> I wonder if anyone has tried work with a theory of finite numbers: where
> BIGGEST+1=BIGGEST or BIGGEST+1=-BIGGEST as in some computers?

There is a group of faculty who address this problem directly in my 
department.  But any general-purpose computer can emulate true, unlimited 
natural numbers (which is what people often do, rather than relying on 
bounded ints).  The only real limitations that make "computer" not-equal-to 
"Turing machine" are memory and the limited patience of humans.  This is one 
reason why people spend more time researching P vs. NP than 
artificially-imposed limits.

When you add bounds to numbers it requires additional proof obligations, 
which makes it more difficult to prove things.  And you can't directly prove 
anything about numbers that exist outside the bounds under which you're 


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