# Thread: Why the set 'N' has infinite elements

1. ## Why the set 'N' has infinite elements

Why the set of all natural numbers (N) has infinite number of elements?

I have started from this....
Suppose N is finite and its largest element is x. Then x + 1 will be a natural number because addition of two natural num make another natural number which is greater than each previous numbers. clearly x+1>x which is contradicting x is largest. So N has no largest element. so N has infinite number of elements.

another way.......
we can build the set N = { x+1 | x is the element of N }
so if 1 is a natural number then 2 is also . as 2 is the element of N then 3 is also and so on 3,4,5,6,7,8,9,10,..........

So N is infinite.

Is these 2 proof is correct? If there there is any strong proof then kindly post them with some explanation.
I dont know where to post this. So I am publishing this here.

2. How do you go from "and so on 3,4,5,6,7,8,9,10,.........." to "So N is infinite"?

When you say "and so on 3,4,5,6,7,8,9,10,.........." you are simply saying that you can get all of N that way. Asserting that it is infinite is just assuming that N is infinite, which is what you are trying to prove.

By the way, what definition of "infinite" are you using?

3. Originally Posted by HallsofIvy

By the way, what definition of "infinite" are you using?
Firstly,Here "N is infinite" means N has unlimited number of elements.

Originally Posted by HallsofIvy
How do you go from "and so on 3,4,5,6,7,8,9,10,.........." to "So N is infinite"?

When you say "and so on 3,4,5,6,7,8,9,10,.........." you are simply saying that you can get all of N that way. Asserting that it is infinite is just assuming that N is infinite, which is what you are trying to prove.
No I am not assuming "and so on 3,4,5,6,7,8,9,10,..........". But as the set definition is recursive so there exists unlimited number of Natural Number if '1' is a natural number.

we can build the set N = { x+1 | x is the element of N }
Here The set definition is important. This is implying that every Natural Number has a next natural number. So N cannot contain a largest element. Doesn't it is enough to say that N has unlimited number of elements

4. Originally Posted by dbngshuroy
Why the set of all natural numbers (N) has infinite number of elements?

I have started from this....
Suppose N is finite and its largest element is x. Then x + 1 will be a natural number because addition of two natural num make another natural number which is greater than each previous numbers. clearly x+1>x which is contradicting x is largest. So N has no largest element. so N has infinite number of elements.

First proof looks good. There are many alternative ways to put it, but it's clear.

another way.......
we can build the set N = { x+1 | x is the element of N }
so if 1 is a natural number then 2 is also . as 2 is the element of N then 3 is also and so on 3,4,5,6,7,8,9,10,..........

So N is infinite.

The second proof is incomplete, I reckon.

Is these 2 proof is correct? If there there is any strong proof then kindly post them with some explanation.
I dont know where to post this. So I am publishing this here.
.

5. Originally Posted by Archie Meade
The second proof is incomplete, I reckon.
Why?

6. Originally Posted by dbngshuroy
Why the set of all natural numbers (N) has infinite number of elements?

I have started from this....
Suppose N is finite and its largest element is x. Then x + 1 will be a natural number because addition of two natural num make another natural number which is greater than each previous numbers. clearly x+1>x which is contradicting x is largest. So N has no largest element. so N has infinite number of elements.

another way.......
we can build the set N = { x+1 | x is the element of N }
so if 1 is a natural number then 2 is also . as 2 is the element of N then 3 is also and so on 3,4,5,6,7,8,9,10,..........

You have the set of natural numbers such that x is a natural number

Now you are building the set containing x+1, such that x is a natural number.
This is infinite "if" the set of natural numbers is infinite.

Your second wording could be interpreted as "the assumption is already made that the set is infinite"

As you add a value to each element in the set of natural numbers,
you get another set with the "same number of elements"

But what you mean to say is that no matter what x is, there is an x+1 which is a natural number,
which brings us back to your first proof.

So N is infinite.

Is these 2 proof is correct? If there there is any strong proof then kindly post them with some explanation.
I dont know where to post this. So I am publishing this here.
You could expand the proof until it is complete.
For example, the set of (x+1)'s are natural numbers,
but it's "largest value" is 1 more than the original set's "largest value".
This contradicts the statement...the set of natural numbers is finite..
"if" you include the very first natural number
that's gone missing from the set of (x+1)'s in the set of (x+1)'s.
However, this process of building new sets must go on to infinity,
so I'd recommend an alternative wording.