Those are theories, which are not generally lost...
I really wish I had the time to explicitly write out the reasons why I believe these examples are compelling reasons to use the usual model of the real numbers. I tried, but I’ve already spent too long and I doubt they would convince you anyway.
People can’t pick out specific examples of numbers that are lost by switching to using nameable numbers,
So? Omega could obliterate 99% of the particles in the known universe, and I wouldn’t be able to name a particular one. If it turns out in the future that these nameable numbers have nice theoretic properties, sure. The effort to rebuild the usual theory doesn’t seem to be worth the benefit of getting rid of uncountability. (Or more precisely, one source of uncountability.)
I think I’ve spent enough time procrastinating on this topic. I don’t see it going anywhere productive.
Suppose someone played idly manipulating small objects, like bottlecaps, in the real world. Suppose they formed (inductively) some hypotheses, picked some axioms and a formal system, derived the observed truths as a consequence of the axioms, and went on to derive some predictions regarding particular long or unusual manipulations of bottlecaps.
If the proofs are correct, the conclusions are true regardless of the outcome of experiments. If you believe that mathematics is self-hosting; interesting and relevant and valuable in itself, that may be sufficient for you. However, you might alternatively take the position that contradictions with experiment would render the previous axioms, theorems and proofs less interesting because they are less relevant.
Generic real numbers, because of their infinite information content, are not good models of physical things (positions, distances, velocities, energies) that a casual consumer of mathematics might think they’re natural models of. If you built the real numbers from first-order ZFC axioms, then they do have (via Lowenheim-Skolem) some finite-information-content correspondences—however, those objects look like abstract syntax trees, ramified with details that act as obstacles to finding an analogous structure in the real world.
I really wish I had the time to explicitly write out the reasons why I believe these examples are compelling reasons to use the usual model of the real numbers. I tried, but I’ve already spent too long and I doubt they would convince you anyway.
So? Omega could obliterate 99% of the particles in the known universe, and I wouldn’t be able to name a particular one. If it turns out in the future that these nameable numbers have nice theoretic properties, sure. The effort to rebuild the usual theory doesn’t seem to be worth the benefit of getting rid of uncountability. (Or more precisely, one source of uncountability.)
I think I’ve spent enough time procrastinating on this topic. I don’t see it going anywhere productive.
Suppose someone played idly manipulating small objects, like bottlecaps, in the real world. Suppose they formed (inductively) some hypotheses, picked some axioms and a formal system, derived the observed truths as a consequence of the axioms, and went on to derive some predictions regarding particular long or unusual manipulations of bottlecaps.
If the proofs are correct, the conclusions are true regardless of the outcome of experiments. If you believe that mathematics is self-hosting; interesting and relevant and valuable in itself, that may be sufficient for you. However, you might alternatively take the position that contradictions with experiment would render the previous axioms, theorems and proofs less interesting because they are less relevant.
Generic real numbers, because of their infinite information content, are not good models of physical things (positions, distances, velocities, energies) that a casual consumer of mathematics might think they’re natural models of. If you built the real numbers from first-order ZFC axioms, then they do have (via Lowenheim-Skolem) some finite-information-content correspondences—however, those objects look like abstract syntax trees, ramified with details that act as obstacles to finding an analogous structure in the real world.