# Set cardinality exercise mistake in a textbook?

During studying set cardinality, I came across on this exercise:

$$2{}^{\aleph_0}2 = {}^{\aleph_0}2 + {}^{\aleph_0}$$

I understand that ${}^{\aleph_0}2$ is just $2^{\aleph_0}$ written in this way, at least previous text implies it. But I am really confused about what $^{\aleph_0}$ is supposed to be as there is no explanation for it whatsoever. There is a solution to the excercise, but it uses the same notation and therefore I do not understand it, even if it's quite short.

I think it's some typographical mistake, but having only brief background in cardinality, I can't figure out what the equation means. I bet it's either meant to be $2^{\aleph_0}$ or simply ${\aleph_0}$. Could you please find an explanation of this typo or post a meaningful version of the equation?

P.S. The textbook is in Slovak, so I won't post the solution here. Though I will translate it if it's necessary.

• Looks like a typo to me. It's hard to guess what was intended without more context. Can you show some of the context? Jul 16, 2017 at 23:30
• It could be related to infinity, $\aleph_0$ is sometimes used for the smallest infinite cardinal I think. Also, this seems to be a notation question I'll add that tag. youtube.com/watch?v=elvOZm0d4H0 and youtube.com/… talk about infinity, and youtube.com/watch?v=SrU9YDoXE88 talks about what's called aleph null.
– user451844
Jul 16, 2017 at 23:44
• I think the book's intention was $2\cdot {}^{\aleph_0}2={}^{\aleph_0}2+{}^{\aleph_0} 2.$ Jul 17, 2017 at 0:04
• @RoddyMacPhee I highly doubt that's the issue. Jul 17, 2017 at 0:41
• Something seems missing from the displayed equation. Jul 23, 2017 at 15:28

I have concluded that the stated equation really is a mistake in the textbook and that the intention was to write $2{}^{\aleph_0}2 = {}^{\aleph_0}2 + {}^{\aleph_0}2$ as Mr. Caicedo pointed out.