Is 1/countable infinity = 0 or not(or is it infinitisimal).I already know that countable infinity/uncountable infinity= 0 and 1/uncountable infinity is also = 0 .On this wiki link-https://en.m.wikipedia.org/wiki/Infinitesimal it is some what mentioned to be = infinitisimal but not clearly. Would be very thankful for any useful input or perspective along with the explanation.

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    $\begingroup$ Infinity is not a real number $\endgroup$ – J. W. Tanner Dec 19 '19 at 14:56
  • $\begingroup$ then please look into this wiki link $\endgroup$ – Anshul Agrawal Dec 19 '19 at 14:57
  • $\begingroup$ en.m.wikipedia.org/wiki/Infinitesimal $\endgroup$ – Anshul Agrawal Dec 19 '19 at 14:58
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    $\begingroup$ Infinite cardinals are not real numbers, nor they extend the real numbers in any kind of way. So asking about $1/{\aleph_0}$ is meaningless in every which way you are trying to put it. $\endgroup$ – Asaf Karagila Dec 19 '19 at 14:59
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    $\begingroup$ Division is defined over fields, or rings, and things like that. But cardinals do not make anything remotely similar to this. For example $\aleph_0+\aleph_0=\aleph_0+5=\aleph_0+1=\aleph_0$, but $\aleph_0\neq 5\neq 1\neq 0$. Likewise $\aleph_0\cdot\aleph_0=\aleph_0\cdot 5=\aleph_0$. Division is simply cancelling out multiplication, and you can't quite do that with infinite cardinals. Yes, you can extend the real line by introducing infinitesimals and their inverses (which are "transfinite numbers"), byt there's no reason for these to be compatible with the infinite cardinals. $\endgroup$ – Asaf Karagila Dec 19 '19 at 15:10

Actually, one doesn't "divide by $\infty$", since $\infty$ is not a number. In mathematics, you should always understand division as multiplication by an inverse. In particular, you have to be within some ring.

Moreover, to say that an infinitesimal is $\frac{1}{\infty}$ is a blatant abuse, an oversimplification (unless you decided to name a specific invertible element in a ring $\infty$, but that's ugly). An infinitesimal is a number : you can multiply/add it with/to another number.

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  • $\begingroup$ for what i can understand their is some problem with division so is 0 x countable infinity and 0 x uncountable infinity defined / meaningfull $\endgroup$ – Anshul Agrawal Dec 19 '19 at 16:43
  • $\begingroup$ @AnshulAgrawal multiplication of cardinals $\kappa$ and $\lambda$ is defined as the size of the cartesian product $\kappa\times\lambda$, so assuming that $0=\varnothing$ is the empty set it makes certainly sense: both are equal to $0$. The problem is that cardinal multiplication has no inverse, thus division does not make sense. Actually the same holds for addition; subtraction does not make sense either. It's best not to confuse cardinal arithmetic with real arithmetic, as they are entirely different beasts. $\endgroup$ – Vsotvep Dec 19 '19 at 18:33
  • $\begingroup$ But i have heard that the probability of selecting a specific number(ex- e-2) randomly from the interval 0 to 1 (real number) = 0.So for the whole sample space to sum up to probability 1(have probability 1), Isn't it is required that 0 x uncountable infinity =1? (here) $\endgroup$ – Anshul Agrawal Dec 19 '19 at 18:40
  • $\begingroup$ youtu.be/-6HxjiW_KwA $\endgroup$ – Anshul Agrawal Dec 19 '19 at 18:51
  • $\begingroup$ @AnshulAgrawal What you found out is that probability does not work like that. As far as countably many events are concerned, you can simply sum them together to get an upper bound to the total probability. This is known as sigma additivity. It does not extend to uncountably many events. Even with countable sets there are problems, since there is no uniform distribution on a countably infinite set (this follows immediately from sigma additivity). Lastly, note that probability has to do with measure of sets, not with cardinality of sets. Once again, they are different things. $\endgroup$ – Vsotvep Dec 19 '19 at 19:41

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