(June 16, 2020 at 9:13 am)The Grand Nudger Wrote: You never got an answer to how all of this fits in a platonic sense. It doesn't. Under platonism, it is axiomatically true that all numbers are computable, even if we're incapable of demonstrating as much....ever. If it seems to us as though there are uncomputable numbers, this could only be an error. The flaw being in the computer.
Well, there is a standard definition of what it means to be computable: that some Turing machine can be found that produces the required sequence of digits. It turns out that there is nothing special about 'decimal' in this: if you can find a Turing machine that gives the digits for some other base, you can write a Turing machine for decimal digits and vice versa.
Once you have that definition (equivalent to the Church-Turing postulate), the collection of computable real numbers is countable (because there are only countably many Turing machines). Since the collection of real numbers is NOT countable, MOST real numbers are not computable.
Even those mathematicians that are Platonists agree that most real numbers are not computable. What Platonists would say is that these real numbers still exist in some Platonic realm.
A much more concerning issue for Platonism, in my mind, is the Continuum Hypothesis. This is a question about cardinality of subsets of the real line.
Once again, a countable set is one that can be put into one-to-one correspondence with the set of natural numbers. Countable subsets of the real line include the set of integers, the set of rational numbers, and the set of algebraic numbers.
On the other hand, we know that the set of real numbers is NOT countable. So, we have another cardinality: we say the cardinality of a set is that of the continuum if it can be put into one-to-one correspondence with the set of real numbers. So, the closed interval [0,1] has the cardinality of the continuum. So does the Cantor ternary set, any graph of any function on the real numbers, the collection of complex numbers, and many other interesting sets. ALL of these are much larger than countable sets.
So, the question: is there a subset of the set of real numbers that is not countable and also not the cardinality of the continuum?
So, if we think of the natural numbers as having a 'small' infinite cardinality, and the real numbers as having a 'large' infinite cardinality, we ask whether there is an *intermediate* infinite cardinality.
A Platonist would say that this question has a definite answer.
But, we *know* that we can get *equally consistent* versions of mathematics by assuming either that there are no intermediates, OR alternatively, assuming that there are such intermediates.
We can construct two models of set theory: one answers the question yes, the other answers the question no.
So, whither Platonism?
