
Gödel and the limits of logic (2006) - Pishky
https://plus.maths.org/content/goumldel-and-limits-logic
======
kruhft
I highly recommend getting Godel's proof[1] and reading it. It's an amazing
journey and quite understandable from the start (the first half of the book is
introduction), and I've never been able to read proofs very well. It takes
concentration, but once you get it (at '17 GenR') it's almost like a symphony
going off. The crystalline brilliance of the "System P" decomposition is worth
it just to see how math as a process could be reduced to a such simple, clear
and concise set of symbol manipulations...and then the rest shows how that
could be used topple itself. Incredibly philosophically insightful.

[1] [https://www.amazon.ca/Undecidable-Propositions-Principia-
Mat...](https://www.amazon.ca/Undecidable-Propositions-Principia-Mathematica-
Mathematics-
ebook/dp/B00A44R3E8/ref=sr_1_1?ie=UTF8&qid=1489447527&sr=8-1&keywords=on+formally+undecidable+systems+godel)

~~~
cgh
I try to read Godel's Proof once a year or so. You might also be interested in
Gregory Chaitin's Meta Math! The Quest For Omega, which relates the halting
problem (among other things) to Godel. A word of warning though, the book is
rather idiosyncratic, as you might have seen from the title.

~~~
kruhft
Got it years ago and loved it. Have to give it another read some day.

------
noobee
A light take on the subject can be found in Logicomix:
[https://en.wikipedia.org/wiki/Logicomix](https://en.wikipedia.org/wiki/Logicomix)
\- a cool blend between the dry topic and comics.

~~~
AndrewKemendo
Thanks for the recommendation. I found it on scribd if anyone is interested:
[https://www.scribd.com/document/98921232/Bertrand-Russell-
Lo...](https://www.scribd.com/document/98921232/Bertrand-Russell-Logicomix)

------
laxd
Are there any Gödel-like problems that does not involve self references?

~~~
jpt4
Yablo's Paradox [0], possibly [1].

Regarding [1], "circularity" in an uncountably infinite context seems
different than "self-referential", though the latter is usually geometrically
analogized as the former. I tentatively consider Yablo's Paradox to
demonstrate that an ineradicable cycle of alternating truth-assignments is
equivalent to an infinite (>= aleph-one) regression of alternating truth-
assignments, and thus that circularity does not map coherently to self-
reference in all logic systems.

[0] [http://www.mit.edu/~yablo/pwsr.pdf](http://www.mit.edu/~yablo/pwsr.pdf)

[1]
[http://ferenc.andrasek.hu/papersybprx/jcbeal_is_yablo_non_ci...](http://ferenc.andrasek.hu/papersybprx/jcbeal_is_yablo_non_circular.pdf)

~~~
pacala
Yablo paradox is so cool.

    
    
        Imagine an infinite sequence of sentences S1, S2, S3, ...,
        (S1) for all k >1, Sk is untrue
        (S2) for all k >2, Sk is untrue
        (S3) for all k >3, Sk is untrue
        ...
    

Perhaps another argument in favor of regarding infinites as a logical fallacy.
Assuming our universe is finite, all objects in the universe are finite,
including sets. Yablo's set chain ends at some N, possibly very very very
large. Sentence N is true [there are no further sequences], all other
sentences are untrue, as there exists a true sentence with k > i: the Nth
sentence.

~~~
lmm
Infinities can be handled safely in logic. You just have to be careful about
them.

There may not be physically real infinities. But there are no physically real
perfect squares either (assuming an analogue universe). It's a simplified
model.

~~~
raattgift
> But there are no physically real perfect squares either (assuming an
> analogue universe).

How does analogue vs digital (assuming you mean that analogue) matter here?
You'd also want to specify which "digital physics" you want to rely on.

We don't need to think along those lines to consider a physically real
idealized geometrical object.

Here I'll recast your square as a planar slice through a real object in our
(3d+1d) universe.

With reasonable assumptions about the local behaviour of spacetime, you can't
construct a physically perfect square because of the small-length-scale
behaviour of matter rather than that or the presence of infinities of
infinitesimals. The assumption here is that locally around the square the
spacetime is Minkowskian, which for a small and light square is practically
guaranteed everywhere outside the event horizons of black holes and far from
the hot dense phase of the universe. "Practically guaranteed" here means that
in spite of plausible theoretical objections to the perfectness, there exists
no mechanism to show that the object is not a perfect square (i.e., such
objections are conjectural physics leaning hard on a purely mathematical
argument).

"Digitalization" and discretization arguments run into observational problems
right away.

The wavelength of light is not quantized, for example, and is not clearly
bounded in the infrared. (In the ultraviolet you get pair production and
gravitational collapse, however.) While one could make an argument that at the
emission event of every photon there is not an infinity of available photon
wavelengths, you would have to forbid infinitesimal spacetime intervals to
avoid emitted photons stretching into arbitrary wavelengths under the metric
expansion of the universe, and the only practical way of doing that is through
discretization (with or without emergence).

There is a substantial weight of evidence that suggests that if spacetime (or
whatever it emerges from) is discretized, the discretization length is not
significantly above the Planck length; notably gamma rays, x-rays, light and
radio from violent astrophysical events do not appear to win races against
each other in an ordered fashion.

So there may be infinitesimal spacetime intervals, and so any quantity that
depends on spacetime intervals has an infinite set of values to choose from.

Absent a description of gravity-matter interaction when matter's small-scale
behaviours are relevant, we cannot really preclude the possibility of building
a perfect square in an _unreasonable_ (but nevertheless physical) local patch
of dynamical spacetime, such that the microscopic behaviour of the latter
offsets the odd quantum mechanical contributions to the square. At best we can
try to place limits on how long such a system could survive (it could last a
very long time in the cold sparse phase of the universe).

