Applicability of Maths to the Universe

[polymath257]

Of course, the cardinality of the set of prime numbers is a countably infinite set.  Thousands of mathematical proofs exist that prove such; of course, an acceptance of ZFC and the Axiom of Infinity is necessary to get the balling rolling, so to speak.  Professor Wes Morriston has written extensively in response to WLC, and even quotes Craig who once stated that his ideas will make sense to an individual at least until that person has taken a course in elementary number theory, complex analysis, etc.

I keep hearing about this ZFC stuff but never getting around to reading up on it. ELI5 (or perhaps ELI15): What's it about?

It's Zormelo-Frankl set theory with the axiom of Choice. This is the current standard set of axioms for set theory and is the basis of almost all modern mathematics. it came about because of inconsistencies in 'naive set theory' that were discovered at the end of the 1800's.

The basic ZF axioms say that there is an empty set, that you make a set that is the set containing any two objects, that the set of subsets of any given set exists, that you can take the union of any collection of sets and get a set, etc. The only undefined concept is that of set membership.

The the axiom in dispute here is the axiom of infinity: essentially that there is an infinite set. This is used in an essential way to define the set of natural numbers and, later on, the set of real numbers.

The axiom of choice is one many people have heard of. It essentially says that if you have a collection of non-empty sets, then there is another set that has one element in common with each of the original sets.

The main difference between ZFC set theory and Cantor's original is that you cannot define arbitrary collections as sets. So, for example, the collection of all sets is not a set. Properly speaking, it simply does not exist at all in ZFC. In  sense, it is 'too big' to be a set. This avoids the paradoxes of Cantor's original system while keeping the material on one-to-one correspondences and cardinality.

Pretty much all of modern mathematics can be constructed inside of the ZFC axiom system. It is a rather lengthy process to get to the set of real numbers simply starting with the notion of set membership, but it is possible. And, from there, it is possible to construct all of calculus, differential equations, etc as well as group theory, graph theory, etc.

The one place where ZFC fails to be helpful is when discussing certain aspects of category theory and hence, of algebraic topology. The standard way of doing this is to allow for 'classes' which are 'too big to be sets'. But it is then impossible to talk about the collection of 'all classes'.

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I keep hearing about this ZFC stuff but never getting around to reading up on it. ELI5 (or perhaps ELI15): What's it about?

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Read Dr. James A. Lindsay's book, Dot, Dot, Dot...Infinity Plus God Equals Folly, which is available on Audible, also (read by Dr. Lindsay himself).  He addresses Craig's argument.  That the number "2" (or, its square root) exists in some Platonic realm is silly; does the transcendental number "pi" exist there, also?  But, read (or listen) to Lindsay's book for more details.

There is an even deeper level. Alan Turing defined the concept of a 'computable real number'. In essence, it is one for which some computer program can compute the digits in the decimal expansion of that number.

So, for example, pi *is* a computable number: we have programs that will give as many digits of pi as you want. SImilarly for e and sqrt(2), and pretty much any number you have ever heard of.

But what Turing discovered (and it is not difficult to prove) is that there are real numbers (in the usual axiom system) that are not computable: no computer program can compute their decimal expansions. And, in fact, 'most' real numbers are not computable.

In what sense do these uncomputable real numbers exist in some Platonic realm?