
Kurt Gödel and the romance of logic - daddy_drank
https://www.prospectmagazine.co.uk/magazine/kurt-godel-and-the-romance-of-logic
======
joe_the_user
Actually, Gödel's theory is fairly accessible compared to, say, the General
Theory of Relativity.

All you have to know that the process of proving a statement is a fairly
mechanical, step-by-step process. At that point, "statements that can be
proven true" and "statements that be proven false" are two fairly defined
sets. At that point, "there are some true statements that cannot be proven
true" is moderately clear.

Then you introduce the idea of a model and make the concept fully exact.

The thing to remember is Gödel himself never really liked this simple view and
wanted truth to be more transcendent. He shared with Einstein the quality of
not liking the results of his discoveries, perhaps a source of their
friendship.

~~~
skh
The quality of being “true” is dependent on the model one is using. One can
not talk about “truth” without being in a model. (Assuming we are talking
about standard mathematical logic.). A statement in a first order system is
provable if and only if it is true in all models for that system.

~~~
eximius
You either don't understand the point of the theorems or are being contrarian.

In the context for the Incompleteness theorems, there are two 'kinds' of
truths: a more informal kind used by all of us everyday and the mathematical
kind as in, proven true under a given system.

The entire purpose of the theorems is to establish that there exists theorems
in the first set that are not in the second set, while being expressable in
the system. To deny the existence of the first kind of true statements ignores
the entire purpose of it all.

~~~
skh
I used the term ‘truth’ as used in mathematical logic. In a given model a
statement can be true or false. Under a given axiomatic system a statement is
either provable or not. We don’t use the word “true” when dealing with
statement under an axiomatic system. We do use the word when dealing with a
statement in a given model.

Your third paragraph doens’t make sense. In first order logic a theorem is a
statement that is true in all models and is one that is provable. This is a
result of the Completeness Theorem.

------
craigr1972
" He announced that he had studied the US constitution in detail, and—no
doubt, forensically examining its propositions one at a time and perhaps
testing it against thought experiments against wild possible futures in which
the president was allowed to get out of control—he had discovered how the US
could legally be turned into a dictatorship."

~~~
DyslexicAtheist
sadly this article skips a lot of detail on what happened during and before
the hearing and how Einstein tried to coach him. The New Yorker had a much
better summary on this. Here in all its hilarity:

\----

from [https://www.newyorker.com/magazine/2005/02/28/time-
bandits-2](https://www.newyorker.com/magazine/2005/02/28/time-bandits-2)

So naïve and otherworldly was the great logician that Einstein felt obliged to
help look after the practical aspects of his life. One much retailed story
concerns Gödel’s decision after the war to become an American citizen. The
character witnesses at his hearing were to be Einstein and Oskar Morgenstern,
one of the founders of game theory. Gödel took the matter of citizenship with
great solemnity, preparing for the exam by making a close study of the United
States Constitution. On the eve of the hearing, he called Morgenstern in an
agitated state, saying he had found an “inconsistency” in the Constitution,
one that could allow a dictatorship to arise. Morgenstern was amused, but he
realized that Gödel was serious and urged him not to mention it to the judge,
fearing that it would jeopardize Gödel’s citizenship bid. On the short drive
to Trenton the next day, with Morgenstern serving as chauffeur, Einstein tried
to distract Gödel with jokes. When they arrived at the courthouse, the judge
was impressed by Gödel’s eminent witnesses, and he invited the trio into his
chambers. After some small talk, he said to Gödel, “Up to now you have held
German citizenship.”

No, Gödel corrected, Austrian.

“In any case, it was under an evil dictatorship,” the judge continued.
“Fortunately that’s not possible in America.”

“On the contrary, I can prove it is possible!” Gödel exclaimed, and he began
describing the constitutional loophole he had descried. But the judge told the
examinee that “he needn’t go into that,” and Einstein and Morgenstern
succeeded in quieting him down. A few months later, Gödel took his oath of
citizenship.

\----

Another one: [https://www.quickanddirtytips.com/education/science/when-
g-d...](https://www.quickanddirtytips.com/education/science/when-g-del-almost-
took-on-the-us-constitution)

\---

EDIT: something I also missed in this piece was that Gödel developed rheumatic
fevers as a child and started reading medical books with the age of 8 to learn
more about the condition. He concluded that he had a weak heart :D

Gödel is really worth studying closer and this article just leaves out a lot.

~~~
edmundsauto
Coincidentally, I'm reading When Einstein Walked with Godel, and this anecdote
is in there. Such an interesting man -- he's up there with Von Neumann for
historical figures I'd like to learn more about, as the work they develop is a
form of first principles that a lot of modern thought is derived from.

Any reccs on books to read, more focused on their theories than their lives?

------
infinity0
> rescued the idea that there are truths that humans can never prove

This is a gross misinterpretation of Gödel's actual theorem that helps
perpetuate irrational superstitious attitudes against science, mathematics,
and logic.

What Gödel showed was that proofs are relative to some underlying axiomatic
model and that for _any particular axiomatic model_ there are always truths
unprovable by it.

That doesn't mean "there are truths that humans can never prove", all it means
is that we have to extend our axiomatic systems in order to prove some truths.
(And whether they are true or false "in reality" is another question, to be
determined empirically.)

A more accurate (and just-as-click-baity) way of interpreting Gödel's theorem
is that "mathematicians and logicians will always have a job".

