Folks, Simple questions.

Why are these not elements of W?

(1,2,3,....1000,0,0,0). The finite numbers stop at 1000 and the rest are 0, hence is it not a element of W?

(1,1,1....1, 0,0,0) Not sure of this one?

Thanks

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- Oct 21st 2011, 09:03 AMbugatti79Why are these not elements of W
Folks, Simple questions.

Why are these not elements of W?

(1,2,3,....1000,0,0,0). The finite numbers stop at 1000 and the rest are 0, hence is it not a element of W?

(1,1,1....1, 0,0,0) Not sure of this one?

Thanks - Oct 21st 2011, 10:01 AMHallsofIvyRe: Why are these not elements of W
I think that before anyone can tell you why those "are not elements of W", you will have tell us what W is! How is "W"

**defined**? - Oct 21st 2011, 12:00 PMbugatti79Re: Why are these not elements of W
- Oct 21st 2011, 01:17 PMHallsofIvyRe: Why are these not elements of W
Well, last time I looked, "1000" was pretty darn finite!

Quote:

(1,1,1....1, 0,0,0) Not sure of this one?

Thanks

**is**a number and so finite. - Oct 21st 2011, 01:25 PMbugatti79Re: Why are these not elements of W
- Oct 21st 2011, 01:47 PMDevenoRe: Why are these not elements of W
where are you getting your information, and exactly which words are being used? it feels like we are not gettting the whole story, here.

- Oct 21st 2011, 01:52 PMHallsofIvyRe: Why are these not elements of W
- Oct 21st 2011, 02:01 PMbugatti79Re: Why are these not elements of W
- Oct 21st 2011, 02:16 PMDevenoRe: Why are these not elements of W
are these printed lecture notes? is there more text surrounding these statements? context is important. a symbol like:

(1,1,1,....,1,0,0,0) could mean many things.

the presence of a right parenthesis suggests this is not an infinite sequence, as infinite sequences do not terminate.

even if "what was meant" was: an infinite number of 1's and then 3 zeros, that is still not an infinite sequence.

an infinite sequence is just a function $\displaystyle \mathbb{N} \to X$ where X is some set (the set the terms of the sequence is in).

the trouble with saying (1,1,1,....,1,0,0,0) is an infinite sequence is: which natural numbers are the pre-images of the last 3 terms?

we cannot decide, so the sequence is ill-defined (you can't count up to infinity, and then count 3 more, infinity + 3 is NOT a natural number). - Oct 21st 2011, 02:30 PMbugatti79Re: Why are these not elements of W
- Oct 21st 2011, 02:53 PMDevenoRe: Why are these not elements of W
we still have a problem determining WHEN the entries turn from 1's to 0. assuming this is NOT a "sequence of sequences", it would appear that we have an element of W, but it's not a very precise way of indicating it. to specify a sequence, you should be able (in theory, at least) specify every term.

for example, in your last example, the explicit function was:

f(n) = n, n ≤ 1000

f(n) = 0, n > 1000

with (1,1,1,....,0,0,0,....) how can we say what the 1000th term is, and-more importantly for deciding whether or not we have an element of W-how can we say for sure that there is some finite positive integer k, for which f(n) = 0 for all n > k?

it's not "good enough" to say, well, eventually, every term is 0, unles you specifically say: after THIS number, every term is 0. this goes back to the same problem we have before: we can't just put in "infinite ones" and then say we have "zeros after". that's not a well-defined function on the natural numbers.

so again, what is MEANT by (1,1,1,...,0,0,0,.....). i am not going to be sure it's actually an infinite sequence until i know what's in the first "...." (finite? all ones? something i don't know what the heck?). - Oct 22nd 2011, 11:44 AMbugatti79Re: Why are these not elements of W
- Oct 22nd 2011, 03:02 PMDevenoRe: Why are these not elements of W
yes, a billion is a finite number.