Some questions on Godel's theorem

I am new here and, although I have considerable knowledge in general science, I am definitely not an expert at maths.

Anyone:

Correct me if I am wrong but; two of the implications of the proven Godel's theorem is that:

1, there must be some mathematical theorems that can be made that are unprovable and yet are nevertheless true.

2, we cannot ever know exactly which mathematical theorems are the ones that are both unprovable and yet are nevertheless true.

But my main question which is my thirds question is this;

3, Is there is an infinite number of possible mathematical theorems that are **BOTH** true **AND** **un**provable **AND** which can **NOT** be *proven* to be **un**provable?

4, IF the answer to question 3, is 'yes', can this be deduced from Godel's theorem *alone* or is understanding of some extra deduction required to deduce that the answer to 3, is 'yes'?

Re: Some questions on Godel's therum

I don't like the wording. How do we KNOW if a statement is true if we cannot prove it? I would prefer to say, instead, given a system of axioms, large enough to encompass the integers, that there exist some statements which can neither be proven nor disproven ("disproven" meaning that the negation can be proved). Yes, it follows that there must be an infinite number of statements that can neither be proven nor disproven. That's the case because, if there were only a finite number of such statements, we could add those statements themselves to the axioms, getting a new system of axioms in which all statements can be either proven or disproven, contradicting the original statement.

Re: Some questions on Godel's therum

yes your are right; my wording was quite wrong! Sorry for such a muddled post; I didn't mean to imply that we can know if a statement is true if we cannot prove it!

I think I understand what you say and, I note in particular, the “there must be an infinite number of statements that can neither be proven nor disproven” part which is one thing I really wanted to know -thanks for that!

Now, for what I want to know most:

Is there an INFINITE number of possible mathematical theorems that can neither be proved nor disproved that ALSO have the characteristic that there is no possible proof that can be given that it can NOT be either proved nor disproved (so we can never KNOW that it can**not** ever be proved or disproved)?