Doubts on the axiom of union?

Question

I'm reading Guerino Mazzolla's Comprehensive Mathematics for Computer Scientists 1:

Axiom 3 (Axiom of Union) If $a$ is a set, then there is a set

$$\{x| \text{ there exists an element } b \in a \text{ such that } x\in b \}$$

This set is denoted by $\cup a$ and is called the union of $a$.

• This axiom seems to take elements from inside of a set of sets, for example in the set: $a=\{\{u_1,u_2 \},\{u_3,u_4 \} \}$, there are two elements $b\in a$, then the mentioned set created by the axiom is $\{u_1,u_2,u_3,u_4 \}$.

The axiom seems to imply that the set $a$ must have a set inside of it for it to work, what if it have an element that is not a set? (I'm not sure if this is possible, but I remember of having read something about the difference of sets and elements).

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• There's also an extension of my proposal, In the same book (and also in others) I've seen that the natural numbers are defined as:

$$\begin{eqnarray*} {0}&=&{\emptyset} \\ {1}&=&{\{\emptyset\}} \\ {2}&=&{\{ \{\emptyset\}\}} \\ {3}&=&{\{ \{ \{\emptyset\}\}\}} \\ {\vdots}&=&{\vdots} \end{eqnarray*}$$

Then the singleton $\{ 3\}$ could be seen as $\{ \{ \{ \{\emptyset\}\}\}\}$, which could mean that

$$\cup \{ 3 \}=\cup\{ \{ \{ \{\emptyset\}\}\}\}= \{ \{ \{\emptyset\}\}\}=\{2 \}$$

Then the union could also be seen as the antecessor of a number?

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• Using the framework I provided, I see some ambiguity, look:

$$\{ \{ \{\emptyset\}\}\}=\{2 \}$$

I decided to interpretate it this way, but from the notation it's not clear in which pair of brackets I should stop, I guess that $\{ \{ \{\emptyset\}\}\}=\{2 \}$ but it could also be the case that $\{ \{ \{\emptyset\}\}\}=3$. I could want a singleton with a $2$ inside of it and a $3$ outside a set, no?

Answer 1

For your first point, you talk of a set with elements that are not sets. That simply does not exist, while in set theory, everything is a set.

I don't really see anything wrong or doubbt-casting in your second point. Yes, $\bigcup 4 = \{2\}$, even more, for any $n$ defined this way, $\bigcup n = n-1$, so yes, it is the antecessor.

For your third point, it is of course true that $\{2\} = 3$, but why would that be a contradiction? There is no ambiguity because if you say $\{2\}$, there is exactly one set you can describe in this way. Ambiguity does not arise from multiple descriptions of the same thing (for example, the phrase "Paris, France" is not ambiguous just because it means the same thing as "the capital of France"). Ambiguity comes from one description meaning more than one thing (the phrase "Paris" is ambiguous because there are multiple things named Paris), which does not happen here.