What is the solution to this puzzle?

How about this: dfnul2 - Metamath Proof Explorer

Does the null object (the x:~(x=x)) equal itself?

I dont see the problem with it equaling itself, as it has no elements.

therefore, ~((the x:~(x=x))=(the x:~(x=x))) is a logical necessity.

If there is such an x, but there isn't such an x because the empty set is the set of all such x. I still dont see the problem, if you've proved that all x not equal to themselves aren't equal to themselves, there's no conflict with the empty set equaling itself. Is there?

I did prove that the described individual object (the x:~(x=x)), does not equal itself.

This sentence is false.

That which is not equal to itself is equal to itself, or, it is false that (that which is not equal to itself is equal to itself)..is a tautology of the form (p v ~p).

This sentence cannot be false.

"This sentence is false." Doesn't that seem like a self-referential paradox?

6.1271 It is clear that the number of the 'primitive propositions
of logic' is arbitrary, since one could derive logic from a single
primitive proposition, e.g. by simply constructing the logical product
of Frege's primitive propositions. (Frege would perhaps say that
we should then no longer have an immediately self-evident primitive
proposition. But it is remarkable that a thinker as rigorous as Frege
appealed to the degree of self-evidence as the criterion of a logical
proposition.)


6.13 Logic is not a body of doctrine, but a mirror-image of the world.
Logic is transcendental.


6.2 Mathematics is a logical method. The propositions of mathematics are
equations, and therefore pseudo-propositions.


6.21 A proposition of mathematics does not express a thought.


6.211 Indeed in real life a mathematical proposition is never what
we want. Rather, we make use of mathematical propositions only in
inferences from propositions that do not belong to mathematics to others
that likewise do not belong to mathematics. (In philosophy the question,
'What do we actually use this word or this proposition for?' repeatedly
leads to valuable insights.)


6.22 The logic of the world, which is shown in tautologies by the
propositions of logic, is shown in equations by mathematics.


6.23 If two expressions are combined by means of the sign of equality,
that means that they can be substituted for one another. But it must be
manifest in the two expressions themselves whether this is the case
or not. When two expressions can be substituted for one another, that
characterizes their logical form.

There is only one number, it seems. This is implicit in Wittgenstein's quote here, and goes all the way back to Parmenides & Pythagoras.

But negation is also necessary, and yet negation is not a number. Or is it?

What the hell are you talking about?
Where in Wittgenstein do you believe he said or implied "there is only one number"???

Negation is a number??? Are you serious, surely nobody could make this stupid claim.