Problem finding other examples of successor sets. Problem understanding successor sets

elementary-set-theory

I'm reading Halmos Naive set theory (Self-study). I'm at chapter 11 (Set theory).
And I'm very confused by successor sets.

In the book the set of natural numbers w is stated to be the smaller of the successor set. But when I try to think of other successor set I can't seem to find any.

The following website gives the example of $S = \{ 1, \sqrt{2}, 1 + \sqrt{2},…\} $. Here I assume that $0$ can be included in the set S with no consequence.
From here on I am going to include $ 0$ in the set. I assume that this set continues with $ 2, \sqrt{2} + 2…$

But I don't understand that either. The problem that I have with the successor set S are the same problem I have with finding any other successor set than the set natural numbers w:

There are two ways that I"m going on about this:

  • First way; I suppose the following:

$\sqrt{2} + 1 = \sqrt(2)^+ = \{\sqrt{2}\}$

$\sqrt{2} + 2 = (\sqrt(2) + 1)^+ = \{\sqrt{2}, \sqrt{2} + 1\}$

…etc

But then I can only conclude that $\sqrt(2) = \emptyset $, But I think that is absurd because both $ 0 $ and $\sqrt(2) $ are included as separate elements in S

  • Second way; I assume the following:

$ 0 = \emptyset$

$1 = 0^+ = \{0\}$

$\sqrt(2) = 1^+ = \{0, 1\}$

$ 2 = \sqrt(2)^+ = \{0, 1, \sqrt(2)\}$

…etc

But then I would get: S $\cap$ w $ = \{0, 1\}$, because $2_S$ != to $2_w$ neither is $3_S$ and $3_w$…

So the intersection is not a successor set.

So I have to be missing something. But I can't figure out what. Can you explain? Can you give examples of other successor sets? And How is S a successor set? What would be the intersection (I believe it should be w)

Thank you

Best Answer

The site where you found your example defines a "successor set" as

A set $S\subset \mathbb{R}$ is called a successor set iff

  • $1\in S$

  • $\forall n\in S$; $n+1\in S$

Note that the second of these conditions say $n+1$ -- not $n\cup\{n\}$. And it explicitly says $S\subset \mathbb R$, so in this setting the elements of a successor set are numbers rather than abstract sets. You're assumed to know how to add real numbers to each other already.

You then manage to confuse yourself when you try to pretend this is the same definition you have seen elsewhere, which is based on $n\cup\{n\}$ to make a successor.

In a set-theoretic setting, with the $n\cup\{n\}$ definition we can make a different successor set by brute force -- just take $\omega$ plus something that is not already in $\omega$, such as $\{2,5\}$ and add all the successors of the latter:

$$ \begin{align} S = \{\; & 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,\ldots, \\ & \{2,5\}, \{2,5,\{2,5\}\}, \{2,5,\{2,5\},\{2,5,\{2,5\}\}\}, \ldots \} \end{align}$$

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