What is the mathematical definition of infinity?

Question

Please explain it in laymans terms.

Answer 1

Infinity is very tricky. Infinity is more of an idea than a thing. In grade school and in calculus, sometimes we talk about infinity as though it were just some huge number, or "as far as you can go on the number line", or something like that. A child's first exposure to infinity might be "it's the biggest possible number". That's not really infinity, that's a cop-out.

This is, in my opinion, the best way to get a grasp of infinity. Think about the adjective "finite". Finite describes any quantity which can be completely counted or measured. Think of a huge number. That is finite. Square it, cube it, multiply it by the number of grains of sand in the world, and it's still finite. Nobody could physically count this number in a lifetime, but in theory if we began the process of counting, the process would have a beginning and an end.

Infinite just means "not finite". An infinite quantity cannot be completely counted or measured; it is larger than any finite quantity. If we take our unimaginably large (but finite) number from before, and we multiply it by any finite number, and we do it finitely many times, an infinite quantity is still larger than our number.

That is your definition of infinity. Infinity is not a quantity or a number; infinity is a property which is possessed by every infinite quantity. Being infinite simply means that no matter what finite quantity you come up with, the infinite one is bigger.

One last note: beware of the sideways-eight that often represents infinity. Sometimes people assume that infinity is somehow a fixed value because it has its own symbol. That symbol is very useful, but it represents the idea of infinity, not just a variable that happens to be enormously large.

EDIT: Many of the answers above me are correct, and they rely on set theory. Although I chose to omit those aspects for the sake of simplicity, the books I recommended do address those topics in clear terms.

Answer 2

Infinity Math Definition

Answer 3

Infinity Definition Math

Answer 4

by my understanding mathematical definition 4 infinity is some thing that doesnt end or is continuous. for example i linear programming.. x>0 &Y>0, the arrows r goin into infinity why means they dont stop at a particular place..lol [eg they dont stop on 4,0 or 4,4 or 100,50..its continuous]

and also numbers are infinitive because there is no end to numbers..after a thousand is million then billion etc..never ends.

Answer 5

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RE:
What is the mathematical definition of infinity?
Please explain it in laymans terms.

Answer 6

if you count the sheep in a field you establish a one to one relation - each sheep has a number. In ordinary situations one would run out of sheep - but lets assume that someone brings in one more sheep then you have to add a number. Now think of the largest possible number that you can conceive of. a number larger than all the atoms in all the mass in all the universe. Now bring in one more sheep. No matter how big the number you can always add one more. So their is an infinity of numbers. An infinity is by definition uncountable. Because you can never run out of numbers.

Answer 7

When a number, or set of numbers never repeats itself. The perfect mathematical example is PI

Answer 8

Georg Cantor and his Alephs. Your beginning infinity is unbounded but countable - the number integers. An infinitely bigger infinity is unbounded and uncountable - the number of points on a line. We can go infinitely bigger still - the number of functions through a point.

Which infinity is big enough for you?

Answer 9

.. A lot of numbers?



Infinity is a number that basically goes on and on and doesn't ever stop.

Answer 10

A set is said to be infinite if it can be put into one-to-one correspondence with a proper subset of itself.
The set of counting numbers is an infinite set because it can be put into one-to-one correspondence with the set of all even counting numbers.
2 is assigned to 1, 4 is assigned to 2, 6 is assigned to 3 and so on; any even number n is assigned to the counting number n/2.