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Except the number you get this way is not between 3 and 4. It's essentially infinite with no functions in the model that separate it from the finite numbers (an
by shasta 15y ago
Except the number you get this way is not between 3 and 4. It's essentially infinite with no functions in the model that separate it from the finite numbers (and this can be proven in PA: if a number isn't 0..N, then it's greater than N).
Generally, I don't think it's fair to characterize this story as something studied by mathematicians. Really, it's an exercise in reasoning about nonsense.
- cperciva 15y agoDepends what your definition of < is. Some definitions leave unspecified what the relationship between two values not in {0, 1, 2, ...} is.
- shasta 15y agoInequality can be defined in Peano arithmetic in a total way. Maybe you're talking about defining some relation on the model? But what justifies calling such a relation an inequality?