Which of the following statement holds?

Question

Let $A$ be a set. What does it meant for $A$ to be uncountable.

1. There is no way to assign a distinct element of $A$ to each natural number.

2. There exist elements of $A$ which cannot be assigned to any natural number at all.

3. There is no way to assign a distinct natural number to each element of $A$.

4. There is bijection between $A$ and real numbers $\mathbb{R}$.

I am totally confused in $3$ and $4$.

Answer 1

You have to see what is the meaning of each statement:

1 means that there is no surjection from $A$ to $\mathbb{N}$;

2 means that there is an injection from $A$ to $\mathbb{N}$;

3 means that there is no injection from $\mathbb{N}$ to A;

4 means that there is a surjection from $\mathbb{R}$ to A.

Now, 1 and 2 would mean the cardinality of $A$ is smaller or equal to the cardinality of $\mathbb{N}$, and that would mean $A$ is countable;

And as said in the comments, 4 would mean the power set of $\mathbb{R}$ would not be uncountable, which is equivalent to say that the power set of $\mathbb{R}$ would be countable, which is a nonsense.

So,the correct answer is 3.