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

In summary, there are multiple notions of infinity, each with its own rules and definitions. In some of these notions, the statement (∞ - 1) < ∞ is true, while in others it is not. The class of cardinal numbers is an example of a notion where this statement does not hold, while in other number systems such as the affine real line or the surreals, it may be true or not depending on the specific definition.
  • #1
ajayraho
6
0
Is this true?
( - 1) <
 
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  • #2
No. Infinity is not a number, so ordinary arithmetic doesn't apply.
 
  • #3
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.
 
  • #4
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.
 
  • #5
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.
 
  • #6
micromass said:
There are more notions of infinity than the cardinal or ordinal numbers.
What do you mean?
 
  • #7
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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Likes aikismos
  • #8
Thank you. (definitely not meant ironic; those somehow esoteric concepts didn't come to my mind)
 

1. What is "Inequality with Infinity"?

"Inequality with Infinity" refers to mathematical inequalities that involve the concept of infinity. This can include expressions such as "greater than infinity" or "less than infinity". In these cases, infinity is used as a theoretical concept rather than a specific numerical value.

2. How can inequalities involve infinity?

Inequalities can involve infinity when dealing with infinite sets or values that approach infinity. For example, in the expression "x > infinity", x could represent a value that is continuously increasing and approaching infinity.

3. Are there any rules or limitations when working with inequalities and infinity?

Yes, there are rules and limitations when working with inequalities and infinity. For example, infinity is not a real number and cannot be used in equations or calculations as such. It is also important to consider the context and meaning behind the use of infinity in an inequality.

4. What are some real-world applications of "Inequality with Infinity"?

Inequalities with infinity can be used to model and analyze various real-world scenarios, such as population growth, economic trends, and the behavior of physical systems. They can also be used in mathematical proofs and in the fields of calculus and analysis.

5. Can inequalities with infinity ever be equal?

No, inequalities with infinity can never be equal. This is because infinity is not a specific value or number, but rather a concept that represents something infinitely large or continuously increasing. Therefore, it cannot be compared or equated to a finite value or number.

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