
The Director’s Cut: On Gödel’s first incompleteness theorem - lordgrenville
https://inference-review.com/article/the-directors-cut
======
zentiggr
The logical formulae are outside my learning, at this point in my life, but I
do understand the general gist of the theorems from prior exposure and
discussion.

If this were my first exposure, the overly florid, 'artsy' presentation would
have sent me packing before the end of the first section, thinking "Oh, great,
now I'm going to have to unpack his prose before I can even follow the
discussion".

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mellosouls
_...to have the privilege of walking home with Gödel_

Albert Einstein, on why he still went in to work at Princeton.

~~~
DyslexicAtheist
my favorite Gödel, Einstein story is when he was prepped by Einstein for an US
immigration hearing, and then told the judge he found some loopholes in the
constitution. Was it his character shaping his work, or his work and
surrounding shaping (making) him, and if so what shapes a man like Gödel (or
Freud, Einstein, Popper, Russel, ...) many others of that time. It seems there
was a huge density of brilliant people concentrated in Europe at the time.
Hardly anyone who died within the past 30 years will be remembered as those
from that generation I'm sure.

Now everyone is writing AI papers which all look identical. Sure academia has
changed.

A Contradiction in the U.S. Constitution
[https://jeffreykegler.github.io/personal/morgenstern.html](https://jeffreykegler.github.io/personal/morgenstern.html)

Gödel’s Loophole
[https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2010183](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2010183)

~~~
seventhtiger
The fact that we are overlooking our Einsteins and Gödels says more about us
than about them. For a moment in history scientists were looked up to and
mythologized. Today they're doing the same work, but they've been put in a
different place in society.

~~~
mellosouls
Examples?

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salty_biscuits
Oh god, just read Nagel and Newman instead!

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ouid
This was awful to read.

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PaulDavisThe1st
ah, another generation bites down hard on Gödel, without even a citation of
Hofstadter. Too pop-science? Too "overly florid, 'artsy' presentation"? Or
just too good, too deep, too wide to make it worth revisiting this topic in a
blog post measured in a few pages of printed text?

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thedudeabides5
A bit too much inscrutable math here, but something every AI/ML researcher
should know by heart.

Imagine today we make the perfect modeling machine for reality.

Now imagine that machine has to model not only reality yesterday, but today,
now that we have the machine.

Additionally, it will have to model the fact that humans are going to change
their behavior based on the fact that this machine is modeling reality...

[Machine Halts...]

aka, all of these problems below from seemingly different domains are actually
isomorphic (aka the same problem in different shapes)

[https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...](https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems)
[https://en.wikipedia.org/wiki/Uncertainty_principle](https://en.wikipedia.org/wiki/Uncertainty_principle)
[https://en.wikipedia.org/wiki/Lucas_critique](https://en.wikipedia.org/wiki/Lucas_critique)
[https://en.wikipedia.org/wiki/Halting_problem](https://en.wikipedia.org/wiki/Halting_problem)

~~~
throwlaplace
>A bit too much inscrutable math here, but something every AI/ML researcher
should know by heart.

If it's inscrutable to you how do you know it's important enough for every
AI/ML researcher to understand?

------
mcguire
" _For all natural numbers, m = n if and only if S(m) = S(n)._ "

Hmm, er, ah, ... That's an odd way to put it.

~~~
tialaramex
The purpose of this is to make the successor function S an injection. That is,
it assures us that all natural numbers are the successor to exactly one other
natural number, except that zero is first and so it isn't the successor to
anything (in the natural numbers). The number line you saw in primary school
is fundamental to arithmetic.

Without this rule we might worry that perhaps there can be other numbers
between five and six, or that if we all count up from one some of us might
never reach a hundred because we inadvertently divert and get to six million
early. These concerns seem intuitively nonsensical, but Peano doesn't want to
rely on intuition, he wants to use logic.

~~~
mcguire
But normally it would be written "For all natural numbers, S(m) = S(n) if and
only if m = n."

"if and only if" is symmetric, but English isn't, really.

~~~
lidHanteyk
Lucky 10000: "If and only if" [0] is a particular phrase in English with a
particular meaning. It is symmetric, so that your version has the same logical
content as the original.

[0]
[https://en.wikipedia.org/wiki/If_and_only_if](https://en.wikipedia.org/wiki/If_and_only_if)

~~~
mcguire
The same logical content, yes, but...

Do you write the Fibonacci function as "n * (n-1) = fib(n)"?

~~~
Dylan16807
I don't know why you think it's backwards.

It makes lots of sense to put "m=n" first. It's giving the definition of
equality. "[term] is [definition]".

