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- Newsgroups: alt.usage.english
- Path: sparky!uunet!caen!uwm.edu!psuvax1!news
- From: plu@math.psu.edu (Mr. Hypothetical)
- Subject: Re: quite unique
- In-Reply-To: nadeau@bcarh1ab.bnr.ca's message of Wed, 18 Nov 1992 22:14:51 GMT
- Message-ID: <BxxwBv.IEM@cs.psu.edu>
- Sender: news@cs.psu.edu (Usenet)
- Nntp-Posting-Host: jacobi.math.psu.edu
- Organization: Penn State Department of Mathematics
- References: <BxuK87.176@ccu.umanitoba.ca> <1992Nov17.181046.21137@nas.nasa.gov>
- <1992Nov18.192304.15503@nas.nasa.gov>
- <1992Nov18.221451.14168@bcrka451.bnr.ca>
- Date: Thu, 19 Nov 1992 01:33:31 GMT
- Lines: 41
-
- In article <1992Nov18.221451.14168@bcrka451.bnr.ca> nadeau@bcarh1ab.bnr.ca (Rheal Nadeau) writes:
- In article <1992Nov18.192304.15503@nas.nasa.gov> asimov@wk223.nas.nasa.gov (Daniel A. Asimov) writes:
- >
- >Come to think of it, consider the following two uniquenesses:
- >
- >a) 2 is the unique integer that is an even prime number.
- >
- >b) 1/3 is the unique real number x satisfying the equation 3x = 1.
- >
- >Since there are infinitely more real numbers than integers,
- >perhaps it *does* make sense to say that 1/3 is "more unique"
- >than the number 2, in the above contexts.
-
- Wrong - there are not infinitely more real numbers than integers. If I
- had my university notes, I could trot out the proof, but in the
- meantime: there are infinite numbers of integers and of real numbers.
- "Infinite" being an absolute term, you can't say that one infinite set
- is larger than the other (and certainly not infinitely larger).
-
- No, in fact you can say that the set of real numbers is larger than
- the set of integers. I shouldn't try to go into too much detail here,
- but if Z is the set of integers and R the set of real numbers, then
- one can prove that there is no function f from Z to R such that every
- real number r is equal to f(z) for some integer z. In other words,
- any function from Z to R has to "miss" part of R (in fact, almost all
- of R). Since we can construct a function from the positive integers
- to Z that doesn't miss any of Z, the integers are said to be
- "countably" infinite - they can be counted, 1,2,3,.... But R is
- uncountably infinite. That said, I don't think I would say 1/3 as a
- real number was more unique than 2 as an integer (but that's just my
- opinion).
-
- Speaking of "uniqueness", I once took a (math) class in which the
- notion of "strong uniqueness of best approximation" was introduced;
- also, the class used a book whose author used the aberrant "unicity"
- instead of "uniqueness". (I think unicity comes from the French word
- for uniqueness [unicit\'e?], though the book was written in English by
- a German.)
-
- - Todd Andrew Simpson
-
-