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By extending the real number line in both directions, I know that $\infty$ or $\aleph_0$ or whatever else you want to call it has a negative, i.e. $-\aleph_0$ is a thing.

Now, of course, $\beth_0 = \aleph_0$, so $-\beth_0$ is also a thing in this sense. My question is, do $-\beth_1$ and $-\beth_2$ etc. exist as well, or are the Beth Numbers greater than $\beth_0$ all unsigned, as Computer Scientists would say?

Ahsim Nreiziev
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  • How is $-\aleph_0$ defined? – Adam Jaffe Mar 05 '16 at 02:22
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    The pressing issue is this: Cardinal addition isn't cancellative, so we can't create inverses. That is: What is is $(\aleph_0+1) + (-\aleph_0)$? – Milo Brandt Mar 05 '16 at 02:23
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    As to extending the real line, I don't want to call the two added points $\aleph_0$ and $-\aleph_0$. – André Nicolas Mar 05 '16 at 03:09
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    Ahsim "knows that $\infty$ or $\aleph_0$ ... has a negative." Well, I don't know that. Perhaps Ahsim should provide a reference for this amazing thing! – GEdgar Mar 05 '16 at 03:46
  • Ok. So maybe I didn't word my question correctly then. I am nothing more than an amateur at Mathematics, so it's entirely possible I misunderstood something. So, let's ask an intermediate question to further my understanding. When people talk about "$-\infty$", which happens very, very often when talking about infinities (the countable one at least) what do they mean if not "$-\aleph_0$"? – Ahsim Nreiziev Mar 05 '16 at 03:50

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