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$\infty$ (Infinity) is not a number, but infinity is considered to be defined, right?

There are expressions in mathematics such as: $\frac x0,0^0, \frac\infty\infty,$ which are not defined because they do not have a certain place on any domain, and sometimes because there is no single value that satisfies all functions yielding such expression in a limit.

How can infinity be defined?


Maybe the argument would hinge on how 'defined' is defined?

Perhaps this requires a circular(infinite) argument?

Or is my assumption that infinity is considered to be defined; false?

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    $\begingroup$ I trust you've looked at the previous question, What exactly is infinity? $\endgroup$ – user856 Jan 28 '13 at 13:09
  • $\begingroup$ Yeah I visited a number of similar questions including that one (I didn't read every single word) and even posed the question in chat, but I did not find the answer to my question. $\endgroup$ – Elements in Space Jan 28 '13 at 13:16
  • $\begingroup$ Have you considered the extended complex no.s definition of $\infty$? $\endgroup$ – Ishan Banerjee Jan 28 '13 at 13:31
  • $\begingroup$ I haven't, Is that just that $\infty\equiv\frac10$ ? $\endgroup$ – Elements in Space Jan 28 '13 at 13:40
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I assume that you're talking about the $\infty$ that appears in analysis and calculus, when calculating limits. In general, $\infty$ is just a symbol, however it intuitively looks like to the countable infinite $\omega$ cardinal. For instance, when you think about Riemann integrals and you tends a partition $P$ to infinity, $P$ is always finite , then it's at most countable. When you take a limit walking on the real line and it goes to $\infty$, then it just means that it's unlimited and since you're computating it by finite approximations, then it intuitively "looks like" $\omega$ (but, of course, it's not $\omega$).

Another way of thinking about infinity in analysis is to think about it as the element that makes the real line compact (the Alexandroff compactification). If you're thinking about limits over $\mathbb{C}$, then I think that the Riemann sphere is the more intuive way of seeing $\infty$.

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$\infty$ is just a symbol to denote such quantities which are bigger than any real quantity and nothing else.

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  • $\begingroup$ So it is defined as being bigger than any real number? $\endgroup$ – Elements in Space Jan 28 '13 at 13:09
  • $\begingroup$ something like this? $$\forall x \in \mathbb R (\infty>x)$$ $\endgroup$ – Elements in Space Jan 28 '13 at 13:19
  • $\begingroup$ May be if you consider $\infty\in\mathbb{R^*}$ i.e. extended real number system. $\endgroup$ – user45099 Jan 28 '13 at 13:21
  • $\begingroup$ I think @user1709828 have the same explanation i was going to give. $\endgroup$ – Abhra Abir Kundu Jan 28 '13 at 13:23
  • $\begingroup$ $$\sup \mathbb R$$ ? $\endgroup$ – Elements in Space Jan 29 '13 at 11:52
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We know what is something finite. In the context of the real numbers it simply means that the value is "a real number"; in the context of sets it means "There is a natural number $n$ such that the set has a bijection with $\{0,\ldots,n-1\}$."

Something is infinite if it is not finite. There is circularity, once we know what it means to be finite we simply define infinite as its negation.

If numbers measure some sort of quantity, and we can identify the real numbers with measuring "length" then an infinity is something longer than any finite length. In the context of set theory there are several notions of infinities (ordinals and cardinals) which extend things which are measured by the natural numbers. However where in real analysis there is pretty much one notion of infinity, in set theory we have a whole class of numbers which represent infinite objects.

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  • $\begingroup$ I downvote because the answer seems to conflate 'infinity' with something infinite, which does not agree with the use of these notions in mathematics. E.g. using $\infty$ in real analysis, you have to add actual objects $-\infty$ and $\infty$ to the set $\mathbb{R}.$ We can let $\infty := \{ \{ \{ \emptyset \} \} \}$ and $-\infty := \{ \infty \}.$ In either case the cardinality of $\pm \infty$ is $1,$ which is finite. Indeed it would seem silly and uneconomical to let $\infty$ be anything infinite, unless the intent is extremely exotic. $\endgroup$ – Tommy R. Jensen Aug 10 '19 at 11:03

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