3

Is there any theory that takes account of the difference in, for example, $1 \over 0$ and $-\ln 0$ and treat them as different infinities with different notations (and maybe useful theorems)? Cantor's theory only specializes in set cardinality but is there a more general theory on the concept of infinity itself?

Michael Hardy
  • 1
cr001
  • 12,598
  • Although it's not clear exactly what you want, you may be interested in nonstandard analysis. – GPhys Aug 10 '16 at 00:30
  • If you mean "analytic" infinity, not really; there's the extended real numbers and the Riemann sphere and that's about it, in the sense of standard analysis. There is also Robinson-type nonstandard analysis (aka the hyperreal numbers), which involves a whole family of infinite real numbers....but their exact role in the theory is not exactly what you might expect from familiarity with standard analysis, because the hyperreals don't really have limits. Also, you may have missed the notion of an infinite ordinal, which is distinct from the notion of an infinite cardinal. – Ian Aug 10 '16 at 00:30
  • Thanks for the answers, I will look into the materials. My motivation was that is there a theory that investigates deeply into for example the reason why the limit of $xlnx$ is zero (In this example the "infinity" generated by $lnx$ is less than the "infinity" generated by $1\over x$). – cr001 Aug 10 '16 at 00:42
  • @cr001 I think you should add your motivation (mentioned in your most recent comment) to the post. It clarifies what sort of answer you're looking for, and (I think) makes the question a good one. In its current form it doesn't really look like a real question (hence the downvotes and vote to close; btw I upvoted). – Noah Schweber Aug 10 '16 at 00:56
  • Note the typographical difference between $x ln x$ (coded as "x ln x") and $x\ln x$ (coded as x\ln x). The latter is standard. $\qquad x ln x \qquad x\ln x$ $\qquad$ – Michael Hardy Aug 10 '16 at 01:26
  • The down-votes and the vote to close the question are rash. The fact that the poster doesn't know standard terminology and concepts should not be a reason for that. $\qquad$ – Michael Hardy Aug 10 '16 at 01:28

1 Answers1

4

Your comment suggests that what you are looking for is the theory of growth rates of functions.

See, for example, the Hardy hierarchy, although you may be interested in fast-growing hierarchies in general.

In the area of the abstract study of growth rates, you may be interested in Hausdorff gaps, or in some of the related cardinal characteristics of the continuum (e.g. the dominating and bounding numbers).

Noah Schweber
  • 245,398
  • I don't know if this is also something you might be interested in, but there are at least two different ways to add an infinity symbol to the real number line, see (1) https://en.wikipedia.org/wiki/Extended_real_number_line which adds two symbols, plus or minus infinity, and (2) https://en.wikipedia.org/wiki/Projectively_extended_real_line which adds only one symbol. As a result, while $1/0$ is still ambiguous in (1), it makes sense in (2) -- in fact, it can even be used to make the function $1/x$ continuous on all of $\mathbb{R}$, even 0. – Chill2Macht Aug 10 '16 at 01:44