[PREV - WHATS_ZFC_DONE_FOR_ME] [TOP]
June 8, 2019
There are many and various popular descriptions
of Goedel's Incompleteness Theorem. This one is
pretty good, I think:
https://blog.infinitenegativeutility.com ~/Dust/Attic/Prog/getty_ritter-2018-why_i_dont_love_goedel_escher_bach-infinitenegativeutility.html
"... Gödel’s incompleteness theorem. The high-level,
slightly handwavey description of this theorem is https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems
that, for any sufficiently expressive mathematical
proof system, there are more true facts within the
system than there are proofs for facts, which in
turn means that not every fact can be proved: in
short, that not every mathematical fact has a
corresponding mathematical proof. My short
explanation papers over a number of important
features of this theorem, such as what I meant by a
‘sufficiently expressive mathematical system’ ... "
The phrase "handwave" seems to come
up in these discussions quite often.
A phrase I picked up from
someone at slashdot: dismissing
the possibility of artificial Though really, that just works
intelligence with a "handwave against the algorithmic
to Goedel". understanding style of AI, not
the microcosmic god variety
that's conquering the world.
One explanation of the the implications for this:
https://www.prospectmagazine.co.uk/magazine/kurt-godel-and-the-romance-of-logic
~/End/Thought/RAW-GOEDEL
"'The First Incompleteness Theorem' states the
following. Given any consistent axiomatic system of
arithmetic, there will be at least one statement about
numbers which cannot be proven in that system, and
neither can its negation. Moreover (though this is not
actually part of the theorem, but of Goedel's
reflections upon it), that statement has to be
true. In other words, the Russell-Whitehead project
was doomed, and so too were the hopes of anyone else
who wanted to derive the whole of arithmetic from a
system of logic. There is, and can be, no such thing
as a consistent and complete axiomatisation of
arithmetic. Moreover, that there cannot be such a
system is itself a truth that can be inferred from a
theorem of mathematical logic."
Now, the question arises: couldn't you take the
one thing that can't be proved, and call it an
additional axiom? Actually, this does you no
good, because there will always be another point
like it, another truth you'd like to have
established on your foundations that can't be...
It could be it's a silly idea, but the wonky thought
occurs to me that it might be the trouble is the idea
that you need to have a fixed, finite number of known
"postulates":
What if you had an automatically self-extensible
set of "axioms"? One of your assumptions could be
that anything which is not provable shall be
assumed to be true... or false?
It could be the system would need to
bifurcate to follow both branches, It's difficult to get
in one the proposition is assumed true away from the need
in the other, false. for *additional* uses
of human intuition to
make sense of things
like this.
--------
[NEXT - WILDBERGER_MATH]