4 ms·
Retrofitting the set theoretic definitions is trivial. Take the set of cauchy real numbers, remove all reals equal to a real-that-would-be-a-rational, i.e. cons
by voxl 4y ago
Retrofitting the set theoretic definitions is trivial. Take the set of cauchy real numbers, remove all reals equal to a real-that-would-be-a-rational, i.e. constant sequences. Union together this set with the set of rationals defined as ratios of integers. Repeat as needed with smaller sets.
Obviously these reals are isomorphic to the cauchy reals. The reality, of course, is that no one actually works with foundations, they work with the intuitive understanding that 1:N is 1:Z is 1:Q is 1:R.
- klodolph 4y agoIf your real numbers are Cauchy sequences of rational numbers, and your rational numbers are a subset of real numbers, then your rational numbers are a subset of Cauchy sequences of rational numbers (which violates the axiom of foundation). This is not as trivial as it sounds, which is why we invented all these different tools for explaining what “is” or “equals” means in mathematics without resorting to set equality (equality, isomorphism, equivalence, etc).