Talk:Q1445491

aleph null = N

@Horcrux: That $\aleph _{0}=|\mathbb {N} |$ is obvious.

Yet, it is also $\aleph _{0}=|\mathbb {N} |=\mathbb {N}$. This is, because $\aleph _{0}$ is the smallest infinite cardinal number and thus the smallest infinite ordinal number. $\mathbb {N}$ is also the smallest infinite ordinal number.

For $\aleph _{0}=\mathbb {N}$ see also Ordnungen, Verbände und Relationen mit Anwendungen. --Eulenspiegel1 (talk) 20:23, 22 October 2018 (UTC)

Can you explain how a number (though cardinal) can be a set? --Infovarius (talk) 08:15, 25 October 2018 (UTC)

Every number is a set:

• $0:=\emptyset$
• $1:=\{0\}=\{\emptyset \}$
• $2:=1\cup \{1\}=\{0,1\}=\{\emptyset ,\{\emptyset \}\}$
• $3:=2\cup \{2\}=\{0,1,2\}$
• $n+1:=n\cup \{n\}=\{0,1,2,...,n\}$
• $\aleph _{0}=\bigcup _{n}\,n=\bigcup _{n}\,\{0,1,2,...,n\}$

See also en:Natural_number#Constructions_based_on_set_theory or Set Theory and its Philosophy: A Critical Introduction, p. 219. --Eulenspiegel1 (talk) 22:53, 25 October 2018 (UTC)

There I read "Von Neumann ordinals" and "Zermelo ordinals". Indeed, I read the same definition in en:Ordinal number#Von Neumann definition of ordinals. Same in the source provided by yourself. --Horcrux (talk) 10:54, 26 October 2018 (UTC)

The natural numbers ($\mathbb {N}$) are a subset of the ordinal numbers. Or the other way around: ordinal numbers are an extension of natural numbers.

"Von Neumann ordinals" and "Zermelo ordinals" are two ways to define ordinal numbers. --Eulenspiegel1 (talk) 11:13, 26 October 2018 (UTC)

I see. However, since "$\aleph _{0}$ is the smallest infinite cardinal number", saying that they are the same thing implies that $\mathbb {N}$ is also a cardinal. --Horcrux (talk) 11:50, 26 October 2018 (UTC)

Yes, of course, $\mathbb {N}$ is also a cardinal. See for example: More Precisely: The Math You Need to Do Philosophy - Second Edition, p. 205 If you say $\{0,1,2,...\}$ is an ordinal number or a cardinal number is just a point of view. It's both. --Eulenspiegel1 (talk) 13:19, 26 October 2018 (UTC)

@Eulenspiegel1: Every number is a set - it is one of many models (opinions). Not everyone agrees with this. --Infovarius (talk) 10:58, 29 October 2018 (UTC)

That's why I use the property "said to be the same" (the item is said to be the same, but the statement is disputed) and not the property "permanent duplicated item". --Eulenspiegel1 (talk) 16:28, 29 October 2018 (UTC)

Please add plwiki page: pl:alef zero

pl:alef zero ("aleph zero") is a redirect to pl:skala alefów ("the scale of alephs"). Please add the former as the plwiki page for this item (aleph null (Q1445491)) and mark it as "intentional sitelink to redirect". --134.191.197.174 21:07, 12 October 2023 (UTC)

Done. Next time, you could use {{Edit request}} so that not only people who happen to watch the item notice it. —Tacsipacsi (talk) 07:23, 13 October 2023 (UTC)