Question about the natural numbers?

[cragar]

My teacher was saying that we can't have a set of infinitely decreasing natural numbers.
What if we started at ω and then worked our way backwards. I realize that is ill defined.
And where ever we start will be a finite number. But if we can have an infinitely increasing set in the natural numbers, why can't we just run this backwards.

 

[micromass]

My teacher was saying that we can't have a set of infinitely decreasing natural numbers.
What if we started at ω and then worked our way backwards. I realize that is ill defined.
And where ever we start will be a finite number. But if we can have an infinitely increasing set in the natural numbers, why can't we just run this backwards.

First of all, $\omega$ is not a natural number. It is defined as something larger than all natural numbers, but it's not a natural number.

Let's say you have an increasing sequence 1,2,3,4,5,6,...
You suggest to run it backwards. Well, what would be the first element of the sequence??
Remember that $\omega$ is not a natural numbers. And $\omega-1$ isn't even well-defined.

 

[cragar]

ok i get what your saying, I was just wondering if there was some clever way to get around it.

 

[micromass]

ok i get what your saying, I was just wondering if there was some clever way to get around it.

No, there isn't. An easy intuitive way to see this is as follows:

Let's have an infinite decreasing sequence of natural numbers. Let the first element be n. But then you can only have n elements after it.

For example, let the first element be 3224. Then the subsequent elements can only be 1,2,...,3223. So you can only have a finite number of elements after. So you can never infinitely decrease.

 

[cragar]

ok, I might be beating a dead horse. But i just want to say, we can have infinitely decreasing sets in the positive rationals, so why can't we just map these rationals to natural numbers using prime bases like $5^p7^q$ . There is probably something I am missing about the rationals.

 

[micromass]

ok, I might be beating a dead horse. But i just want to say, we can have infinitely decreasing sets in the positive rationals, so why can't we just map these rationals to natural numbers using prime bases like $5^p7^q$ . There is probably something I am missing about the rationals.

Sure: 1/n is a good infinitely decreasing sequence. So we can find it in the rationals.

But how would you transfer this sequence to the set of rational numbers? What is the map you use?
And more importantly: will the image under this map still be decreasing? I claim that it will not end up being a decreasing sequence.

 

[Number Nine]

ok, I might be beating a dead horse. But i just want to say, we can have infinitely decreasing sets in the positive rationals, so why can't we just map these rationals to natural numbers using prime bases like $5^p7^q$ . There is probably something I am missing about the rationals.

Your mapping will not generate an infinitely decreasing sequence in the natural numbers; it won't even be decreasing. Given rational numbers 1/p < 1/q, the former is mapped to a greater natural number than the latter.

 

[cragar]

ok so well start with small rationals and then go up to bigger rationals . Why can't we just order the set later.

 

[micromass]

ok so well start with small rationals and then go up to bigger rationals . Why can't we just order the set later.

What do you mean??

The best way to see why it fails is to try the idea out. So try it out on a example and post your results.

 

[cragar]

ok i see why it fails now, it seems like you might have been able to carefully pick rationals in such a way as to make it happen, but apparently not. Interesting to think about though.