Is infinity plus or minus one possible?

[swerdna]

Can you have infinity plus or minus one? Assuming infinity is possible of course.

 

[arildno]

In our ordinary number system(s), infinity is not a number. Thus, you cannot add it to anything.

it is perfectly possible to create a number system in which infinity I is a number, that has its own special and unique properties, just like the other numbers have theirs.
In such a system, for example, as a closure of the positive integers, I is a number that, given any number "a" has the property I+a=I for all "a".

 

[swerdna]

In our ordinary number system(s), infinity is not a number. Thus, you cannot add it to anything.

it is perfectly possible to create a number system in which infinity I is a number, that has its own special and unique properties, just like the other numbers have theirs.
In such a system, for example, as a closure of the positive integers, I is a number that, given any number "a" has the property I+a=I for all "a".

In a newly created number system would it be possible to say that the current number of monkeys in the world is I (infinite) and when a monkey gives bisth it is I + 1?

 

[arildno]

In a newly created number system would it be possible to say that the current number of monkeys in the world is I (infinite) and when a monkey gives bisth it is I + 1?

And since I+1=I, there are "I" monkeys there.

 

[swerdna]

And since I+1=I, there are "I" monkeys there.

So "I" is defined by the current number of monkeys? Doesn’t this "anything is possible" approaoch remove any particle meaning and use of the term infinite?

In other words hasn’t the very meaning of infinite been changed and not just the numbers system?

ETA - Isn’t infinity merely a theoretical lack of a beginning and end?

 

[AlephZero]

When you started learning arithmetic, you probably got the idea that "numbers" have something to do with the "real world" (whatever that is), and that "two apples" have something to do with the number "two".

If you want to progress with math, sooner or later you have to realize that those real-world ideas are irrelevant. Concepts like "number" or "infinity" mean whatever mathematicians define them to mean. If the definitions don't seem to match up with "common sense", then the math definition wins and common sense loses.

If you want to explore some simple mathematical ideas about infinity, Google for the "Hilbert's Hotel" paradoxes. It's an imaginary hotel with an infinite number of rooms - the so-called paradoxes are about what can happen when every room is already occupied but more people (even infinitely more people) want to check in. (The hotel is imaginary, but Hilbert was a real mathematician, working about 100 years ago).

 

[gb7nash]

For the extended real numbers, yes. There is + and - infinity. In the extended complex plane however, there's just infinity. It depends on the context.

 

[swerdna]

Thanks for all the posts. Merry 2010/2011 and beyond.

 

[TylerH]

Infinity is usually first [formally] encountered when learning of infinite limits, so it would also be appropriate to think of ∞±1 as the limit n±1 as n becomes infinite. As n becomes infinite, the ±1 would become negligible, thus the limit of n±1, as n becomes infinite, is ∞. This can be generalized to say that ∞±c, where c is any constant, is always just ∞.

 

[sadaf2605]

infity means INFINITY, dude! Just plusing or minusing some lil value rather than another infinity doesn't really bother to hamper its value at all...
For example when you fill a bottle with sea water, the water level does not really change that negligible, right? And infinity is much more huge than any ocean :p that's why we don't count them: infinity +/- 1 = infinity

 

[Phrak]

I think a consistent number system is possible, and really more natural, where I+1≠I. That is, I+1 is irreducible much in the same way that vector elements, as an ordered set, are not reducible to a single element.

For example,

$$ax + b\frac{1}{dx}\ ,$$

or generalized to "higher orders" of infinity,

$$ax + b\frac{1}{dx} + c\frac{1}{(dx)^2} + d\frac{1}{(dx)^3} + ... \ .$$

 

[g_edgar]

In cardinal numbers, $\aleph_0 + 1 = \aleph_0$, but subtraction is undefined.

In ordinal numbers, $1+\omega = \omega$ but $\omega+1 \ne \omega$ .