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Double-Vision Crossword-Wise

With great creativity, Jim Henle, Smith College, created this "pretty small" crossword puzzle. The odd thing is that it has two solutions...entirely different!

Can you find both solutions?

ACROSS
3. Philosophical logician with contributions to proof theory and modal logic.
4. Mathematical logician who studied the natural numbers, and also the existence of solutions to certain equations.
5. Mathematical logician who proved half the independence of the continuum hypothesis.

DOWN
1. Mathematical logician, contributed to the foundations of infinity; middle-European, but the name sounds sort of Italian.
2. Mathematical logician who also wrote children's books.

Answer: Combined answers in alphabetical order: Belnap, Bolzano, Boolos, Carroll, Cohen, Davis, Dodgson, Godel, Peano, Zermelo

Source: Mathematics Magazine, February 2011, p. 115