I don't think we can say this. That is one of the implications of incompleteness, as long as we are talking about Godel.
I think we can only say that things are mathematically and logically possible. We can't say anything else, because we don't have access to thought that is not grounded in mathematical and logical possibility.
We can't say there is nothing that is mathematically and logically impossible because we don't even know what that means.
Note that this is a moot point since it just implies that in any discussion any two humans ever have for the rest of time, it will be about stuff that is mathematically and logically possible. And any computers we build will suffer from the same issues. Any two intelligent agents that function according to the laws we have discovered, and have yet to be discovered, will suffer from the same issues.
Tiger Tiger. burning bright,
In the forests of the night;
What immortal hand or eye.
Could frame thy fearful symmetry?
In what distant deeps or skies.
Burnt the fire of thine eyes?
On what wings dare he aspire?
What the hand, dare seize the fire?
And what shoulder, & what art,
Could twist the sinews of thy heart?
And when thy heart began to beat.
What dread hand? & what dread feet?
What the hammer? what the chain,
In what furnace was thy brain?
What the anvil? what dread grasp.
Dare its deadly terrors clasp?
When the stars threw down their spears
And watered heaven with their tears:
Did he smile His work to see?
Did he who made the lamb make thee?
Tiger Tiger burning bright,
In the forests of the night:
What immortal hand or eye,
Dare frame thy fearful symmetry?
The Tiger
William Blake