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December 21st, 2016, 10:34 PM   #1
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Singularity with continuity; function

Can all numbers be proved unique, i.e. singular? Are real numbers both continuous and singular? Can arithmetic operations act upon all numbers?
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December 21st, 2016, 10:47 PM   #2
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I think a more realistic question is can all numbers be distinguished from one another within the lifetime of the universe.

Suppose you are given two real numbers whose values are stably encoded in the atoms of two cubes each a few hundred thousand light years to a side.

Can you verify these two numbers are different?

Now to get really fun.... Suppose these cubes require enough matter so that their mass collapses the local space into a black hole. Their values are shielded from the outside world now so you'll never be able to verify that they are distinct.

I'm not sure what you are asking with the last question. Are you asking if there are numbers out there that have some property that render standard operations like addition invalid? Sounds like Greg Egan's stories Luminous and Dark Integers.
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December 22nd, 2016, 04:22 AM   #3
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Why does the title include the word "function"?

If n is an integer, n - 1 and n + 1 are also integers. Any two integers that aren't the same differ by another integer that is at least 1. In contrast, there is no minimum difference between arbitrary real numbers that aren't the same.

The answer to the third question is "no", as, for example, division of a number by zero isn't defined.
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December 22nd, 2016, 05:27 AM   #4
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Can all real numbers be proved unique?
All real numbers are unique by definition. If two real numbers are equal, they are the same number. Romsek's treatment of the question is probably more interesting.

Can arithmetic operations act upon all numbers?
The real numbers can be defined as the closure of the rationals (they form the set containing the limit of every sequence of rational numbers). The properties of limits mean that the reals are closed under addition, subtraction, multiplication and division (except by zero).

The closure of the rationals under the for operations is easy enough to demonstrate. You can probably find proofs online.
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Last edited by v8archie; December 22nd, 2016 at 06:15 AM.
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