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If I take infinity and subract one from it,then I'll get infinity, but what if i take that result and subtract one again and I repeat that process an infinite number of times. Shouldn't I get infinity. but isn't that equal to $\infty-\infty $ which isn't defined?

Basil Hallaq
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    How long did it take you to subtract one an infinite number of times? – Angina Seng Feb 06 '18 at 18:39
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    Sorites paradox. A heap less a grain is still a heap. – Doug M Feb 06 '18 at 18:46
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    You cannot subtract $1$ from $\infty$. Even, if we allow the ordinal number $\omega$, there is no predecessor of $\omega$ – Peter Feb 06 '18 at 18:46
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    Why are there so many downvotes? This is a good question, that comes from a common misunderstanding of the concept of infinity. Why discourage people from asking such questions? – idok Feb 06 '18 at 18:54

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Infinity isn't a number. You can't add or subtract it, and you can't subtract something from it either.

It is a concept involved in the notion of limit, which is more subtle than you think, because you can tend to infinity at very different speeds.

It is also involved in counting the number of elements in a set, but then again there are many different sizes an infinite set can be, and you still need to be careful when manipulating them.

Arnaud Mortier
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  • Is then incorrect to write $\infty!$ to represent the $1\cdot2\cdot ... \cdot n=n!$ ? where $n$ goes to $\infty.$ – Verónica Rmz. Jul 25 '21 at 00:58
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    @VerónicaRmz.If you are willing to write "$f(\infty)$" as a shortcut for "$\lim_{x\to +\infty}f(x)$" and if it is clear for the reader then it is not incorrect per se. But I would not recommend it as it can only bring confusion to unexperimented readers. – Arnaud Mortier Jul 26 '21 at 08:33
  • Ok. This piece actually came from a result that I saw once: $\infty!=\sqrt{2\pi}$. Did they write the notation of $\infty!$ so that it looked striking right? But they just meant to indicate that the factorial holds for all naturals. – Verónica Rmz. Jul 27 '21 at 16:24
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    @VerónicaRmz. I see. This equality is of the same vein as $1 + 2 + 3 + \ldots = -\frac{1}{12}$ and has to do with complex analysis and analytic continuation of special functions. See this question. – Arnaud Mortier Jul 28 '21 at 17:55