A gap in Cantor's diagonal argument ?
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From: Arturo Magidin on 6 Aug 2010 22:52 On Aug 6, 6:58 pm, netzweltler <reinhard_fisc...(a)arcor.de> wrote: > I see, you misunderstood my statement. Please learn to quote the message you are replying to. You give no context, it makes it impossible to understand you. You are posting from mathforum; use the "quote original" button and edit down the quote to what is necessary. As to me "misunderstanding" your statement, no; what I did was take your staement *as written*. If you did not write what you meant, then the fault lies entirely with you, and not with me. > All I wanted to state is, that I cannot turn a terminating decimal like 0..3 into a non-terminating decimal like 0.333... (1/3) just by adding a number n of 3´s, where n is a natural number. Might be no news for you. This statement is about as confused as your previous one. If you mean that you adding a finite number of digits to a number with only finitely many digits cannot turn it into a number with infinitely many digits, all I can say is that bringing up decimals, terminating or otherwise, only confuses things. And if you did not mean that, then it's not what you wrote. -- Arturo Magidin
From: Tim Little on 6 Aug 2010 23:01 On 2010-08-06, Reinhard Fischer <reinhard_fischer(a)arcor.de> wrote: > Which of the following statements is true/false? > > 1. Terminating decimals are always countable, no matter how many > decimal places they consist of. "Countable" is a term that refers to a set. A terminating decimal is not usually considered to be a set. Do you mean "the set of all terminating decimals is countable"? If so, then yes. > 2. Decimals are uncountable, if they consist of infinitely many > decimal places. The set of all decimals consisting of infinitely many decimal places is uncountable, if that's what you mean. > 3. Numbers are countable, if they consist of a finite number of > decimal places. What sort of numbers? There are many kinds. However, any set of numbers representable by a finite number of decimal places is countable so it doesn't matter very much. > 4. Numbers are uncountable, if they consist of an infinite number > of decimal places. In this case it does matter what sort of numbers. Rational numbers can have infinitely many decimal places, but the set of all rational numbers is countable. Real numbers are not. > 5. Natural numbers are countable, if they consist of a finite > number of digits. The "if" is unnecessary: all natural numbers have a finite number of digits when represented decimally. The set of natural numbers *defines* countability, so the answer is obviously true. > 6. Natural numbers are countable, even if they consist of an > infinite number of digits. This is vacuously true: there are no natural numbers consisting of an infinite number of digits (unless you count leading zeroes). - Tim
From: Tim Little on 6 Aug 2010 23:08 On 2010-08-07, netzweltler <reinhard_fischer(a)arcor.de> wrote: > I can see similarities. A terminating decimal can approach > 0.333... but never reach it. This sounds to me like working with > limits, where n approaches infinity. Real numbers can be represented as classes of certain ("Cauchy") infinite sequences of rational numbers, where the real number is actually the limit of the sequence. So in at least that sense, yes. There are other representations though, and it usually only comes up when demonstrating a formal foundation for real analysis. - Tim
From: netzweltler on 7 Aug 2010 06:03 > On 2010-08-07, netzweltler > > > I can see similarities. A terminating decimal can > > approach > > 0.333... but never reach it. This sounds to me like > > working with > > limits, where n approaches infinity. > > Real numbers can be represented as classes of certain > ("Cauchy") > infinite sequences of rational numbers, where the > real number is > actually the limit of the sequence. > > So in at least that sense, yes. There are other > representations > though, and it usually only comes up when > demonstrating a formal > foundation for real analysis. > > > - Tim Infinity is called a concept, not a number, so infinity is not part of the set of real numbers. What if 0.333... is also just a concept? Any terminating decimal can only approach infinity as it can only approach 0.333... or pi. My idea is, that we might erroneously place terminating and non-terminating decimals in the same set. Is it possible, that this means mixing up terminating decimals and concepts in the same set? Reinhard
From: Arturo Magidin on 7 Aug 2010 16:49
On Aug 7, 9:03 am, netzweltler <reinhard_fisc...(a)arcor.de> wrote: > > On 2010-08-07, netzweltler > > > > I can see similarities. A terminating decimal can > > > approach > > > 0.333... but never reach it. This sounds to me like > > > working with > > > limits, where n approaches infinity. > > > Real numbers can be represented as classes of certain > > ("Cauchy") > > infinite sequences of rational numbers, where the > > real number is > > actually the limit of the sequence. > > > So in at least that sense, yes. There are other > > representations > > though, and it usually only comes up when > > demonstrating a formal > > foundation for real analysis. > > > - Tim > > Infinity is called a concept, By whom? >not a number, so infinity is not part of the set of real numbers. There are any number of meanings of the word "infinity"; while it is true that it is not an element of the set of "real numbers", there are many things which are not elements of the set of "real numbers" and yet are numbers (e.g., nonreal complex numbers; surreal numbers; extended real numbers, etc). >What if 0.333... is also just a concept? Please define "concept", then define "just a concept". Without definitions, you aren't doing math. > Any terminating decimal can only approach infinity What do you mean by "approach infinity"? > as it can only approach 0.333... or pi. Please define "approach 0.333..." and "approach pi". > My idea is, that we might erroneously place terminating and non-terminating decimals in the same set. Depends entirely on what you want to do. Note that according to the axiom of unions and some other axioms of set theory, given any two sets A and B, there always is a set whose elements are exactly those things that are elements of A or elements of B. >Is it possible, that this means mixing up terminating decimals and concepts in the same set? There is certainly a lot of mixing up going here, but I think it is mostly happening in your head. Try to give *precise* definitions of the concepts before philosophizing more; you will find that it clarifies things immensely, while the approach you are taking currently can only obscure them. -- Arturo Magidin |