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Consider the extended reals, $\mathbb{R} \cup \{\infty\}$. Suppose we ask the question of whether $\infty$ is an integer.

  1. It isn't an integer - that is, it is some non-integral value. But then, by the Archimedean property, there exists an integer larger than it, which is contradictory.

  2. It is an integer. But again, by the Archimedean property, there is an integer greater than it. Also a contradiction.

That is, the statement "$\infty$ is an integer is false", but so is the statement "$\infty$ is not an integer". This stands in contradiction to the law of excluded middle, however. Now, I have no knowledge about logic, so I wanted to ask how this might be resolved (is the initial question even valid? It certainly seems well-defined but I am unsure).

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    $\begingroup$ Since when do the Archimedean property works with the extended reals ? $\endgroup$
    – LL 3.14
    Oct 5 at 20:16
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    $\begingroup$ This isn’t a contradiction. Rather, it’s a proof that the extended reals aren’t Archimedean. $\endgroup$
    – David H
    Oct 5 at 20:16
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    $\begingroup$ It's not hard to get a contradiction if you're claiming "but then by the Archimedean property" in a non-Archimedean set :P $\endgroup$
    – Jam
    Oct 5 at 20:16
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    $\begingroup$ The arquimedian property says that for every real number there is a integer number larger than it. This doesn’t apply to $\infty$, that it is not a real number. $\endgroup$
    – azif00
    Oct 5 at 20:17
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    $\begingroup$ The Archimedean principal is about the real numbers. The very notation $\mathbb R\cup\{\infty\}$ suggests what you should already know, that $\infty$ is not a real number. $\endgroup$ Oct 5 at 21:09

2 Answers 2

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quick answer: No. There is a standard definition of "integer" and $\infty$ is not a member of the set of integers. Longer answer: You can use words any way you like. If you wish to redefine "integer"to include $\infty$, go ahead. But you have a moral obligation to tell readers in the introduction to your journal paper or book or to tell your students in your introductory lecture that you are using that word with your own, non-standard definition.still longer answer: If you add $\infty$ and/or $- \infty$ to $\mathbb Z$ you destroy the property of being an abelian group. My purely subjective opinion is that you are paying too high a price for whatever benefits you might gain.

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It's not an integer.

The extended reals don't have an arithmetic structure that makes them a field, so saying that the order structure in the extended reals is not Archimedean is not a contradiction.

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