A gap in Cantor's diagonal argument ?
in [Math]
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From: Arturo Magidin on 8 Aug 2010 18:05 On Aug 8, 3:16 am, netzweltler <reinhard_fisc...(a)arcor.de> wrote: > > On Aug 7, 6:39 pm, netzweltler > > <reinhard_fisc...(a)arcor.de> wrote: > > > > > > Is it also valid to say: > > > > > There are any number of meanings of the word > > > > > "number"? > > > > > Indeed; "natural number", "rational number", > > > > "real > > > > number", "complex > > > > number", "cardinal number", "surreal number", > > > > "nonstandard real > > > > number" are just over half a dozen that come to > > > > mind > > > > in a few > > > > nanoseconds. > > > > What is common to all these different meanings of > > > the word number? > > > They all refer to specific concepts, defined in a > > precise, explicit, > > and careful manner? They all refer to concepts called > > "number"? They > > are all made up by human beings? > > > I have no idea what you are driving at, but your > > question is > > hopelessly vague, much like most of your musings in > > this thread so > > far. > > A number is a mathematical object used in counting and measuring. That´s what you can read in Wikipedia. And of course, if it is in Wikipedia, it must be true! Note also that even if we take it at face value, such a statement would not *define* numbers, but merely mention two things for which numbers *may* be used. It does not even exhaust all possible uses for "numbers". > Is it valid to measure the distance between New York and Paris and to define, this is a number now? No. All you can say is, you can measure the distance between New York and Paris and your result is a terminating decimal. Really? That would be interesting. What units will you be using? Are you asserting that in all units you are to measure this distance, the distance will result in a fraction which, when reduced, will have a denominator which is equal to a power of 2 times a power of 5? Please, do so! Now, granted, you *can* say whatever you want (you have so far); but most of the things you *can* say also happen to be false. > So, why is pi called a number? All you can say is, there is an algorithm to determine pi as precisely as you need it. No, I can say a lot of things, and that happens to not be one of the ones I would choose to say. > And again, your result is a terminating decimal. This terminating decimal is the number, not pi (as a non-terminating decimal). Sigh. Given the natural numbers (which are assumed to be a primitive notion), one can *define* integers; given integers, one can *define* rationals; given rationals, one can *define* real numbers. "pi" happens to satisfy the definition of a real number, and as such *is* a real number, BY DEFINITION. Nothing whatsoever to do with your vague fuzzy musings. It seems to me that you are far more interested in pontificating than in doing math; might I suggest sci.philosophy? They take far more kindly to those who enjoy speaking pontificating on topics on which they are ignorant than folks in sci.math tend to be. > By the way, why is> > What is common to all these different meanings of > > > the word number? > > a vague question? Extremely vague. It would have surprised me that you do not realize this if I hadn't already seen the vast amount of vague insipid musings you have seen fit to spew over the newsgroup. >Is it allowed to define any concept and call it a number? Of course you can; whether or not this notion will be seen as useful by others is of course a different matter. >There must be something, they all have in common, like "they are all used in counting and measuring". There are *lots* of things they have "in common"; that's why your question is so hopelessly vague, mired as it is in your fuzzy thinking. -- Arturo Magidin
From: FredJeffries on 8 Aug 2010 18:24 On Aug 8, 1:16 am, netzweltler <reinhard_fisc...(a)arcor.de> wrote: > > By the way, why is> > What is common to all these different meanings of > > > the word number? > > a vague question? Is it allowed to define any concept and call it a number? There must be something, they all have in common, like "they are all used in counting and measuring". See what Wittgenstein said about "Family Resemblance" http://en.wikipedia.org/wiki/Family_resemblance http://www.hum.utah.edu/~phanna/classes/ling5981/autumn03/web/webnotes/29sept/node3.html http://fact-archive.com/encyclopedia/Philosophical_Investigations#Natural_language.2C_meaning_and_use
From: FredJeffries on 8 Aug 2010 19:34
On Aug 8, 1:16 am, netzweltler <reinhard_fisc...(a)arcor.de> wrote: > A number is a mathematical object used in counting and measuring. That´s what you can read in Wikipedia. There's a third major use of numbers and it's rather ironic that it's not mentioned in the Wikipedia article because without it Wikipedia wouldn't exist: Encoding information, which (seems to me) is reducible to neither counting nor measuring. For instance ASCII codes: http://www.asciitable.com/ |