# An infinite amount of integers?

1. Aug 23, 2006

### daniel_i_l

This is probablly a silly question but it has been bothering me for a while.
How can there be for example an infinite amountof integers? each integer has an integer that is smaller than it by one and an any number that is one bigger than a finate number is also finate. doesn't this prove that all integers are finate?

2. Aug 23, 2006

### Alkatran

You're right, in that all integers are finite, but infinity is not an integer. It's just what you get when you increase without bound. For example, the 1/x increases without bound as you approach 0 from the positive side, so the limit of 1/x as x approaches 0 from the positive side is infinity.

$$\lim_{x\rightarrow0^{+}}\frac{1}{x} = \infty$$

Last edited: Aug 23, 2006
3. Aug 23, 2006

### matt grime

who has said that there is an integer that is 'infinite'? Note your own argument proves that the cardinality of the set of integers, whatever it may be, is not an integer.

4. Aug 23, 2006

### JonF

If the set of integers were finite, there would have to be a greatest integer. Can you see how that would cause a contradiction?

5. Aug 23, 2006

### shmoe

Is this a confusion between the statements "there are infinitely many integers" and "any given integer is 'finite'"?

6. Aug 24, 2006

### daniel_i_l

yes thats exactly the problem, i still don't understand how both of those statments can be true.

7. Aug 24, 2006

### Alkatran

What do you think of the fact that there are infinitely many real numbers between but not including 0 and 1? Then we have infinitely many numbers, no highest number, and yet all of them are obviously finite.

The size of a set has nothing to do with the size of the elements.

8. Aug 24, 2006

### mathwonk

the finiteness of an integer refers to the fact that the set of integers les than it is finite.

i.e. 10 is finite because 10 counts the number of integers betwen 1 and 10. but 10 does not count the integers laregr than 10.

i.e. given each (positive) integer n, there are only finitely many integers smaller than it, namely n of them, but infinitely many more larger than it.

9. Aug 24, 2006

### Hurkyl

Staff Emeritus
Well, let's look at the opposite question. Why do you think one of them has to be false?

10. Aug 24, 2006

### WhyIsItSo

Try this approach:

Any given integer is a finite value.

The range of possible integers is infinite.

Whatever integer you give me, I can add 1 to it. Both the number you gave me and the number I created by adding 1 are finite values.

But we can keep adding 1 forever. There is no limit. We never reach a point where we can not add just 1 more. The range is therefore infinite.

Does that help?

11. Aug 24, 2006

### matt grime

Just because all elements of S have property P this does not imply that S has property P. Indeed there is nothing that makes it remotely sensible for S to even have property P. Invent you own counter examples at will, and with whatever level of silliness you wish: every penguin is a bird, the set of all penguins is not a bird. Every president of the United States is a man, the collection of all presidents of the united states is a not man. Need I go on?

12. Aug 25, 2006

### CRGreathouse

All sets are sets, but the collection of all sets is... a proper class. :tongue2:

13. Aug 26, 2006

### daniel_i_l

Atleast what i understand is that the value of an integer is basiclly the "place" of the integer - the integer 10 is the tenth integer. So if there were 100 integers the last one would be 100. so if there're infinate integers, doesn't that imply that one of them is infinate.
What i understtod from the answers so far is that even though there is an infinate amount of integers, all the integers are finate. I still don't understand how this can be, it seems like one of the sources of my confusion is the boundry between the finate and the infinate, when does "a lot" turn into infinity?
So i still don't understand, how can the integer in the "infinity" place be finate?
Is there something wrong with my definition of infinity?
Thanks.

14. Aug 26, 2006

### shmoe

There is no "last" integer, nor an "infinity place".

If "a lot" means some finite amount, then it is never infinity. You can keep adding one more element to your set: {1},{1,2},{1,2,3},... but no set on this list has infinitely many elements.

15. Aug 26, 2006

### daniel_i_l

So are you basicly saying that you can never "get to" the integer with the infinate place on the set of integers?
Thanks

16. Aug 26, 2006

### StatusX

When you talk about the "infinite place on the set of integers", you are inherently assuming infinity is a number. It is not, and there is no "infinitieth" integer nor any infinite integers. When we say the set of integers in infinite, we just mean that for any number you could pick, there are more integers than that.

17. Aug 26, 2006

### d_leet

No, he is saying that there is NO integer in the infinite place as you put it.

18. Aug 26, 2006

### matt grime

we are saying nothing of the kind, since the notions of 'getting to' and 'infinte place on the set of integers' are meaningless things (to us): they are terms chosen by you to try to express something, though I'm not sure what.

Last edited: Aug 27, 2006
19. Aug 27, 2006

### Alkatran

That is the best description I've heard in awhile.

20. Aug 27, 2006

### arevolutionist

I had someone explain some concepts of infinity once. He was explaining the cardinality of an infinite set. How does a cardinality exist for an infinite set?