Can Infinity Minus X Ever Be Larger?

[jumbogala]

I understand that infinity - X = infinity.

However, what I don't understand is this. If I have: infinity - 1 = infinity, AND infinity - 5 = infinity, is the first infinity larger than the second?

If so, how can that be? Because they are infinite, how can one really be larger than the other?

Also, you can't add to infinity, right? Or can you? *confused*

Thanks!

 

[PowerIso]

I understand that infinity - X = infinity.

It does? When did we define infinity to be a number instead of a concept?

The problem with using infinity as a number is that you run into problems like this one. Infinity plus or minus can be anything you want it to be, but the point is that infinity is not a number. Infinity is simply a concept (for the most part). Using it as an actual number leads to results like these. It doesn't work.

 

[nicktacik]

Looking for a value of infinity minus X is like looking for a value of Apple divided by Cat. It's just not defined.

 

[PowerIso]

Looking for a value of infinity minus X is like looking for a value of Apple divided by Cat. It's just not defined.

Never had a cat with apple filling? You're missing out.

 

[Hurkyl]

Infinity is what it's defined to be.

In the arithmetic of real numbers, there is no infinity. So statements like

infinity - X = infinity​

are gibberish.


It is possible to consider different systems of arithmetic. Two practical examples are the projective real numbers and the extended real numbers. In the extended real numbers, for example, it is true that
$$+\infty = +\infty - x$$
whenever x is not $+\infty$. The arithmetic of the extended real numbers is a different system; in particular, the statement

if x > 0, then x + y > y​

is simply not true.


Another useful number system is the ordinal numbers. There is no ordinal number called "infinity", but there are lots of infinite ordinals. The smallest infinite ordinal is $\omega$. In the ordinal numbers, the following statements are true:
$$1 + \omega = \omega$$
$$\omega + 1 > \omega$$
(note that addition is not commutative! If you add in different orders, you usually get different ordinal numbers)

The ordinals are related to orderings, and sequences. $\omega$ is the ordinal that describes the sequence of natural numbers. $1 + \ometa$ is what you get by prepending an extra thing to the natural numbers... clearly the sequences

0 < 1 < 2 < ...
* < 0 < 1 < ...​

have the same order type. The ordinal number $\omega + 1$ is what you get if you append an extra thing to the natural numbers:

0 < 1 < 2 < ... | *​

Every natural number appears in this sequence before the *, and there is no number that comes immediately before *. This sequence is clearly different than the other two above.

(The | is my own notation for this sort of thing. It's like a parenthesis: it's meant to indicate that the stuff on the left is to be taken as one "group", and the stuff on the right is to be taken as another "group")


I suppose the simple answer is, if you intend to learn about the infinite, you should forget everything you "learned" about it from nonmathematical sources.