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May 10th, 2017, 09:00 AM   #101
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Quote:
Originally Posted by zylo View Post
You ask if 101010.... is a natural number.
I didn't ask that. I gave 101010101... as an example that doesn't satisfy the definition of a natural number.

Quote:
Originally Posted by zylo View Post
It is for a finite number of digits.
Yes, because the definition of a natural number is satisfied for any finite number of digits. It isn't, however, satisfied when the number of digits isn't finite, which was the case in the example I gave.

Quote:
Originally Posted by zylo View Post
So as usual I ask, for what number of digits does it cease to be a natural number?
It doesn't cease to be a number for any natural number of digits. In all the finite cases, it's a natural number. In the infinite case, it isn't, because that case doesn't satisfy the definition of a natural number. That's consistent with "infinity" not being a natural number.

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Originally Posted by zylo View Post
101010.... converges in the sense that it converges to a very specific natural number, distinguished by the fact that it consists of a countably infinite sequence of unique digits, which is exactly what distinguishes the real numbers.
Whatever you meant by "converges to", you haven't justified calling it a specific natural number; 101010101... has infinitely many digits and doesn't satisfy the definition of a natural number, so it's certainly not a specific natural number.

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Originally Posted by zylo View Post
. . . consists of a countably infinite sequence of unique digits, which is exactly what distinguishes the real numbers.
No, you simply made that up, and it's clearly incorrect, as each natural number is also a real number. Even if I ignored that error, asserting that my example is a real number wouldn't justify asserting that it's a specific natural number.
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