#203 God and Gödel
March 07, 2011Hi Dr Craig,
I'm a university mathematics lecturer, and have always been interested in the idea of God being infinite in some way. I've noticed you have explored the idea of infinity, and its implications, in many of your debates. (It is mainly from your debates that I am familiar with your work by the way.)
I have just read your answer to Question 197 (Does God know an actually infinite number of things?), and might take a little more time to digest the ideas. But it also stirred up in me a question I have long thought about.
In mathematics, there are famous theorems stating that not all mathematical truths can be known - I'm sure you are familiar with Gödel's Incompleteness Theorems. But what's more surprising is that it's actually possible to give particular examples of unknowable truths - for example the Continuum Hypothesis (which, interestingly enough, concerns infinite sets). When I say "give examples", of course I mean it is possible to write down a statement, and its negation, and know that one of them is true, but also know it is impossible to say which one is.
I guess my question, in its simplest form, is: "Does God know the answer to these questions?"
Note that, for example, the Continuum Hypothesis is either "true" or "false". Either answer is consistent with the axioms of set theory, but there IS an answer!
I would very much appreciate hearing your thoughts on this one.
Cheers from Sydney Australia,
James
Australia
Dr. craig’s response
A
Wonderful to hear from a professional mathematician, James! This week I’ve been reading about non-standard models of arithmetic. It’s so interesting and strange! The Peano Axioms for arithmetic are incomplete in the way you describe; there are arithmetic truths that can’t be decided in a first-order logical language on the basis of the axioms alone.
As I understand the incompleteness theorems, however, what they show is not the unknowability of certain mathematical truths but rather their underivability from the axioms of the relevant theory. In some cases we do know that the statements are true, even though they can’t be proved on the basis of the axioms. In other cases, like the Continuum Hypothesis you mention, we may not know whether the statement is true or false, but that doesn’t preclude its having a truth value. If it does, then it will be known to God, since God, as an omniscient being, has the essential property of knowing only and all true propositions. He doesn’t need to derive the statement from the axioms; He already knows them if they are true and knows as well whether or not they are derivable.
I say, “If they are true.” It is only on the Platonist view of mathematics that there cannot be truth value gaps with respect to some of the sentences of mathematics. The Continuum Hypothesis, which states that the next greatest number after ℵ0 (the least of the infinite cardinal numbers) is the number of points on a line, need have no truth value at all if we are anti-realists about mathematical entities. If, as I am strongly inclined to do, we regard set theory as an exercise in make-believe, according to which we treat the axioms of set theory as proposals to be imagined true and then derive the consequences, then in the fictional world of set theory there just is no truth about the Continuum Hypothesis, any more than there is a truth about the size of Sherlock Holmes’ shoes. That’s just left open by the story! Fictional entities are thus radically incomplete, and the fact that neither the Continuum Hypothesis nor its negation is derivable from the standard axioms of set theory is just what we should expect from fictional entities which haven’t been completely characterized. In a case like this, there is no truth to be known by God, just as there is no truth for Him to know about the size of Sherlock Holmes’ shoes.
- William Lane Craig