Cardinal Numbers and the Concept of Infinity in Mathematics

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In summary, the concept of subtracting infinity from infinity does not have a mathematical meaning unless we are considering specific contexts such as the extended real line or cardinal numbers. In most cases, it is undefined or results in the same value.
  • #1
Holocene
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Does subtracting infinity from infinity leave you with zero?

Or could you subtract infinity from infinity and still have infinity?
 
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  • #2
Subtracting infinity from infinity has no mathematical meaning. Unless we are talking about limits and orders of magnitude.
 
  • #3
Werg22 said:
Subtracting infinity from infinity has no mathematical meaning. Unless we are talking about limits and orders of magnitude.

You can't subtract but you can add infinity from infinity.

Let N be the natural numbers and R be the real numbers. And card(X) denote the cardinality of X.

card(R) + card(N) = card(R)
card(N) + card(N) = card(N)
 
  • #4
Holocene said:
Does subtracting infinity from infinity leave you with zero?

Or could you subtract infinity from infinity and still have infinity?
What "infinity" are you talking about? What notion of subtraction are you talking about? Your question really cannot be answered unless these are specified. (Though we can guess at what you meant, in order to give an answer)

Some examples:
The extended real line contains two points "at infinity": [itex]+\infty[/itex] and [itex]-\infty[/itex]. The extended real line also comes with a notion of subtraction that's defined for most, but not all arguments. [itex](+\infty) - (-\infty) = +\infty[/itex] and [itex](-\infty) - (+\infty) = -\infty[/itex], but [itex](+\infty) - (+\infty)[/itex] and [itex](-\infty) - (-\infty)[/itex] are undefined. (And any combination involving at least one finite number is defined)

In the cardinal numbers, there are lots of infinite cardinals. (literally, too many to count) Subtraction makes little sense for them, because if [itex]\alpha \leq \beta[/itex] and [itex]\beta[/itex] is infinite, then [itex]\alpha + \beta = \beta[/itex].

The hyperreal line contains many infinite and infinitessimal numbers, and in a certain sense, the hyperreals behave exactly like the reals. (e.g. you can subtract any hyperreal from any other hyperreal)
 
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  • #5
JasonRox said:
You can't subtract but you can add infinity from infinity.

Let N be the natural numbers and R be the real numbers. And card(X) denote the cardinality of X.

card(R) + card(N) = card(R)
card(N) + card(N) = card(N)
But, in order to do that, you have to be talking about "cardinal numbers", not the regular real numbers- which I'm pretty sure is what the OP was talking about. "Infinity", in any sense, is not a real number and so neither addition nor subtraction (nor, for that matter multiplication or division) is defined for "infinity".
 

What is infinity minus infinity?

Infinity minus infinity is a mathematical expression that represents the result of subtracting one infinite value from another. It is often used in discussions about the concept of infinity and its properties.

Is infinity minus infinity equal to zero?

No, infinity minus infinity is not always equal to zero. In some cases, it may result in zero, but it can also result in other values such as infinity or undefined.

Can infinity be subtracted from infinity?

Yes, infinity can be subtracted from infinity. However, the result of this subtraction may vary depending on the type of infinity being used and the mathematical context in which it is being used.

Why is infinity minus infinity undefined?

Infinity minus infinity is undefined because it violates the rules of arithmetic. In traditional arithmetic, any number subtracted from itself results in zero. However, infinity is not a number and does not follow the same rules.

What are the different types of infinity and how do they affect infinity minus infinity?

There are different types of infinity, such as countable and uncountable infinity, and they can affect the result of infinity minus infinity. For example, subtracting an uncountable infinity from another uncountable infinity may result in an undefined value, while subtracting a countable infinity from an uncountable infinity may result in an uncountable infinity.

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