For me Gödel's completeness theorem is the miracle. Every valid statement has a proof. Amazing!
Aim a little higher, every true statement, and there might not be a proof. It is no surprise to me that this is true. It is a big surprise to me that Gödel was able to prove it; ordinary proofs are hard to find, and proofs of the limits of provability presumably even more deeply hidden.
Non-standard models of arithmetic are weird. Theorems that are true of the standard model of arithmetic and false in some non-standard model must surely be convoluted and obscure. The first order version of the Peano axioms nail down the integers, not perfectly, but very well. Restricting one-self to theorems that are true in all models of them, even the weird, non-standard ones, feels like a very minor restriction. Gödel's completeness theorem raises the possibility of writing a computer program to find a proof of every theorem that isn't convoluted and obscure. Gödel completeness theorem is the really big deal.
Except it isn't. That computer program turns out to be one of those wretched tree search ones that soon bogs down. The real problem turns out to be the combinatorial explosion inherent in unstructured search through the Herbrand universe. One needs Unification and one needs a still missing ingredient to give search a sense of direction. The interesting questions are about the "sense of direction" that lets us find some of the deeply hidden proofs that do exist. Will LLM's help? The answer will be interesting, either way.
Unfortunately, Gödel's proof method per se only shows an example that is not so meaningful, involving self-reference. He builds a number-theoretical formal system which can talk about its own formulas, by encoding them as integers (something we do with computers now as a daily matter: all computer program text and other data is arithmetically encoded into binary, which has an interpretation as a number). In the context of Gödel's work, we call this arithmetic encoding "Gödel numbering".
Whether a proposition is true is reformulated as a number-theoretical property: instead of asking, is this proposition or equation true, we ask, is the arithmetic encoding of this proposition an integer which belongs to the set of theorem-integers; is it a theorem-number?
Within this framework, Gödel shows that a proposition can be made which says "G is not a theorem-number", such that this very propostion's own Gödel number is G! In other words, a kind of Quining is going on, whereby the proposition embeds a coded reference to itself. Essentially, Gödel introduces the idea that we can make a statement which says "I am unprovable", in formal, rigorous way. If that statement can be derived from the axioms, then a contradiction results: it was derived, yet it asserts the falsehood that it is not derivable, and so a falsehood was derived from the system's axioms. If it is true, then it points to incompleteness: there is a truth that can be expressed in the syntax of the system, yet cannot be derived.
Thus if we have any system expressive enough to encode the "G is unprovable" statement where that statement itself is G, that system is either inconsistent (allows a falsehood to be derived) or incomplete (allows true statements to be written which cannot be derived).
* There are axioms, there are models, and there are theorems.
* A model is a particular structure with objects and relationships. The "standard model of arithmetic" is just the natural numbers 0, 1, 2, ... with normal rules of addition and subtraction and so on. A different model could be the real numbers, or one that includes infinitesimally small numbers, or so on.
* A set of axioms are rules about how a model can work. For example, we can have an axiom for any set of objects called "numbers" with an operation called "addition" that the operation must be commutative (x+y = y+x). The standard model above is one model that satisfies this axiom.
* A theorem is a fact that can be true or false about a given model. For example, "the model has infinitely many objects." If we can prove a theorem from a set of axioms, then that theorem is true for every model that satisfies the axioms. However, there can also be theorems that are true for one model that satisfies the axioms but false for a different model.
Godel's completeness theorem says that if a theorem is true for every model that satisfies a set of axioms, then one can prove that theorem from the axioms.
Godel's first incompleteness theorem says that in any axiomatic system (sufficiently complex) there are theorems that are neither always true nor always false. In other words, there is a theorem that is true for some model of the axioms but false for some other model of the axioms.
e.g. that Godel didn't think this scrapped Hilbert's project totally:
>Gödel believed that it was possible to redefine what we mean by a formal mathematical framework, or allow for alternative frameworks. He often discussed an infinite sequence of acceptable logical systems, each more powerful than the last. Every well-formulated mathematical question might be answerable within one of them.
Yes there are undecidable statements (eg the continuum hypothesis) but that doesn’t change the fact that the vast vast majority of statements in any axiomatic system are going to be decidable, and most undecidable statements are going to have “niche” significance like that.
[1] eg https://www.imperial.ac.uk/study/courses/postgraduate-taught...
To understand the above Gödel result, the essential point is
the principle of elementary logic that a contradiction
Ā A implies all propositions, true and false.
(Given any two propositions A and B, we have A ⇒ (A+B),
therefore Ā A ⇒ Ā (A+B) = Ā A + Ā B ⇒ B.)
Then let A = {A1, A2, ..., An,} be the system of axioms
underlying a mathematical theory and T any proposition,
or theorem, deducible from them:
A ⇒ T.
Now, whatever T may assert, the fact that T can be deduced
from the axioms cannot prove that there is no contradiction
in them, since, if there were a contradiction, T could
certainly be deduced from them!
This is the essence of the Gödel theorem, as it pertains to
our problems. As noted by Fisher(1956), it shows us the
intuitive reason why Gödel's result is true. We do not
suppose that any logician would accept Fisher's simple
argument as a proof of the full Gödel theorem; yet for most
of us it is more convincing than Gödel's long and
complicated proof.
There is usually a 'not sufficiently complex' clause in that definition. Presburger arithmetic is complete: https://en.wikipedia.org/wiki/Presburger_arithmetic
Natalie mentions the Newman & Nagel's text "Gödel's Proof," a
(//the//?) 1958 classic on the subject. [[ 1 ]] Having left IBM
in December 1990, I spent a month with the text, dipping into
mild insanity, taking to strange wines to relieve myself of the
fear that my previous years long study of Whitehead & Russell's
"Principia Mathematica" [[ 2 ]] was useless.
I really appreciate the inclusion of Alvir's statement on
whether or not Gödel thought he proved all logical systems
undecidable and incomplete. About 80% into the article is her
quote:
>> Often people will speak as if the CH is the smoking gun that
>> shows sometimes mathematical questions have no answer. But
>> in my opinion, this situation provides very little evidence
>> that there are “absolutely undecidable” mathematical
>> problems, relative to any given permissible framework.
Though I would have added a reference to Infinitary Logic
[[ 3 ]] after dropping the reference to L-omega-1-omega. I
suspect most readers would find discussion of higher-order and
modern logic a bit confusing without a pause for further study.
But a guide post pointing in the appropriate direction would be
good.
That this is the only critique I have of the article speaks to
Wolchover's skill in communicating complex ideas for a lay
audience. I really liked this article, so thank you @baruchel
for posting the reference to it.
:: References
1. https://search.worldcat.org/title/1543160023
2. https://search.worldcat.org/title/933122838
3. https://en.wikipedia.org/wiki/Infinitary_logic