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So recently I have been thinking about infinity, and one of the things that I thought of was if you were able to get a defined value for the reciprocal of a transfinite (cardinal) number. So, I plugged $\frac{1}{א_0}$ into WolframAlpha and it said the following: img1 Why is this the case? Shouldn't this be similar to this case? $$\lim_{x\to\infty} \frac{1}{x} =0$$ Aren't $\infty$ and $א_0$ equal to the same value in this context? What am I missing here?

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    $\begingroup$ You cannot divide by a number that has no multiplicative inverse. $\endgroup$ – Peter Foreman Mar 18 '20 at 19:24
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    $\begingroup$ What is $f$? Why did you put $\infty$ in the function and not $x$? Division is not defined to be a limit, especially in a set-theoretic context rather than an analytic context. In a set-theoretic context, it would only make sense to divide one cardinality by another if you can partition the one set into subsets of size equal to the other set. Which you can't do with $1$ and $\aleph_0$. $\endgroup$ – runway44 Mar 18 '20 at 19:26
  • $\begingroup$ I have now edited the formula to what I think you mean. Feel free to edit and clarify. $\endgroup$ – G. Chiusole Mar 18 '20 at 19:32
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    $\begingroup$ Probably some infinitely many more questions explaining what are the different notions of infinity. Please search the site before posting. $\endgroup$ – Asaf Karagila Mar 18 '20 at 21:31
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For one thing, in $\lim\limits_{x\to\infty}f(x)=L$ the symbol $\infty$ does not represent a number of any kind, cardinal or real, and the expression says something very different from $\lim\limits_{x\to c}f(x)=L$. The latter is a statement about two numbers, $L,c$, and a function $f$, that says values of $f$ get really close to $L$ when its inputs are really close to $c$. The former is a statement only about $L$ and $f$, that says the bigger the $x$, the closer $f(x)$ is to $L$. Note that it is not a statement about values of $x$ getting closer to some value: "bigger" does not mean "closer to $\infty$" (since "closer" only has a defined meaning between real numbers in this context).

There is, of course, such a thing as $\aleph_0$, but it is emphatically not a real number; it belongs to an entirely different class of numbers, with an entirely different ordering relation, while convergence to a limit is defined only in terms of real numbers and their relationships. So real division by $\aleph_0$ makes no sense.

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