
Picking holes in mathematics: Finitely stated, undecidable propositions - ColinWright
http://plus.maths.org/content/picking-holes-mathematics
======
dkarl
Mathematicians have been dealing with this problem in concrete ways for longer
than implied. For instance, the necessity of the parallel postulate for
Euclidian geometry and the existence of non-Euclidean geometries where it
doesn't hold were demonstrated in the 19th century. The question of how many
and which axioms are required as the basis of a field of mathematics has long
been known to be a valid, meaningful question.

What Gödel proved is that in the formal systems we use to model mathematical
propositions and proofs, there is no finite number of axioms that enable you
to prove or disprove all of the propositions that can be formulated. For
someone who grew up in the post-Gödel world, this isn't shocking at all and
looks like a natural progression from previous results. You need a certain
number of postulates. To get Euclidean geometry, for example, you need the
parallel postulate. It's natural to ask how many postulates you need to decide
all interesting mathematical questions, and there's no reason to assume the
answer to that question is finite. Apparently, that attitude reflects a lot of
hindsight bias, but there is no way for us to experience the shock of Gödel's
work the way it was experienced by mathematicians at the time, since the books
we learn from now are written by mathematicians who have already digested
Gödel's discoveries.

------
ionfish
As he posted on the FOM mailing list earlier this month, the 'final' version
of Friedman's book on boolean relation theory is now up on his website. [1] It
clocks in at around 900 pages. Not for the faint of heart.

[1] <http://www.cs.nyu.edu/pipermail/fom/2011-May/015383.html>

~~~
timtadh
For those interested, the crux of Friedman's argument for pursuing examples of
"Concrete" Incompleteness is:

> The second rationale for pursuing Concrete Mathematical Incompleteness
> preserves ZFC as the ambitious target. The idea is that normal mathematical
> activity up to now represents only an infinitesimal portion of eventual
> mathematical activity. Even if current mathematical activity does not give
> rise to Concrete Mathematical Incompleteness from ZFC, this is a very poor
> indication of whether this will continue to be the case, particularly far
> out into the future.

page 8 in <http://www.math.osu.edu/~friedman/pdf/0EntireBk051411.pdf>

------
timtadh
tldr; Gödel proved the incompleteness of finite axiomatizations of arithmetic.
Further more there is no maximum set of unprovable sentences in arithmetic.
Therefore, there must be interesting and unprovable sentences in arithmetic.

The article discusses a couple such sentences but unfortunately does not
actually state the sentences, which is the interesting part.

Edit: unsurprisingly: "The extent to which we have concrete incompleteness and
what the wider implications are, will not be clear for hundreds of years, if
not longer." -- Friedman

------
JeanPierre
I'm curious: If you add a new axiom in order to make a undecidable proposition
decidable, will this new axiom add more than one undecidable proposition, or
is the number of new undecidable propositions unknown? Or are there infinitely
undecidable propositions for every set of rules?

If the former is true, couldn't we theoretically just apply axioms in order to
solve undecidable propositions which have a practical application if solved?
Or will it practically be hard to find these axioms?

~~~
DaveInTucson
It's been a long time since I had my Theory of Computation class, but as I
recall, the proof that there are computationally undecidable questions uses
the same diagonalization technique[1] used to demonstrate that the real
numbers are not countably infinite.

The same technique can be used to show there are still computationally
undecidable questions if you have a Turing Machine with an Oracle.

This proof technique suggests not only are there undecidable questions, there
are uncountably many of them. Even if you have an Oracle.

[1] c.f. <http://en.wikipedia.org/wiki/Cantors_diagonal_argument>

------
ColinWright
If you like this, you'll like that:

<http://news.ycombinator.com/item?id=2567222>

~~~
pbhjpbhj
Links to HN post on "Searching for the missing truth: Axioms and infinities in
mathematics".

