An infinite amount of integers?

[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?

 

[Alkatran]

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?

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.

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

 

[matt grime]

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?

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.

 

[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?

 

[shmoe]

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

 

[daniel_i_l]

yes that's exactly the problem, i still don't understand how both of those statements can be true.

 

[Alkatran]

yes that's exactly the problem, i still don't understand how both of those statements can be true.

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.

 

[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.

 

[Hurkyl]

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

yes that's exactly the problem, i still don't understand how both of those statements can be true.

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

 

[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?

 

[matt grime]

yes that's exactly the problem, i still don't understand how both of those statements can be true.

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?

 

[CRGreathouse]

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?

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

 

[daniel_i_l]

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

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.

 

[shmoe]

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.

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.

 

[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

 

[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.

 

[d_leet]

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

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

 

[matt grime]

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

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.

 

[Alkatran]

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.

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

 

[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?

 

[CRGreathouse]

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?

You define it, like for anything else.

Take the natural numbers, for example. Call their cardinality $$\aleph_0$$. It can be shown that the cardinality of the integers and even of the rational numbers is the same. The real numbers, on the other hand, have a greater cardinality.

 

[WhyIsItSo]

Daniel,

Try looking at it this way.

We say the universe is infinite. No matter how far you travel, you cannot reach the end of it, because it has no end.

That is a mind boggling concept to be sure.

And yet, no matter how far you travel, you still have some definable position in the universe.

Your position is finite, but the possible positions you could move to are inifinite.

 

[yenchin]

Daniel,

Try looking at it this way.

We say the universe is infinite. No matter how far you travel, you cannot reach the end of it, because it has no end.

That is a mind boggling concept to be sure.

And yet, no matter how far you travel, you still have some definable position in the universe.

Your position is finite, but the possible positions you could move to are inifinite.

I don't mean to confuse the discussions, just pointing out that it is possible for the universe to be finite without boundary, like the surface of a sphere (you cannot reach any end while walking on it, but it is finite). But again, that's out of topic :tongue:

 

[Hurkyl]

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?

All that really matters is that there's a way to tell when two sets are the "same size". (see bijection)

The cardinal numbers are defined simply so that there is a (unique) cardinal number for each possible "size".

We say the universe is infinite. No matter how far you travel, you cannot reach the end of it, because it has no end.

An infinite universe can have an end. (e.g. consider the extended real numbers)

But this concept of infinite is different than the one being discussed in this thread. (But they share the same salient feature -- something is infinite if, in some sense, it is bigger than any integer)