Can Arithmetic Operations with Infinities Be Undefined?

[AxiomOfChoice]

Am I correct in assuming that you can make sense of \infty + \infty and \infty + c for any c\in \mathbb{R} (both evaluate to \infty), but that we can make no sense of \infty - \infty?

Are there any other arithmetic operations one can perform with infinities that are undefined?

 

[rasmhop]

Am I correct in assuming that you can make sense of \infty + \infty and \infty + c for any c\in \mathbb{R} (both evaluate to \infty), but that we can make no sense of \infty - \infty?

Are there any other arithmetic operations one can perform with infinities that are undefined?

What do you mean make sense of? I can define \infty -\infty = 8 if I want. We would lose some properties of our numbers, but there would be nothing contradictory about it. Usually we choose one of two approaches:
1) Never do arithmetic with infinities, but instead speak about infinity using the concept of limits (that is a number can't be \infty[/tex], but a sequence can approach \infty).<br /> 2) Introduce some operations using infinity like you have. We normally call the set of real numbers together with \infty,-\infty the extended real line. In the extended real line we have defined all the usual operations for all values except \infty - \infty[/tex], 0\times(\pm\infty), (\pm \infty)/(\pm \infty), and the real values at which they are normally not defined (such as 0/0). However it comes at a cost. For instance we can usually say that if a+b = a+c, then b=c (this is known as the cancellation law), but it&amp;#039;s not true in the extended reals.&lt;br /&gt; &lt;br /&gt; Personally I have only seen one use of the extended real line and that was in measure theory. Apart from that I think people should stick to the real numbers unless the extended real lines provides distinct advantages over the real number system.&lt;br /&gt; &lt;br /&gt; EDIT: I may inadvertently have given the impression that the extended real line is the only way to add \infty, but this is wrong. Another common way is known as the real projective line. I&amp;#039;m not really familiar with this except as an example from topology so I haven&amp;#039;t looked into its arithmetic properties, but it differs in the way that there is only one extra element (-\infty=\infty). I still believe \infty - \infty is left undefined though, but on the real projective line \infty+\infty is also undefined. You can look it up on for instance Wikipedia if you want more in depth information.