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How might one show that (aleph_null)! = aleph_1?

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- Thread starter Doom of Doom
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- #1

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How might one show that (aleph_null)! = aleph_1?

- #2

HallsofIvy

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One might start by **defining** the "factorial function" for aleph_null!

- #3

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So, consider two sets, A and B. Then |A|*|B|=|A x B|, where AxB is the cartesian product of A and B.

Thus, consider N_m={1,2,3,...m}, and |N_m|=m.

Then (aleph_null)! = |N_1 x N_2 x N_3 x ...... |.

So how can I find a bijection from N_1 x N_2 x N_3 x ...... to, say, P(N), the power set of the naturals?

- #4

CRGreathouse

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This shows that the set has cardinality [itex]\beth_1[/itex], not [itex]\aleph_1[/itex] unless you have the CH.

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