Which set is considered to be well defined in set theory?

Question

I just started learning set theory and in my book a lot of questions start with "Let A be the set defined as ...", but it in most cases turns out that the set $A$ (for which they claim is "defined") is actually not defined, but just satisfies some statements.

I am looking for a set of rules which will help me determine if a given statement is "completelly defining" a set or just "describing" it. Here are some examples to help me better explain what I mean.

For example, statement $A=\{\}$ completelly defines set $A$. There is exactly one set $A$ for which it is true (and obviously $A$ is empty set). However, statement $A=A\cup\left\{\{\}\right\}$ just describes set $A$. It tells us that $A$ contains empty set, but it doesn't tell us anything more about set $A$. We cannot, for example, determine if $\left\{\left\{\{\}\right\}\right\}\in A$ or not.

However, there are also the third type of sets which are contradictory to itself. For example $A\in A$ tells us that there is no set $A$ which satisfies that statement.

So, I'm looking for a way to determine if a given statement which includes some set $A$ is defining it, describing it or making it inconsistent. It may look obvious, but there are some sets I really cannot figure out are they well-defined or not. For example, lets see some test suites:

$$A=\left\{a\mid a=\{\}\lor\exists b\in A\left(a=\{b\}\right)\right\}$$

Is the above set well-defined? I mean, is there exactly one set $A$ which satisfies the above statement? I think it is well-defined, but I cannot prove it. Also, lets see another example I came up with:

$$A=\{a\mid a\notin A\}\cap\left\{\right\}$$

Is the above set well defined? I really have no clue what to think in this situation. Our book is focusin on pretty pragmatical examples and doesn't cover these edge cases. But, I'm just curious how to determine if some set is well-defined or not.

I know we cannot determine for every set is it well-defined because of Godel theorem. But, I am pretty sure these two examples can be proved to be either well-defined (exactly one set $A$ satisfies it), just described (more than one set $A$ satisfies it) or inconsistent (there is no such set $A$).

So, my questions has two parts:

1. How to generally determine if a statement is well-defining a set?
2. Does the first example I posted well-define set $A$? Does the second example well-define it?

Thank you in advance.

Edit

To avoid misunderstanding, I'm using ZFC set of axioms.

Answer 1

In practical usage-- that is, actual published mathematics papers and textbooks -- definitions take a wide variety of shapes. I think the closest one can come to pinpointing what we accept as a "definition" is to say something like:

A definition is a particular kind of description.

More precisely, a description becomes a definition when you can prove (from whichever axioms and principles you prove things) that there is exactly one thing in your universe of discourse that satisfies that description.

The practical difficulty with this is in determining when to say you "can prove" that the definition defines something. In most practical cases this proof is tacitly left to the reader, and the author will be somewhat careful only to use definitions of a simple enough form that he expects readers to be able to see for themselves that they work.

Note well, however, that this is not really a technical distinction, but a matter of social expectations between the writer and the readers -- exactly the same question of "which proof steps can I leave implicit?" turns up when we're talking about proofs of other things than "this is a good definition".

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As for your concrete examples:

$$A=\left\{a\mid a=\{\}\lor\exists b\in A\left(a=\{b\}\right)\right\}$$

Whether this can be proved to pinpoint a particular $A$ depends, as far as I can figure out offhand, on whether you're working in a set theory that has the Axiom of Regularity (or something similar). The proof is in any case not trivial, and I would expect of an author who wants to use this property to define $A$ that he gives some kind of explicit argument that there is a unique $A$ that meets this description.

$$A=\{a\mid a\notin A\}\cap\left\{\right\}$$

It is quite easy to see that this property is satisfied by exactly one $A$, namely the empty set. My main objection to actually seeing this as a definition would be that it is a silly and superfluous way to define the empty set. Therefore it is bad writing to use it, but I wouldn't say it is meaningless -- and certainly not dangerous in the sense that accepting that this defines $A$ could lead to concluding a falsehood later on.

Answer 2

If a math book, without further context, contains the statement

$\quad$Let the set $A$ be defined by $A=A\cup\left\{\{\}\right\}\;$...

you would be upset, since the following makes more sense,

$\quad$Let $A$ be any set with the property $A=A\cup\left\{\{\}\right\}\;$...

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$\tag 1 A=\left\{a\mid a=\{\}\lor\exists b\in A\left(a=\{b\}\right)\right\}$

The expression in (1) is recursive. You can make sense of it if you want to express/define a 'minimal construct' set. You should not be exposed to (1) in a math book unless there has been some preparatory exposition.

So, the best idea is to take (1) as defining the countably infinite set

$\{ \{\},\{\{\}\},\{\{\{\}\}\},...,\{\{\{\dots\}\}\},... \}$

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$\tag 2 A=\{a\mid a\notin A\}\cap\left\{\right\}$

If you see $a\notin A$ without additional context you should be suspicious - seems like too much stuff to make sense of. So the expression (2) makes no sense, and it is pointless to claim that it defines the empty set.

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Although you may have a formal introduction to ZFC, when approaching problems first use 'circumspect set theory', and see if it provides intuition or useful logic/foundation/principles.