A gap in Cantor's diagonal argument ?
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From: Virgil on 6 Aug 2010 16:25 In article <547001489.67058.1281099777043.JavaMail.root(a)gallium.mathforum.org>, Reinhard Fischer <reinhard_fischer(a)arcor.de> wrote: > Yes, I can see. If a natural number had an infinite number of digits, > infinity would be a natural number. And this is not possible. This makes > sense even without the arguments of set theory. This in turn means, a > terminating decimal at no time will convert into a nonterminating decimal, no > matter how many decimal places you add. Thanks. That depends on your definition of a non-terminating decimal. If your definition of "non-terminating" requires infinitely many non-zero digits, you are correct, but if it allows only finitely many non-zero digits (with, necessarily, infinitely many zero digits in that case), you are wrong.
From: netzweltler on 6 Aug 2010 15:54 You might have misunderstood my statement, Virgil. 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. Reinhard
From: netzweltler on 6 Aug 2010 15:58 I see, you misunderstood my statement. 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. Reinhard
From: netzweltler on 6 Aug 2010 16:08 You might have misunderstood my statement. 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. I am thinking about the difference between terminating and non-terminating decimals. And I am wondering if non-terminating decimals, decimals with infinitely many places, could also be treated as a concept like infinity. 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. Is there some kind of relationship between infinity and non-terminating decimals? Reinhard
From: Arturo Magidin on 6 Aug 2010 22:49
On Aug 6, 7:08 pm, netzweltler <reinhard_fisc...(a)arcor.de> wrote: > You might have misunderstood my statement. 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. Sigh; if you add a natural number of 3's, then you still have a finite decimal. > Might be no news for you. No, but the value of the "terminating" decimal 0.3 happens to be *exactly the same* as the value of the "non-terminating decimal" 0.2999999.... > > I am thinking about the difference between terminating and non-terminating decimals. And I am wondering if non-terminating decimals, decimals with infinitely many places, could also be treated as a concept like infinity. This is so muggy it will be hard for you to come up with any reasonable answers. The concepts seem to be too fuzzy in your head. First, you need to figure out exactly what you mean and how to say it correctly and accurately. You might also want to learn to quote the message you are replying to, at least in part, to give context. Not everyone reads sci.math in an interface that makes looking up previous posts easy. -- Arturo Magidin |