Did I construct an infinite set equal to $\{1\}$?

Question

Okay, I'm trying to understand the argument that NJ Wildberger gives in the following video: https://www.youtube.com/watch?v=5CiiGdaYEPU

He tries to explain why he things infinite sets don't make sense. And I've written his argument in my own words to better understand what he is saying.

So to my suprise his argument seems totally valid to me. Don't understand me wrong, I'm humble and young enough to understand that probably I just didn't spot the weakness in his reasoning. So I've outlined my little bit more mathematical version of his argument below, and I invite people to find the mistake in my reasoning where I try to deduce that the set $\{1\}$ must have infinite many elements.

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Consider the the natural numbers $\Bbb{N}=\{1,2,3,4,...\}$. We are going to construct three new sets $A,B,C$ using those natural numbers.

The construction goes as follows. Begin with $A_{1}:=\Bbb{N}$ and $B_{1}:=C_{1}=:\varnothing $.

Then given $A_{k},B_{k},C_{k}$ for some $k\in \Bbb{N}$ we construct $A_{k+1},B_{k+1},C_{k+1}$ the following way:

Consider $A_{k}$.

• Suppose $k+1\in A_{k}$. Set $a_{k}:=k+1$ and take 3 other nonequal natural numbers $\{b_{k},c_{k},d_{k}\}\in A_{k}$ and construct the following sets using those 4 numbers:

  1. $A_{k+1}:=A_{k} \setminus \{a_{k},b_{k},c_{k},d_{k}\}$
  2. $B_{k+1}:=B_{k} \cup \{b_{k},c_{k},d_{k}\}$
  3. $C_{k+1}:=C_{k} \cup \{a_{k}\}$

• Suppose $k+1\not \in A_{k}$. Then by construction $k+1$ must be in $B_{k}$ (*). Take 4 nonequal natural numbers $\{a_{k},b_{k},c_{k},d_{k}\}\in A_{k}$ and construct the following sets using those 4 numbers: :

  1. $A_{k+1}:=A_{k} \setminus \{a_{k},b_{k},c_{k},d_{k}\}$
  2. $B_{k+1}:=(B_{k} \setminus {k+1}) \cup \{a_{k},b_{k},c_{k},d_{k}\}$
  3. $C_{k+1}:=C_{k} \cup \{k+1\}$

So the first sets of these constructions are:

1. $A_1 = \Bbb N, B_1 = \varnothing, C_1 = \varnothing$
2. $A_2 = \Bbb N \setminus \{2,b_1,c_1,d_1 \}, B_2= \{b_1,c_1,d_1\}, C_2 = \{2\} $

From here on it becomes a little bit a mess, but the idea is that you make sure that $$A_k \cup B_k \cup C_k = \Bbb N$$ for any $k\in \Bbb N$ and that you slowly construct all natural numbers in $C_k$ except $\{1\}$.

So now the question is, what are the following sets:

$$A:=\lim_{k\rightarrow \infty }A_{k} \quad B:= \lim_{k \rightarrow \infty }B_{k} \quad C:=\lim_{k\rightarrow \infty }C_{k}$$

It seems to me that $B$ is an infinite set as $B_{k+1}$ contains 3 more elements than $B_{k}$ does for any $k\in \Bbb N$.

On the other hand, and this is the confusing part, it also seems that except for the number $1$ any $k\in \Bbb{N}$ can't be in $B$. The set $B_{k}$ can't contain the number $k$ by construction and sets $B_{n}$ where $n>k$ are writable as an union of $B_{k}$ together with subsets of $A_{k+1},...,A_{n}$ which don't contain the number $k$. So $B:= \lim_{n \rightarrow \infty }B_{n}$ can't contain the number $k$. But we chose $k\in \Bbb{N}\setminus \{1\}$ arbitrarily, so except for the number $1$ any $k\in \Bbb{N}$ can't be in $B$.

And by construction $B\subseteq \Bbb{N}$. So we get that $B$ must be equal to $\{1\}$.

(*), the reason that $k+1$ must be in $B_{k}$ in that case, follows from the construction. Surely $A_{k}\cup B_{k}\cup C_{k}= \Bbb{N}$ for each $k\in \Bbb{N}$. And $k+1\not \in C_{k}=\{2,3,4,...,k\}$. So therefore $k+1$ must be in $B_{k}$.

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Okay, I understand now that my $B$ is not well defined, but I'm still not really satisfied with this. I think I do have that:

$$\bigcup_{n\in\mathbb{N}}\,\,\bigcap_{k=n}^{\infty} A_k = \varnothing =\bigcap_{n\in\mathbb{N}} \,\, \bigcup_{k=n}^{\infty} A_k $$

$$\bigcup_{n\in\mathbb{N}}\,\,\bigcap_{k=n}^{\infty} C_k = \Bbb N - \{1\} =\bigcap_{n\in\mathbb{N}} \,\, \bigcup_{k=n}^{\infty} C_k $$

And therefore $A:=\lim_{k\rightarrow \infty }A_{k} = \varnothing $ and $ C:=\lim_{k\rightarrow \infty }C_{k} = \Bbb N - \{1\} $.

But on the other side I have that $A_k,B_k,C_k$ form a partition for $\Bbb N$ for any $k\in \Bbb N$ and $\lim_{k \rightarrow \infty} |B_k|=\infty$.

If I look at all those facts at the same time, I do feel like conlcuding that Wildberger is a little bit right about that infinite sets doesn't really make sense. At least it is weirder than I even thought it was.

Answer 1

The definition $$B := \lim_{k \rightarrow \infty} B_k$$ is the problem. A limit of a sequence of sets is not well defined in general. You can define the limit superior and limit inferior of a sequence of sets:

$$\liminf_{n\rightarrow\infty} B_n={\bigcup_{n=1}^\infty}\left({\bigcap_{m=n}^\infty}B_m\right)$$ $$\limsup_{n\rightarrow\infty} B_n={\bigcap_{n=1}^\infty}\left({\bigcup_{m=n}^\infty}B_m\right)$$

But even for those, it is not true that if the $B_k$ have increasing cardinality, the limit has infinite cardinality.