
Waiting for Gödel (2016) - keiferski
https://www.newyorker.com/tech/annals-of-technology/waiting-for-godel
======
ProfHewitt
[Gödel 1931] is _not_ applicable to the foundations of computer science for
the reasons stated here:

[https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3603021](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3603021)

~~~
caleb-allen
Thank you for this link. A few months ago I stumbled across your paper,
_Common sense for concurrency and strong paraconsistency using unstratified
inference and reflection_ and found it entirely remarkable. I look forward to
reading the new paper you linked.

~~~
ProfHewitt
You are _very_ welcome!

Comments and suggestions for the linked article are greatly appreciated!

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smitty1e
Title alludes to Beckett =>
[https://en.m.wikipedia.org/wiki/Waiting_for_Godot](https://en.m.wikipedia.org/wiki/Waiting_for_Godot)

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p0llard
> There are true statements that are unprovable within the system.

I really don't like this, and it verges on being flat-out incorrect: the first
incompleteness theorem does _not_ say this at all. It says that there are
sentences (and indeed it constructs one, the so-called _Gödel Sentence_ ) that
_cannot be proven or refuted_ within the system.

For first-order theories it follows from the contraposition of Gödel's
_completeness_ theorem that there exist (classical) models of the theory where
this sentence holds and models where it doesn't: the existence of models where
the sentence does not hold means that this must be a very different meaning of
"true" to the one used in common parlance.

In higher-order logics, which do not have a completeness theorem, it makes
sense to talk about true statements which are unprovable: there can exist
tautologies, sentences which hold _in all models_ of a theory, which cannot be
proven from the axioms of a higher-order theory (and you don't need the
incompleteness theorem for this); on the other hand, in a first-order theory
all tautologies are provable.

Sentences which cannot be proven or refuted within a theory are said to be
_logically independent_ of the theory. Famously, the Axiom of Choice is
independent of the axioms of Zermelo–Fraenkel set theory: it's up to
mathematicians to decide whether they accept the Axiom of Choice. If they do,
they can work in ZF+C; if not, they can work in ZF. Neither system is "more
true" from a purely logical perspective, so I really don't like describing
logically independent sentences as "true but unprovable": it almost certainly
doesn't mean what people think it means.

The first incompleteness theorem could perhaps be stated better for a lay
audience as:

    
    
        No recursive set of axioms can capture our notion of arithmetic it its entirety.
    

This is a limitation on how we can use axiom systems to represent mathematical
objects: even more informally, we might say:

    
    
        Truth is subjective in sufficiently complex systems.

~~~
natcombs
> so I really don't like describing logically independent sentences as "true
> but unprovable":

I don't see a particular problem with it. It can be "true but unprovable"
within a given system. I feel like you may be arguing semantics, but the
sentence is still clear and accurate to me, while your lay audience definition
is less clear and steps further away from the theorems than necessary

~~~
p0llard
> I feel like you may be arguing semantics

Yeah, I am; it really depends on how you define "true". I prefer this to be
interpreted as "true in all models" so sentences are "true" when they are
tautological consequences of a theory.

Using this definition, all "true" sentences are provable in first-order logic.

The (usual) Gödel sentence is true in the intended model of arithmetic, but I
don't really like this property being referred to as "truth" without
qualification.

> "true but unprovable" within a given system

Not sure about this: I don't think you can really say something is "true in a
system" if it isn't provable. You can only assert its truth by saying it's
"true in the intended model" without qualification, or by doing some meta-
reasoning in a more powerful system outside the original one.

~~~
natcombs
Thanks for the response. After reading it I think I may be less clear than in
my initial reply now.

Question: Can a sentence be provably true in one arithmetic system but not
another?

If so that means there are provably true sentences which exist, but not
provably true with the axioms that I have at my disposal right now?

~~~
p0llard
> Question: Can a sentence be provably true in one arithmetic system but not
> another?

The answer is yes!

    
    
        ZFC |- AC

but

    
    
        ZF |/- AC
    

and both ZFC and ZF can encode arithmetic.

But there's an issue here: no-one really talks about the "truth" of the Axiom
of Choice as though it's a concrete thing: it's a very controversial axiom,
and although most mathematicians accept it, quite a few don't. Constructivist
mathematicians don't accept it, and it's provably equivalent to the law of the
excluded middle, so it can't be used in intuitionistic logics.

Now you might counter and say that AC isn't the Gödel sentence for ZFC, and
the Gödel sentence for PA is true in the intended model. But that's a
different matter from whether it's provable from an axiomatic foundation. The
reason I think this matters is because mathematicians work with proofs! Most
mathematicians aren't working in foundations, and rely on proofs to the extent
that they don't even consider the truth of statements which cannot be proven.

> If so that means there are provably true sentences which exist, but not
> provably true with the axioms that I have at my disposal right now?

The issue is that provability is completely contingent on the set of axioms
you use: provability isn't a universal notion. Of course if you add something
which is unprovable (such as the Gödel sentence) to PA, you get a new system
which can prove more things: but this system has its own unprovable Gödel
sentence.

I'd also question what you mean by "provably true": a sentence is provable
from a theory when there exists a derivation in some proof calculus of the
sentence starting from the axioms of the theory. "True" is much harder to pin
down, and we wouldn't usually say "provably true". "True" can mean "true in
the intended model", "true in all models", or even just "provable from a set
of axioms".

The Gödel sentence is "true but unprovable" only in the sense that it is true
in the intended model: it is not a tautology.

I think most people who have had a brief exposure to mathematics would
consider "true in all models" (i.e. tautological truth) to be what is meant by
"true", so I don't like it being used to mean "true in the intended model"
without qualification.

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dang
Discussed at the time:
[https://news.ycombinator.com/item?id=12008020](https://news.ycombinator.com/item?id=12008020)

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ur-whale
[https://archive.is/IAsgM](https://archive.is/IAsgM)

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KineticLensman
This Gödel's killing me

