The geometry theorem has two steps. The first step is to show that statements in Euclidean geometry can all be rephrased as statements about systems of multivariate polynomials equalities or inequalities with real coefficients. I don’t know a reference for the first step. The second step is to prove that there is an algorithm for solving such systems. This theorem is by Tarski, and goes under the name “quantifier elimination” for real closed fields. A book on model theory would be a good starting place.

I know there’s a book devoted specifically to this subject, but I don’t remember the name. Searching Amazon, it might be Mechanical Geometry Theorem Proving by Shang-Ching Chou.

Okay, good to know – I’ve always suspected the version of the “the computer discovered it” story I’d heard was apocryphal / misleading. It’s too basic a proof for it to be plausible that it’d been overlooked for so long.

But if you wait long enough, it will churn out the incompleteness theorem, and all other interesting theorems.

In some sense, but given that there are infinitely many logical constructions that could be proven or disproven, it’s also possible the computer would never get around to proving something you didn’t already know (assuming finite computational speed). Some of this depends on programming intuition into the computer about what propositions to pursue first, and what corollaries to ignore.

A fact that deserves to be better known is that Euclidean geometry is complete: there is algorithm to determine whether or not any statement in Euclidean geometry is true.

Do you happen to have a link or source to recommend, for a development of this? I’d be interested in its history, the implementations, etc. With such a resource and with regard to “deserves to be better known,” as a teacher, I can probably do my part :-)

A fact that deserves to be better known is that Euclidean geometry is complete: there is algorithm to determine whether or not any statement in Euclidean geometry is true.

If you are asking about Turing reducibility, i.e. an “oracle” computer that can solve problems, then the answer is: the computer would “solve” a given problem by reducing it to an elementary logical combination of problems that are known to have solutions. Once the problem has been so reduced, the computer would recognize the reduction is complete, and report that reduction to us in a logic-symbolic language which we’re presuming we would understand.

However, I noticed you asked about “Gödel had never lived and the rest of us maybe never suspected the incompleteness theorem”—It is important to note that (1) a universal oracle that would be capable of such a task has not been realized in real life, and probably couldn’t be, and (2) even if realized, there’s no guarantee such a computer would pursue proving the kind of theorems we’re interested in, without guidance as to what answer it’s trying to find (I believe all theoretical Turing models assume that a specific and falsifiable question is asked of the oracle to prompt its operation).

In other words, if none of us suspected the incompleteness theorem, how would a computer know to characterize it and pursue its proof, instead of one of the infinitely many other theorems that can be characterized within the system? If we wanted a computer to determine what proofs are important and then prove them, instead of mucking about arbitrarily, we’d need to induce in it some intuitive guidance, which gets us into neural networking and artificial intelligence and ideas I’d rather not speculate about; my thoughts on the matter would be even more useless than average.

Relatedly, there exists a canonical story, possibly mythical, that the nowadays-common classical trigonometry proof that angle ABC = angle ACB implies that AC = AB (completed via AA similarity of triangle ABC and triangle ACB using CB = BC) was first discovered by an oracle computer designed to find geometry proofs. So oracles designed to perform specific tasks, moving from theoretical to applied territory, do exist and have been used. The standard examples are the various proofs of the four-color map theorem. I don’t think these practical-purpose oracles start from any kind of set-theoretic axioms, though; I think at the very least computation on the real numbers or at least the integers (as a field) is naively assumed in any such computer model.

Ok, some potentially interesting resources for you (and anyone so inclined). All PDF links:

A very readable explanation of Turing reducibility can be found this paper by Martin Davis.

On the other end of the theoretical-practical spectrum, if the controversy surrounding applied computational proofs interests you, check out this paper E. R. Swart.

If the technical aspects of how such a proof is constructed interest you, here’s a very readable summary of the combinatorial reducibility of the four-color map problem, written by Robin Thomas, who offers this caveat with surely no small amount of sly humor: “[V]erifying all of this without a computer would require an amount of persistence and determination my coauthors and I do not possess.” :-)

So then it seems to me then if you were to run the algorithms of set theory on a computer to find the incompleteness proof and other new, undiscovered proofs of mathematics and metamathematics, then the real problems are that (1) it takes an enormous amount of computation time, (2) the computer wouldn’t know when it has run past a completed proof, in order to alert you, i.e. it doesn’t know where to insert “endpoints” (which would be different than the halting problem,) and (3) you might not be able to reduce it to another language, in order to comprehend it yourself. Are these correct?

I may have to leave it here in order to do other stuff, but thanks again.

Cantor’s diagonalization argument is straightforward to express in ZFC, and is how you prove that the power set of a set is bigger than the original set.

And there were, and still are, at least two compelling, a priori reasons to hold onto belief in the rationality and discoverability of the world:

The first reason is sometimes called Leibnitz’s Law of Sufficient Reason (although Leibnitz certainly wasn’t the first to notice it)—i.e., that everything that happens, happens for some prior reason. There are counterexamples such as emissions of subatomic particles, but because they are repeating and regular in form, we are led to suppose that there may be reasons. (And there are things like extreme and absolute counterexamples, such as the beginning of the universe, or why is there anything at all, instead of nothing.) Some events have reasons but might not be predictable (such as the moment an apple falls from a tree,) or else we have to use probability, but these still fall under the law.

The second reason for belief in a rational world is that we are able to construct finite typologies of things—there are only so many subatomic particles, only so many natural elements, only so many species of apple tree, etc. Nobody has an explanation for this either, but of course it’s related to Occam’s razor and it propels reductionism.

These two reasons are put together and react powerfully with a third: the millennial Great Chain of Being was turned upside down. That is, we no longer believe (most of us, anyway) that the universe emanates downward from a Creator, with no gaps in the creation. This was rather precisely inverted in the modern period to the evolutionary view that things are constructed upwards from tiny bits, atoms or whatever, and that there are lacunae in the picture.

Gödel very clearly decided that the world is rational, that math can stay, and so science has the wrong metaphysics and needs a rewrite.

The semantic primitives are all expressible in ZFC. Of course, it still requires an outsider to say, “yes, that’s what I really mean by that semantic primitive”, but that’s true of the semantics of 2+2 just as well.

I am not studied in set theory, but I thought that “set theory” proper is just ZFC, or the like. Gödel demonstrated the incompleteness of the axioms of ZFC, only if his same recursive extensions are introduced. In other words you still need Peano arithmetic, i.e. with multiplication. Arithmetic without multiplication (Pressburger arithmetic) on the other hand is complete: it is also, I would gather, expressible as set theory. Is this correct?

I thought ZFC set theory is more or less a set of rules for thinking, for framing the methods of proof of mathematics, and that it has its own theorems—but it doesn’t have the primitives of your subject, which you still must introduce, whether it’s arithmetic with or without multiplication, or the order of the lepidoptera, or all the rules of a nation’s judicial system.

Now I have read that set theory by itself has unresolved paradoxes, and that ZFC is consistent with or without the continuum theorem, and that Wittgenstein derided parts of set theory as being as close to mysticism as you can get. (Not that I don’t like actual mysticism.) Certainly set theory allows hierarchies constructible to the absolute, which looks like the source of some of its paradoxes.

But to have it be about something, you still need to add the semantic primitives, and you still need to judge, as an outside observer, whether what you are doing with set theory follows the meaning of those primitives. (Which is where the later Wittgenstein jumped in to show that natural language doesn’t follow a scheme like that.) I have never heard of set theory algorithms that are able, of themselves, to logically lead to Gödel’s proof, without being preprogrammed to do so. I would think that if formal languages could do so, we’d have general, self-discovering artificial mathematical intelligence by now.