~~~
lmm
> How does analogue vs digital (assuming you mean that analogue) matter here?
> You'd also want to specify which "digital physics" you want to rely on.

In the kind of "digital physics" where you have a fixed grid it would be
possible to create a truly perfect square, that's all I was referring to. (I'm
well aware that such physics would be incompatible with Lorentz invariance and
thus unlike our real universe).

~~~
raattgift
> In the kind of "digital physics" where you have a fixed grid it would be
> possible to create a truly perfect square

Discretization is not really the problem: you instead have to fix the definite
position and momentum of the microscopic components of the square such that
all the parts of it stay in place relative to one another and to the total
arrangement of the shape they collectively form.

Switching to a taxicab geometry doesn't change the tendency of matter to move
around; it only defines the points in which matter might be found in
principle, and where it can never be found, ever.

In effect, the dynamism of spacetime is a red herring (we have difficulties
building ideal spheres in flat space, theoretically speaking), and spacetime
geometry (in the sense of minimal lengths vs infinite differentiability) is at
best a marginal contribution to the problem.

While Lorentz invariance is essentially what blocks the quantization of
energy, you can still have a discretized spacetime that preserves Lorentz
invariance. One broad approach to that involves a Lorentz contraction on the
spacings between the allowed points, so that observers may disagree on the
lengths of objects marked off in Cartesian coordinates on the fundamental
lattice spacings. In particular, even in the absence of gravitation, one
observer will conclude a small object has a definite lattice length while a
generic accelerated observer (and at least some boosted observers) will
conclude that the same small object's length is in superposition.

Additionally, one may quantize lengths and times as in a Snyder spacetime, and
still recover Lorentz invariance (or at the very least do away with preferred
frames; things are tricky where boosts are high and ultimately it is likely
impossible to recover the Lorentz subgroup in its entirety) in a quite natural
fit with a deformation of Special Relativity.

Unfortunately we still have GR with its strong successes and several viable
models which have minimum length scales. So at least at the energy limits we
have access to now, we can't really make a concrete choice about which better
describes our universe. In part that's because almost all these minimal-length
theories are carefully designed to reproduce General Relativity in some limit
_because_ of its strong successes.

------
personjerry
This reminds me of the Berry Paradox:
[https://en.wikipedia.org/wiki/Berry_paradox](https://en.wikipedia.org/wiki/Berry_paradox)

------
tmsldd
I like very much the program x virus example/citation.. but I would put it in
the other way around: "A program can find all virus, unless the operating
system is a virus itself."

------
DigitalPhysics
"Digital Physics" (the movie) is now free on Amazon Prime. It is also
available for rent/purchase on iTunes and Vimeo, in case you are kind enough
to support an independent filmmaker:) Come explore the world of self-
reference, infinity, complexity, Gödel & Turing, and psychedelics with
Khatchig!!

------
470b23bb7c9c44b
Why have I never had to debug a Gödel paradox in my day-to-day programming? Is
it because of types? Finite memory? Can someone enlighten me? Thanks

~~~
neel_k
You almost certainly have had to debug a Gödelian paradox in your day-to-day
programming. It's just that programmers call it "nontermination" or "infinite
loops".

In 1952 the logician Martin Löb invented something called "provability logic"
as a way to formulate Gödel's theorem more abstractly, so that we could see
how the proof worked without having to work with explicit numerical encodings
of formulas. (Think of this as the difference between working with strings
versus abstract syntax trees.)

His version of the proof is now called Löb's theorem, and its structure is
almost identical to what logicians call Curry's paradox, and which (via the
Curry-Howard correspondence) programmers know as the Y combinator. The Y
combinator is used to implement recursion in the lambda calculus, specifically
including the ability to go into infinite loops.

I blogged about this last year:

[http://semantic-domain.blogspot.co.uk/2016/05/lobs-
theorem-i...](http://semantic-domain.blogspot.co.uk/2016/05/lobs-theorem-is-
almost-y-combinator.html)

------
bitL
Isn't Gödel's incompleteness a consequence of overly ambitious material
implication properties? When you look at the truth table:

A | B | A -> B

0 | 0 | 1

0 | 1 | 1

1 | 0 | 0

1 | 1 | 1

I fail to see why all except for 1 -> 0 = 0 aren't undefined. I understand
reasoning behind why this was done, but mixing unrelated predicates seems like
a generally bad idea, even if it allows mechanical proofs.

Aren't there already "complete" systems like relevance logic? Now with
computers we can perhaps make proofs we need "relevant" instead of being based
on fundamentally incomplete though "simpler" logic?

~~~
cvoss
The first incompleteness theorem shows that, in a sense, the culprit of a
logic's incompleteness is not its "simplicity" but its "complexity": if the
logic is rich enough to include Peano arithmetic, then it is incomplete.
Compared to mathematics in general, a complete logic system is far less
powerful and cannot be used to prove nearly as many interesting things.

~~~
burntrelish1273
Undefined / don't care states also allow for simpler physical implementations.
For those whom did EE/CS undergrad might remember Karnaugh maps.

