3 ms·

Actually the proofs depend on philosophical assumptions that both can and have been questioned. The conclusions likewise. If you go down that path, then "unc

• analog31 5y ago
  Does that imply that the reals are a subset of a countable list?
  • btilly 5y ago
    Well, not necessarily. One of the standard constructions of the reals is via sequences of rationals. A sequence of rationals that looks like it should be converging is called a Cauchy sequence. (I'm avoiding the technical definitions, but they are easy to find.) Two Cauchy sequences that look like they are converging to the same thing, are equivalent. And reals are defined as equivalence classes of Cauchy sequences. This translates pretty well to a constructive approach. For example we can build our mathematics out of things expressible in a programming language. We can define a Cauchy sequence as a function that can be proven by our favorite axiom system to produce a sequence of rationals converging at a specified rate. Again, two functions are equivalent if they can be proven to produce sequences converging to the same thing. We can certainly enumerate all possible programs. But we cannot, thanks to the Halting problem, write a program that is able to select out which possible programs represent reals. Nor can we reliably identify which pairs of programs are equivalent. So in this construction there is really no actual set of reals that can be identified. Nor can we tell whether a real has been listed already. But there is a countable list that has all possible things that might possibly represent a real. Which will include each real many times. Does that clarify what I meant by "self-referential tangle"?
    • wtallis 5y ago
      > But there is a countable list that has all possible things that might possibly represent a real. Haven't you only demonstrated a countable list of all possible things that might possibly represent a computable number, or some similar concept?
      • btilly 5y ago
        No, but you may have to put aside some preconceptions to get the point. https://en.wikipedia.org/wiki/Brouwer%E2%80%93Hilbert_controversy https://en.wikipedia.org/wiki/Brouwer%E2%80%93Hilbert_contro... is a good starting point on the historic debate over the philosophy of math. From within Formalism, which for all intents and purposes won, your characterization is correct. We have the standard reals. And we've constructed a proper subset of the reals. From within Constructivism, the "standard reals" is a piece of sophistry. It is ridiculous to claim the existence inconceivable infinite swarms of non-existent things whose only claim to "existence" is the sheer multitude of numbers that can never be named or constructed. And therefore the "reals" that I described are a sensible thing to call reals, all of whom have an existence that can be established on reasonable grounds. From within either philosophy, the other doesn't make much sense. But, in fact, both philosophies are internally consistent, and no logical argument can ever establish one over the other. (In fact, Formalism won because it is more convenient. And for no other reason.)
        • wtallis 5y ago
          > you may have to put aside some preconceptions to get the point. "Preconceptions" is a rather unhelpful way to describe the actual situation here, which is you using a different definition for real numbers, that is not even remotely equivalent, without making that clear from the outset. You said: > But there is a countable list that DOES include every possible real - you just can't always figure out whether things on that list are reals! Which is pretty clearly incorrect using the usual definition of real numbers. The fact that you don't like how real numbers is usually defined is not sufficient justification for you to start confusing a discussion by mixing in your alternative definition without making an explicit distinction. You've defined a set that can certainly be the subject of interesting investigation, even in a context where the usual definitions about real numbers and uncountable infinities are still accepted. Choosing to inject a naming conflict is counterproductive and suggests you're more concerned with making smug claims about being able to do things mainstream mathematicians consider impossible, rather than having a productive discussion about how to construct most of familiar mathematics without allowing uncountable infinities.
• a1369209993 5y ago
  > But there is a countable list that DOES include every possible real - you just can't always figure out whether things on that list are reals! A (hopefully) helpful way to look at it: Cantor's diagonalization argument doesn't work if every function from natural numbers to reals, that includes all reals, is partial. Which, per the halting problem[0], is exactly the behaviour you get if your correspondance function involves doing arbitrary computations decoded from the input natural. 0: The class of propositions "Turing machine #N halts.", in addition to true and false propositions, also contains infinitely many propositions that are neither true nor false[1], so compacting out the non-halting naturals doesn't help. 1: aka infinitely many counterexamples to the axiom of excluded middle