Paradox?

Originally posted by bobbob1056th
You say infinity is not a big number (but an "infinitely big" number), I say that is a matter of semantics. You say "when you divide 1 by this colossus, you don't get a very small number, or a very, VERY small number, you get something vanishingly small."

One could say 0.0001 is a large number when comparing it to numbers like 10^-(10^10^10^10) or -10 (that brings me to another thing, 0.000...1 isn't an infinitesimal number (maybe this matter is open to debate). That honor goes to -999... but it is the smallest number larger than 0.). The same is true with large numbers. One could compare any number (not counting repeating numbers, or with an infinite type factor involved) and always find a number smaller/larger than itself, so saying a number is big or small is not helpful, unless it has a context, such as saying there are 6 billion people on this earth ("Wow that's a lot compared to (key phrase: compared to) what it used to be 100 years ago"😉 Of course there is no standard for numbers from which to compare them to each other. This probably didn't clarify anything but I'm going to post it anyways. And for tomorrow I have a very interesting mathematical concept to post on a new thread.

Originally posted by PBE6
I think the problem here is that you have defined a function to be the difference between two numbers of indefinite and infinite magnitude. When you do that, strange things happen.

For example, as you have framed the question the number of seeds left after some finite number of plantings n is given as the difference between the following two sums:

seeds ...[text shortened]... infinite bird snacks.

I know this explaination was a little long winded, but I hope it helps.

Thanks PBE6!

That was exactly the type of explanation I was hoping for. I think it's especially confusing when the number of remaining seeds is constant for all n, but the limit is 0 when approaching infinite.

In my example, the composed function "seeds left" (SL) is SL(n)=10 for any given n, but the limit when n approaches infinity is 0.
🙄
I think understand why now (after your explanation using L'Hopital Rule) but it's still completely unintuitive.


Sorry I can't rec you, though. :'(

Originally posted by Palynka
Sorry I can't rec you, though. :'(

Why not?!? Do you want me to tap-dance? I'll do it! Just gimme one of those delicious little rec's...

Originally posted by bobbob1056th
You say infinity is not a big number (but an "infinitely big" number), I say that is a matter of semantics. You say infinity is a number, I say infinity is not a number. You say "when you divide 1 by this colossus, you don't get a very small number, or a very, VERY small number, you get something vanishingly small." I say that is semantic ...[text shortened]... nclusion to the thred although I'm sure some people may want to post a responce. Good night 😴

Hi bobbo,

No, we do not hate you, or take offence to your debate. In fact it's fun trying to explain concepts that maybe we've never had to, learning something ourselves in the process.

Now, when I posted that little diatribe about infinity I was very careful not to call infinity a number. I always called it something else (colossus, something, etc...). The limit, however, is a number. There is a big difference.

We can say the function f(x) = 1/x is defined for every real x (except x=0), but we can't technically say that it's defined at infinity because we never get there! However, lim[1/x]|x-->inf is defined, and it's equal 0. This is what I understand 1/inf to mean (although I'm sure I'd lose marks on a math test for writing it that way - so you may technically be right there, bobbo). We prove the existence of these limits using delta-epsilon proofs. Here's an entry from the Wikipedia that describes them in more detail:

http://en.wikipedia.org/wiki/Limit_of_a_function

I guess what I really want to get across is that searching for infinity by thinking up the biggest number you can and then trying it in an equation is not going to get you anywhere. This is what I believe you are trying to do by saying 1/inf = 0.00000....1. It's like trying to apple something into an orange. Large numbers and infinity are in different classes, and never the twain shall meet.

Anyway, it was fun debating this topic. As a side note, I always thought of calculus as applying the concept of infinity (and infinitessimality <-- is that a word?) to all the math you know, breaking through barriers of understanding and opening up a whole new world in the process. Like living on a line and suddenly discovering a world of curves (sound like puberty, doesn't it?). I may be a huge nerd for thinking that way, but at least I'm a happy nerd. But that's for different thread!
🙄

Originally posted by PBE6
Why not?!? Do you want me to tap-dance? I'll do it! Just gimme one of those delicious little rec's...

I can do the tap-dancing part, but us non-subscribing leeches are not allowed to rec.

No, I'm not a stingy ba§tard, I need the self control of 6 games maximum.

bobob

0,000...1 is not a number. The '...' means there are infinitely many zero's there. You cannot put soething after an infinite row, in other words, you never get to that 1.

Please read my earlier post, I proved that your X has to be 0. If you're convinced the proof is flawed, indicate where the problem is.

Originally posted by davegage
No, your analysis is wrong. Bird B eats all of the seeds.

[b]This 'paradox' only excists because you claim that infinity - infinity IS defined and equal to 0.

Please be good enough to point out precisely where I made the statement that 'infinity - infinity is defined and equal to 0.' I am well aware that infinity - infinity is undefined beca ...[text shortened]... of cardinality -- you can create a 1-1 mapping between an infinite set and a subset of itself. [/b]

In the case of bird A you don't make a 1-1 mapping, in the case of bird B you do. You can compare what happens but the results will have no meaning.

By saying that the bird will eventually eat 'all' seeds, you say that at one point he has eaten infinitely many seeds. Where as the farmer has planted infinitely many seeds. Your reasoning is then that there remain no seeds. How is this not infinity-infinity?

Why do you insist on looking at how many seeds the bird eats? Why not look at how many seeds are left at any given time?

The number of seeds left grows and grows, and will keep growing.

Both bird eat the same amount of seeds, But that is where the comparison ends; different mappings, diferent rules.

Originally posted by TheMaster37
In the case of bird A you don't make a 1-1 mapping, in the case of bird B you do. You can compare what happens but the results will have no meaning.

By saying that the bird will eventually eat 'all' seeds, you say that at one poi ...[text shortened]... is where the comparison ends; different mappings, diferent rules.

Your post is really confusing (although not as confusing as bob's). Your first sentence makes it seem like you agree:

In the case of bird A you don't make a 1-1 mapping, in the case of bird B you do.

And so does your last paragraph:

Both bird eat the same amount of seeds, But that is where the comparison ends; different mappings, diferent rules.

But in between you disagree. BTW, davegage is not saying that "infinity-infinity=0". I don't why that keeps getting brought up. But Bird B still eats all the seeds, leaving none in the field.

The reason you can't use a function of "how many seeds are left in the field" is because you trying to take the difference between two diverging infinite sequences. You can do this for any finite number of terms, but for an infinite number of terms the result is indeterminate. I try to explain why in a previous post.

I think the simplest way to think about it is this: what does it mean if there are seeds left in the field? It means that there are seeds the bird couldn't get to.

Are there any seeds Bird A couldn't get to? Yes, there are. For every 10 seeds the farmer plants, the bird will pass by 9 of them, never coming back for them. They're in the field for good.

What about Bird B? Well, Bird B eats every seed in order. Will he get to the 100th seed? Yes. Will he get to the 1,000,000th seed? Yes. Will he get to the 1E+99th seed? Yes. In fact, there is no seed you can say is off limits to Bird B. So how can any seed be left in the field? This is an example of the surjective (1-1 and onto) mapping you mentioned in your own post.

That's my story and I'm sticking to it.