Re: compactness in angels/devil problem

On Feb 1, 10:06 pm, David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:
On Fri, 1 Feb 2008 01:29:08 -0800 (PST), pauldepst...@xxxxxxx wrote:
The literature on the angels-and-devil problem often refers to a
"compactness argument" for passing from conclusions about finite
boards to conclusions about the infinite case.  What is this
"compactness argument" and which topology is the compactness concept
being applied to?

What's the angels-and-devils problem?

Quite possibly the compactness being referred to is from
logic: If S is a collection of formulas and every finite subset
of S has a model then S has a model.

I could tell you what compact toplogy that's connected
with, at least in the case of propositional logic, but it
will take a little space. So first tell me what the d/a problem
is and what sort of assertions about the problem you're
talking about - the theorem I have in mind could be irrelevant.

Thank you,

Paul Epstein

David C. Ullrich

Hi David,

Informal version of angels/devil theory: The angel is on a square
chessboard which is infinite in all directions (Z x Z in other
words). For each square of the chessboard, there is a finite set of
squares which the angel can visit on the next move (you get a
different problem or question for each rule determining the finite
set). The devil and the angel take alternate moves. The devil moves
by eating one non-occupied square and therefore preventing the angel
landing on it. Can the devil run the angel out of moves? (As stands,
not a meaningful problem because I haven't given a rule determining
the angel's possible moves.)

Concrete example: Suppose the angel moves like a king in the version
of chess played in the US, England, Eastern Europe and elsewhere (in
fact, the most globally widespread version of chess.) Then the devil
can trap the angel.

Yes, David, thank you so much for your offer of explaining the related
topological notion of compactness. I keenly await. (But don't neglect
the backgammon.)

Paul Epstein
.

Relevant Pages

  • Re: compactness in angels/devil problem
    ... boards to conclusions about the infinite case. ... "compactness argument" and which topology is the compactness concept ... The angel is on a square ... The devil and the angel take alternate moves. ...
    (sci.math)
  • Re: compactness in angels/devil problem
    ... boards to conclusions about the infinite case. ... "compactness argument" and which topology is the compactness concept ...  Thedeviland the angel take alternate moves. ... by eating one non-occupied square and therefore preventing the angel ...
    (sci.math)
  • Re: compactness in angels/devil problem
    ... boards to conclusions about the infinite case. ... "compactness argument" and which topology is the compactness concept ... Now M is a finite set; give M the discrete topology, ... suppose that a given angel moves like a chess king. ...
    (sci.math)
  • Re: compactness in angels/devil problem
    ... For each square of the chessboard, there is a finite set of squares ... the angel take alternate moves. ...  The devil moves by eating one ... non-occupied square and therefore preventing the angel landing on it. ...
    (sci.math)
  • Re: compactness in angels/devil problem
    ... chessboard which is infinite in all directions. ... the angel take alternate moves. ... The devil moves by eating one ... non-occupied square and therefore preventing the angel landing on it. ...
    (sci.math)