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A “number” with an infinite number of digits is a natural number?

Is $58654965542\ldots\in \mathbb{Z}^+$ ?

In General, for any number $N$:
$N:=a_0 a_1 a_2 ...a_{n-1} a_n a_{n+1} ...$ "to infinity" : $ a_i \in {0,1,2,...,9}$ $ \forall i \geq 0$. Is $N \in \mathbb{Z}^+$ ?

If the answer "not" : Is $\exists A ( N \in A)$ : $A$ is a set of numbers ?

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marked as duplicate by Arturo Magidin, Eric Naslund May 20 '12 at 20:02

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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There is no number in the positive integers that contais infinitely many digits. –  Michael Greinecker May 20 '12 at 12:18
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The sum is not readable. What is $m\mathbb{Z}^+$? A question should contain a verb or a symbol that can be read as one. –  Michael Greinecker May 20 '12 at 12:27
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It's an integer if and only if all but finitely many of the $a_i$ are zero. –  Gerry Myerson May 20 '12 at 12:28
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You might want to ask yourself what is $\mathbb{Z}^+$? If you use the Peano axioms, then $\mathbb{Z}^+$ is exactly the set of numbers which can be reached by added $1$ to itself a finite number of times. A consequence is that no integer can have an infinite string of digits. –  Eric Naslund May 20 '12 at 12:29
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As for the second question, it is incomprehensible. What does $\exists A(N\in A):A$ mean? –  Gerry Myerson May 20 '12 at 12:29
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1 Answer 1

The set $\mathbb{Z}_+$ is simply the set $\{0,1,2,3,\ldots\}$. In practice, one treats the numbers as basic objects. We write them with finitely many digits. It is of course perfectly possible to create a strange set-theoretic construction that allows the question to be meaningful.

For eaxample, you can write every number in $\mathbb{Z}_+$ in the form $a_n 10^n+\ldots+ a_2 10+a_1$ for some, uhm, positive integer $n$ and $a_i\in\{0,1,\ldots,9\}$. What you can do is reverse the order and add infinitely many zeros. So you can write, say, $113$ as $311000000\ldots$ and adapt the rules of arithmetic accordingly. In that case, it is possible to write an element in $\mathbb{Z}_+$ as an infinite sequence of digits. The sequences that will denote such numbers are exactly those that have only finitely many terms not equal to $0$. But this is a highly artificial construction.

And without a somewhat artificial construction, the question makes essentially no sense.

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"you can write, say, 113 as 311000000…", why ? –  AHH May 20 '12 at 12:59
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Why not? I just do it. I don't claim it is useful or sensible, but it certainy can be done. –  Michael Greinecker May 20 '12 at 13:01
    
Are you mean 113=...000113 ? –  AHH May 20 '12 at 13:03
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What I mean is that there is no unique way to formalize $\mathbb{Z}_+$ in set theory, and one such formalization is as a certain sequence of digits. Unless you provide a specific set-theoretic constructions of $\mathbb{Z}_+$, the question cannot really be answered. –  Michael Greinecker May 20 '12 at 13:07
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