Was being deliberately inaccurate here. Hard does mean more than a limited multiplier. Sudden means that there’s an appreciable change over an infinitesimal variation aka discontinuous.
Lookup overflow, underflow, and “principle of permanence” in Goldblatt for why I’d do that. Also called overspill and underspill. The basic idea is “as above, so below” except this link is 2 way. Say some internal function has all infinitesimals in its range. Then it must have non infinitesimals too, since the set of all infinitesimals is known to be external, and images of internal functions over internal sets are internal. This is an example of overspill. Infinitesimal behavior has spilled over into the appreciable domain.
Was being deliberately inaccurate here. Hard does mean more than a limited multiplier. Sudden means that there’s an appreciable change over an infinitesimal variation aka discontinuous.
Lookup overflow, underflow, and “principle of permanence” in Goldblatt for why I’d do that. Also called overspill and underspill. The basic idea is “as above, so below” except this link is 2 way. Say some internal function has all infinitesimals in its range. Then it must have non infinitesimals too, since the set of all infinitesimals is known to be external, and images of internal functions over internal sets are internal. This is an example of overspill. Infinitesimal behavior has spilled over into the appreciable domain.
“as above, so below” tends to be two way, atleast outside of mathematics.