Can infinity be more than infinity?

Question

Would all the numbers over 5 (infinity) be the same as all the numbers over 10 (also infinity)? I'm trying to prove to my math teacher that it would be. Does anybody have a credible link or something?

Answer 1

What is infinity?

Infinity is not a number; it is the name for a concept. Most people have sort of an intuitive idea of what infinity is - it's a quantity that's bigger than any number. This is sort of correct, but it depends on the context in which you're using the concept of infinity.

There are no numbers bigger than infinity, but that does not mean that infinity is the biggest number, because it's not a number at all. For the same reason, infinity is neither even nor odd.

The symbol for infinity looks like a number 8 lying on its side.

Now for the fun part! Even though infinity is not a number, it is possible for one infinite set to contain more things than another infinite set. Mathematicians divide infinite sets into two categories, countable and uncountable sets.

In a countably infinite set you can 'number' the things you are counting. You can think of the set of natural numbers (numbers like 1,2,3,4,5,...) as countably infinite.

The other type of infinity is uncountable, which means there are so many you can't 'number' them. An example of something that is uncountably infinite would be all the real numbers (including numbers like 2.34.. and the square root of 2, as well as all the integers and rational numbers).

In fact, there are more real numbers between 0 and 1 than there are natural numbers (1,2,3,4,...) in the whole number line!

http://mathforum.org/dr.math/faq/faq.large.numbers.html

Answer 2

First and foremost, infinity and infinite are concepts - they are NOT real. Infinite and infinity do not occur in reality.

When we think we can conceptionalize an infinite number of anything - some one will always ask why you can't add another to it, or square it, or double it or a similar argument.

Infinity is a paradox. How much is infinity plus one? Does the resulting sum mean the first infinity wasn't really infinite?

Another paradox occurs when we convert a fraction into a decimal:

1/3 = 0.33333 to infinity, hense
1/3+1/3+1/3 then must equal 0.99999 to infinity
but it doesn't, 1/3+1/3+1/3=1

An infinite amount of zeros after the decimal point ending in a 1 would have to added to our 0.999999 to infinity it to make it equal to 1.

On the other hand, if 0.99999999 to infinity does indeed equal 1, as "proven" by the above mathmatical addition, then an infinite number of zeros following the decimal point with a 1 at the end would have to equal zero.

I think you can get the picture. Your question to your math teacher is a paradox - it can't be answered, let alone proven.

It is, however, a useful concept - for example, it seems reasonably true that one could travel around a circle an infinite number of times and not find the end. There are zero ends and zero times infinity is zero. We think.

Arguments concerning infinity are in the same catagory as asking someone if God can do anything. When they natually answer, "yes," You then ask them, "Can God can make a stone large enough so that he can't lift it?"

Answer 3

Try this:

  6   7   8   9  10  11...
11 12 13 14 15 16...

This establishes a 1-1 and onto relationship between your two sets - so they have the same number of elements.

Answer 4

okay, nothing can actually BE infinity, things can only APPROACH infinity because by definition infinity is a number that never ends...

So, you can say that 10 x infinity is greater than 5 x infinity, but because infinity is not a number, there is really no such thing.

By the way, I am really very impressed that you are challenging your math teacher!! Keep on doing that, it shows strength of mind and character!

A second by the way, if you are thinking this way, you will probably really enjoy calculus, in which you get to think about numbers as they approach infinity and as they approach zero. It changes they way that you can manipulate numbers, and makes problems much easier to solve.

Answer 5

If you have two series of numbers like the ones below, you can say that the lower one B is growing to infinity faster then the first one A.
A=1+2+3+4+.....
B=2+4+6+8+.....

Both these series are growing to infinity, but you can still say that B/A=2, since every number in B is twice as much as in A.

Answer 6

Infinity isn't a number, but a kind of behaviour. But there are different kinds of infinity, some of which are 'bigger' than others.
To understand this, you have to learn about sets, and then about cardinality of sets (the number of members of a set).

But your suspicions are correct: the set of counting numbers > 5 has the same cardinality as the set of counting numbers > 10. You have to set up a 1-1 correspondence (a special kind of function, called an isomorphism) between the first set and the second set.

Answer 7

All numbers over 5 are the same infinity as all number over 10 because you can make a one-to-one correspondence of those numbers:

f(n)=n+5

So:

n = 6, 7, 8...
| | | |
f(n)=11, 12, 13, ...

However, the infinity of all real numbers is "larger" than the set of all natural numbers. Indeed, lets assume we established a one-to-one correspondence of all real and all natural numbers:

R(n)=m(n).d1(n)d2(n)d3(n)d4(n)...

Where m(n) is an integer, and dk(n) are integers between 0 and 9. For example, for the real number 521.53278 we will have

m(n)=521
d1(n)=5
d2(n)=3
d3(n)=2
d4(n)=7
d5(n)=8
dk(n)=0 for all k>5.

But wait, we can come up with the number

x=M(1).D1(2)D2(3)D3(4)...

Where M(1)=|=m(1), Dk(l)=|=dk(l) for all k, l.
This number is not equal to any R(n) because according to construction of x, its component M(1) if n=1 or component D(n-1)(n) if n>1 is not equal to the corresponding component of R(n), m(1) if n=1 or d(n-1)(n) if n>1.

So we cannot establish a one-to-one correspondence because there is at least one more number in the set of real numbers than in the set of natural numbers. Quod erat demonstrandum.

Answer 8

Contrary to some answers above, there is more than one infinity, and some of these infinities are larger than other infinities. However, your example is not one of these, it is an example of the lowest order infinity. Any set that can be "counted" with ordinary (integer) numbers belongs to that infinity. A higher order of infinity is the set of transcendental numbers; the infinity of that set is greater than the infinity of integers or rational numbers. This subject in mathematics is callled the study of "transfinite" numbers and was introduced by mathematician Georg Cantor in the late 19th century. He was also a pioneer in set theory. See http://en.wikipedia.org/wiki/Georg_Cantor

Answer 9

Not the way I see it. 6,7,8,9 are not over 10, BUT are over 5. So the infinite streams would not be identical, because starting at 5 would have four numbers in it's infinity, that 10 would not have. Now if you ask, are all numbers over ten (infinity) and all numbers under ten (infinity), then yes, both "streams" of numbers would be infinitely IDENTICAL.

Having said that, if we define the word "infinite" as meaning "all inclusive", then both numbers, by definition, would have to be identical. - you would have to envision a "closed loop system" that encompasses "all things" to make this true. - Which in and of itself, for most people, would appear limiting, and by definition, could not be infinite (limitless).

The problem with this "problem" is the definitions are not clearly defined, and the question mixes language and mathmatics.

Now, consider this - If we define "infinity" as a stream that runs in any and all directions, then, YES, all is "captured" regardless of the starting point. In other words, if you define mathmatics as the "pure science", capable of traveling in all directions (not a linear progression), then the answer is easy - They are identical because "all directions and all encompassing" mean exactly that - ALL THINGS. - Another way to say this is "The eye of God".

But I personally believe that 2+2 is NOT four - But "4 and change". - we, as human beings are incapable of "pure thought", let alone "pure science"!

Answer 10

BTW what's "hols!"?? once you're thinking the idea... comprehend too that there are "more beneficial infinities", the infinity of irrational numbers is larger than the infinity of counting or rational numbers. With the adaptation of (e^x) - x², as x-->? the adaptation -->?. were the adaptation (e^x) - (x² + a million) the adaptation is an similar. perhaps to approximate the abstractness, evaluate the infinity of waterdrops in the sea; then upload a raindrop.