> Machines do not have arbitrary-precision integers.
Machines don't have concepts like "integers" at all. They have concrete functions that produce defined outputs with defined inputs, which can be interpreted as fixed size integers, components of arbitrary precision integers, or specifications of glyphs, or numerous other things. But none of those interpretations are inherent in the machine.
I suppose we can get philosophical and say that there is no such thing as a machine word, that when you operate on the memory using different instructions, or jump into the memory, or whatever, this "machine word" becomes something else... But IMO this is a pointless exercise. The fixed-size machine word is a reality in the machines we're talking about. (How that gets translated into a higher level language like C will vary from machine to machine, compiler to compiler, but the analogy still stands and I do not believe it is meaningful to argue such points unless engaged in purposeful misinterpretation.)
Machines don't have concepts like "integers" at all. They have concrete functions that produce defined outputs with defined inputs, which can be interpreted as fixed size integers, components of arbitrary precision integers, or specifications of glyphs, or numerous other things. But none of those interpretations are inherent in the machine.