Is (∞ - 1) < ∞ True for Inequalities with Infinity?

[ajayraho]

Is this true?
(∞ - 1) < ∞

 

[UncertaintyAjay]

No. Infinity is not a number, so ordinary arithmetic doesn't apply.

 

[micromass]

Is this true?
(∞ - 1) < ∞

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.

 

[fresh_42]

There are multiple notions of infinity, some notions where the above is true...

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

 

[micromass]

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

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

 

[fresh_42]

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

What do you mean?

 

[micromass]

What do you mean?

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 $\infty$.

 

[fresh_42]

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