts, using other facts called rules. Rules are a type of fact, which relate atomic facts logically. So, when you say logic determines validity, you seem to be saying that deductive logic determines whether a conclusion is valid, ASSUMING a set of facts, including atomic facts and axioms. Inductive logic, on the other hand, is used to measure the SOUNDNESS of the axioms or rules one applies, by comparing the conclusions of an argument to the reality it seeks to draw conclusions about. When you say my concept of induction is "uninformed, superficial, and confused", I would disagree. In my opinion, those that think it just "springs from the axioms" without a sound logical basis are the confused and uninformed ones. Those who refuse to recognize the inherently recursive nature of such a proof are being superficial. All I have suggeted as a change to inductive proof is that we consider any infinite value to be greater than a finite one, and consider equalities between expressions proven inductively true for al x greater than some finite y to remain true even when x is infinite, since x is still greater than that y. So, there you have it. Tony
From: Tony Orlow on 27 Aug 2006 10:50 Dik T. Winter wrote: > In article <44ef36ae$1(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > > Dik T. Winter wrote: > ... > > > > So you can compute all solutions of a polynomial equation even of > > > > higher than fourth degree in finite time? I doubt that. > > > > > > I did not state that. I said that the numbers were computable, where > > > I use the mathematical sense of computable. > > > > Such that one can specify which finite number of iterations will get one > > within a specific finite range of accuracy, gven a specific method of > > approximation? It's a limit concept, really, yes? > > Computer scientist you are? How wrong you are. The very first definition > is that a number is computable if there is a Turing machine so you can give > it a specific integer n and it will calculate all digits from the first > to the n-th. There is no limit concept involved at all. All algebraic > numbers are computable, as are a host of non-algebraic numbers (e, pi). Dik, you didn't even try to think about that one. Sorry. It was a lame answer. When you say you can calculate to the nth digit, that means you have calculated to within an error of b^-n, where b is the base of your number system. It means that you can approach the limit of pi, or e, or whatever value you are computing, to within any finite accuracy in a specific finite number of steps. What you said is not different from what I said. And, yes, I am a computer scientist, thank you. Have a nice day. Tony
From: stephen on 27 Aug 2006 11:23
Tony Orlow <tony(a)lightlink.com> wrote: > So, is it technically incorrect to say 1 = 3/3 = 1.000... = 0.999...? I > never thought so. Consider the following: int main() { int x=1; float y=1; double z=1; } Are x, y and z the same? If they are the same, why does their internal representation differ? Stephen |