# Inequality with infinity

1. Nov 13, 2015

### ajayraho

Is this true?
( - 1) <

2. Nov 13, 2015

### UncertaintyAjay

No. Infinity is not a number, so ordinary arithmetic doesnt apply.

3. Nov 13, 2015

### micromass

For this you will need to define exactly what you mean with infinity. There are multiple notions of infinity, some notions where the above is true, some where it isn't true. But there is no standard notion of what $\infty$ means.

4. Nov 13, 2015

### Staff: Mentor

How? The smallest cardinal number is countably infinite and it doesn't matter whether you add 1 or not.

5. Nov 13, 2015

### micromass

There are more notions of infinity than the cardinal or ordinal numbers.

6. Nov 13, 2015

### Staff: Mentor

What do you mean?

7. Nov 13, 2015

### micromass

The class of cardinals numbers is one where $\aleph_0 - 1$ doesn't even exist.
There is a number system (e.g. the affine real line $\mathbb{R}\cup \{-\infty,+\infty\}$), where $\infty - 1$ exists an is equal to $\infty$.
There is a number system (e.g. the surreals) where $\infty-1$ exists and is distinct from $\infty$.

8. Nov 13, 2015

### Staff: Mentor

Thank you. (definitely not meant ironic; those somehow esoteric concepts didn't come to my mind)