Does $1 \cup 2$ has any meaning?

Question

Does $1 \cup 2$ has any meaning?

In this answer I used something similar, but it was pointed out $1 \cup 2$ is not same as $\{1,2\}$

Is this only due to convention or is there a deeper reason for it?

Added what about $a \cup b \equiv\{a,b\}$?

Answer 1

Where did you get the notion that $a\cup b$ means $\{a,b\}$? What we do have is $$\{a\}\cup\{b\}=\{a,b\}$$ But that is not the same thing at all.

Answer 2

The union of two numbers is not defined. If you want to express $\{1,2\}$ you can write down $\{1\} \cup \{2\}$ where those are singlets. In fact the union is between sets, not between numbers.

Answer 3

Since $1=\{\emptyset\}$ and $2=\{\emptyset,\{\emptyset\}\}$ by definition, yes.

Answer 4

Just to be the devil's advocate, you can give a meaning to $1\cup 2$ by considering the construction of $\mathbb{N}$ in ZFC. That is, $$0 = \emptyset$$ $$1 = \{\emptyset\}$$ $$2 = \{\{\emptyset\},\emptyset\}$$ Then $$1\cup 2 = 2\ .$$ But unless you specify what you are doing very precisely, I would advise against writing things like $1\cup 2$.

Answer 5

Set theoretically it has meaning and it is defined, remember that the reals (in one of its representations) are Dedekind cuts, so joining two numbers has meaning. Also you can think 1 and 2 as ordinals, so by computing the union you have $1\cup2=2$.

And working in ZFC, $a\cup b=\{a,b\}$ violates foundation.