random

[Lulu of the Lotus-Eaters]

"Alex Martelli" <aleaxit at yahoo.com> writes:
> Moreover, we can now
> derive an infinite number of new theorems of Peano+_I_ that utilize _I_
> in their derivation and that are independent of Peano arithmetic.

|Oh, I didn't know the 'new' theorems will always be infinitely many --
|interesting indeed (is it anything deep or just a trivial by-product of
|considering "_I_ AND (old theorem)" &c as a new theorem and there
|being infinitely many old theorems in arithmetic?).  But definitely

My observation is of the trivial sort you observe.  Probably a lot of
those theorems will have forms other than "_I_ AND (old theorem)"... it
depends on how "good" of an axiom _I_ is in terms of plugging into
proofs.  But being infinite isn't particularly profound.  Then again, it
is quite difficult to distinguish, in a general way, interesting from
uninteresting theorems.

Maybe I went too far with the modelling thing.  Inasmuch as any system
we might create would be written in an finite alphabet, and each
statement would be of finite length, it is trivial to map every
statement to a unique number (it's a little harder to make sure it's
onto, not just into).  And every theorem of the system can thereby be
treated as a fact about numbers.

If that's all you want with a model... arithmetic is plenty.  Actually,
Lowenheim-Skolem even buys you a bit more than this (not that much more,
but it sure is a neat fact that every every system with a model has a
countable model... in the integers, if you like)

|The assertion NOT(<have-complete-info> IMPLIES <can-predict>)
|can never hold, if <have-complete-info> is an impossibility, because
|in that case (<have-complete-info> IMPLIES X) is true for any X,
|so its negation is false. NOT(X IMPLIES Y) *does* imply X is possible.

I don't read the conditional that way.  Doing a quick web search, I find
this URL discussing different types of conditionals:

  http://www.icsi.berkeley.edu/~kay/bcg/lec07.html

The basic point is that conditionals in English (or other natural
languages) are not always material conditionals...  I suppose not even
usually.  The sense of the limitation imposed by the randomness in
quantum states is much more like a epistemic conditional than a material
one.  However one reads it, many counterfactuals should meaningfully be
considered true... even ones where the antecedents are "impossible" in
various ways (there are lots of ways to be impossible too).

So my example:

  NOT (<i-can-lift-1000-lbs> IMPLIES <i-could-lift-my-car>)

is perfectly fine.  A normal speaker considers this statement TRUE (one
she knows that my car weighs more than 1000lbs).  The fact:

  NOT <i-can-lift-1000-lbs>

holds too.  I suppose that number is just on the border of possiblility.
But 2000 lbs is more than any weigh-lifter does, but still less than my
car weighs, so work from there.  So <i-a-human-being-can-lift-2000-lbs>
is in a very real sense *impossible*.

But if you are not happy with mere physical/empirical impossibility, it
works equally well for mathematical impossibles:

  <2+2=5> IMPLIES <2+3=6>

works fine by application of Peano arithmetic.  Using the successor
operator ' and some permitted reasoning:

  Premise: 2+2 = 5
      (1): 2+2'= 5'   # successor of equals maintains equality
      (2): 2+3 = 6    # rewriting terms in conventional manner

The reasoning is valid, so the conditional holds.  The antecedent is in
extremely bad shape--mathematically impossible--on its own.  But so
what?

Yours, Lulu...