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

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    Inequality Infinity
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Discussion Overview

The discussion revolves around the validity of the inequality (∞ - 1) < ∞, exploring the nature of infinity and its implications in various mathematical contexts. Participants examine different definitions and frameworks of infinity, including cardinal and ordinal numbers, as well as other mathematical systems.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that infinity is not a number, suggesting that ordinary arithmetic does not apply to it.
  • Others propose that the truth of (∞ - 1) < ∞ depends on the specific definition of infinity being used, indicating that there are multiple notions of infinity.
  • One participant mentions that the smallest cardinal number is countably infinite, arguing that adding 1 does not change its value.
  • Another participant points out that there are more notions of infinity beyond cardinal and ordinal numbers.
  • It is noted that in certain number systems, such as the affine real line, the expression (∞ - 1) can exist and be equal to ∞, while in other systems, such as the surreals, (∞ - 1) exists and is distinct from ∞.

Areas of Agreement / Disagreement

Participants do not reach a consensus; multiple competing views regarding the nature of infinity and the validity of the inequality remain present throughout the discussion.

Contextual Notes

Participants highlight the importance of defining infinity clearly, as different mathematical frameworks lead to different interpretations and results regarding the expression (∞ - 1).

ajayraho
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Is this true?
(∞ - 1) < ∞
 
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No. Infinity is not a number, so ordinary arithmetic doesn't apply.
 
ajayraho said:
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.
 
micromass said:
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.
 
fresh_42 said:
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.
 
micromass said:
There are more notions of infinity than the cardinal or ordinal numbers.
What do you mean?
 
fresh_42 said:
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##.
 
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Thank you. (definitely not meant ironic; those somehow esoteric concepts didn't come to my mind)
 

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