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In set theory I read that the sets are either finite or infinite. If they are infinite then there are also two categories countably infinite or uncountable. Natural numbers $\Bbb{N}$, integers $\Bbb{Z}$, rational numbers $\Bbb{Q}$ etc. are examples of countably infinite sets , whereas real numbers $\Bbb{R}$, irrational numbers are very well known examples of uncountable sets.

Today I was thinking about counting objects in our day to day life. That time I realize that we can count each and every object.

For example :

1) Suppose I decided to count number of sand particles on a beach, even though the number is huge but one can count them one by one. ( I also want to know that are they infinite or just finite ( a big natural number will be representing their quantity )).

2) Same thing when I think about number of leaves on big tree, they are definitely finite.

3) Stars in the sky ( I read on internet that there are approx $10^{24}$ stars in universe). etc.

Similarly many objects seems to be finite ( or countably infinite* ( * please correct me if I'm wrong)) .

So my question is do we have any object in real world which is uncountable.

Thanks in advance !

Math_Explorer
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    As your examples show, it is not obvious to give an example of any physical collection which is even countably infinite as opposed to being enormously, but finitely, large. – Tom Collinge Jul 05 '17 at 10:32
  • @TomCollinge true . +1 for this. – Math_Explorer Jul 05 '17 at 10:34
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    The structure of space in the real world is not clear to science yet: it may ultimately be discrete or continuous, and bounded or unbounded. These questions matter to the cardinality of the set of points in space. – Ian Jul 05 '17 at 10:39
  • @Ian Thanks for this info :) – Math_Explorer Jul 05 '17 at 10:43

2 Answers2

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While counting real world objects motivated the creation of mathematical theories of natural numbers, the theories have limited applicability in the real world. Even "counting" is problematic when you examine it closely. So it does not make sense to apply theoretical classifications as you mention to real world objects. However, Archimedes speculated in his "Sand Reckoner" about the number of grains of sand to fill a ball the size of the universe, but strictly for a demonstration of how to think about large finite numbers.

Somos
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  • Thank you ! I have added Archimedes' "Sand Reckoner" page of Wikipedia in my bookmarks to read it carefully and understand it. – Math_Explorer Jul 05 '17 at 10:55
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Energy levels of a free particle

Riccardo.Alestra
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