- #1

- 86

- 0

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

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter Doom of Doom
- Start date

- #1

- 86

- 0

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

- #2

Science Advisor

Homework Helper

- 43,008

- 974

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

- #3

- 86

- 0

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

Science Advisor

Homework Helper

- 2,842

- 0

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

- #5

- 17

- 0

Share:

- Replies
- 1

- Views
- 736

- Replies
- 5

- Views
- 774

- Replies
- 5

- Views
- 613

- Replies
- 7

- Views
- 542

- Replies
- 69

- Views
- 7K

- Replies
- 13

- Views
- 620

- Replies
- 1

- Views
- 1K

- Replies
- 4

- Views
- 522

- Replies
- 2

- Views
- 488

- Replies
- 5

- Views
- 1K