~~~
truantbuick
> That doesn't mean "there are truths that humans can never prove", all it
> means is that we have to extend our axiomatic systems in order to prove some
> truths.

If you believe that the only consequence to Gödel's theorem is we need to
"extend our axiomatic system", I do think you've missed the point.

For one thing, I think Gödel's theorem and Gödel's proof are unfortunately
conflated.

Gödel's proof is lovely and elegant, and can be understood with minimal
knowledge of logic, but, ultimately, all it does is provide a counter example.
So when people read his proof and understand it, they tend to be unimpressed
with its power because the counter example is very generic and seems an
uninteresting barrier to our ability to discern "truth". But that says nothing
about there may be lots of other kinds of unprovable statements.

Ultimately, Gödel's theorem tells us one absolute kernel of truth by saying
all but very basic axiomatic models are necessarily incomplete. However which
way you want to make the philosophical leap to connect that to our notion of
"truth" seems far more up to interpretation, but it annoys me when people
disrespect the theorem because they're unimpressed with the counter examples
the proof constructs.

~~~
skh
An important nitpick. His Incompleteness Theorem deals with recursively
enumerable axiomatic systems. The second order Peano Axioms are categorical.
That is, they have only one model up to isomorphism. It’s easy to come up with
a complete axiomatic system for the standard model of the natural numbers.
Just take as your axiomatic system the collection of all true statements. This
ins’t a useful system since there is no procedure for determining if a
statement is an axiom or not.

~~~
whatshisface
How would you be able to take every true statement as an axiom? Without
proving anything I don't see how you could identify any statements as true.

~~~
ginnungagap
Whenever you have a structure M of some language L you can take the so called
complete theory of the structure, denoted Th(M), which is just the set of all
L-sentences true in M. In particular if L is the language of PA and M are the
standard natural number with the standard operations Th(M) is a theory called
true arithmetic. This theory is complete (that's because Th(M) is always
complete) and clearly enough to talk about the integers, but it escapes
Gödel's theorem since it's axioms are not recursively axiomatizable.

You seem to be confusing "true" and "provable", as far as first order logic is
concerned those are equivalent (by another theorem of Gödel, the completeness
theorem), but the first in defined in terms of models and the second is purely
syntactic

~~~
mietek
You seem to be forgetting that constructive truth means provability.

Also, it doesn’t seem right to use the informal “take” when the object in
question is not computable.

~~~
ginnungagap
I specified I'm working in standard first order logic. And I'm not sure what
do you mean with the informal take.

Also intuitionist logic is not my area, but isn't it divided in inference
rules for provability and (Heyting or Kripke) semantic for model theory and
truth just like FOL?

~~~
mietek
You may be working in classical logic, but that doesn’t mean the parent poster
is. You asserted the parent poster seems to be confused, but what the parent
poster said makes sense in a constructive setting. Please see my other
comment:

[https://news.ycombinator.com/item?id=18756974](https://news.ycombinator.com/item?id=18756974)

------
kkylin
Anyone interested in this article may also be interested in this discussion
from a while back:
[https://news.ycombinator.com/item?id=18115696](https://news.ycombinator.com/item?id=18115696)

Also of interest is Godel's AMS Gibbs Lecture, which unfortunately I have not
been able to locate on-line. It can be found in Volume 3 of Godel's Collected
Works, alongside his paper / lecture notes on closed time loops in general
relativity.

------
neokantian
The biggest qualm that I have with the example for the incompleteness theorem,
is the use of two-valued, Boolean/Aristotelian logic, in which "not true"
automatically becomes "false".

If you allow the use of a three-valued logic, for example, with (true,
undetermined, false), the Gödel statement, “I am not provable" amounts to
saying "My provability is false or undetermined".

The same problem occurs in Russell's paradox, "Does the set of all sets that
do NOT contain themselves, contain itself?"

The use of the NOT-operator is degenerated in a cyclic group Z2. It is the
only situation in which the NOT-operator is not set-valued.

In my opinion, if the "isProvable()" predicate allowed for multi-valued logic,
Gödel's incompleteness theorem would look much less paradoxical. In other
words, the paradoxical outcome could simply be the result of Boolean
shoehorning.

~~~
theaeolist
The incompleteness result does not even mention "true" and "false". It says
that "there are statements of the language of F which can neither be proved
nor disproved in F." Whether those statements are true/false is left unsaid.

------
DyslexicAtheist
> _The theoretical physicist and mathematician Roger Penrose, for example, has
> argued that Gödel’s theorem shows that “Strong AI” is false: our minds
> cannot be computers, and that by extension the intelligence of computers
> will never fully replicate them._

Note that there are camps that have been disputing this (e.g. McCullough’s
Objection):

[http://www.deepideas.net/godels-incompleteness-theorem-
and-i...](http://www.deepideas.net/godels-incompleteness-theorem-and-its-
implications-for-artificial-intelligence/)

and

[https://www.iep.utm.edu/lp-argue/#H3](https://www.iep.utm.edu/lp-argue/#H3)

sadly, I'm not even close to figuring out if the Roger-Penrose argument is
valid. Nada, not even a gut-feeling!

------
sorokod
BBC podcast on the subject, interesting for historical background.

[https://www.bbc.co.uk/programmes/b00dshx3](https://www.bbc.co.uk/programmes/b00dshx3)

