Originally posted by adam warlock
I don't think there will be any bell ringing. go see some serious book on set theory and then see if they use the word half when refering to attributes of sets with infinite members. and if you do see some book like that just send me the reference of it!
A serious book on set theory will say something like "The asymptotic density of the even integers in the set of integers is
one half". I still don't like Dr S's use of the verb "halve" before, though, because when we halve a number, we get a unique result, while "halving" the integers by constructing some set with density 1/2 can be done in uncountably many distinct ways.
To prove this last fact, note that there are uncountably many bounded integer sequences, because there is a bijection from the set of integer sequences in which all terms are between 0 and 9 and the real numbers between 0 and 1. Let {s(n)} be a bounded integer sequence. Then the sequence {2n + s(n)} has density 1/2 in the integers. There are thus uncountably many integer sequences of density 1/2 which differ in at least one term.
This means that to say "halve the set Z" can refer to one of uncountably many operations, while the operation "halve the number x" refers to only one thing. I don't like this recycling of words, but I agree with everything else Dr S has said in this thread.
I have decided to call people who use one word to mean two things "Al Gore linguists" in honour of Al Gore's claims of environmentalism.
EDIT Come to think of it, people who say "halve x" where x is a numer are dinosaur linguists. Thus I support Dr S's use of the verb "halve" and reject the use of "halve" in the sense of "multiply by 1/2". The real Al Gore linguist is KellyJay, who began the thread by conflating of these two different concepts on the basis that the same word sometimes gets used for both.