Le 16-avr.-05, à 01:21, Jonathan Colvin a écrit :

At first glance that would seem to be the case. But isn't there a problem?

If we consider worlds to be the propositions of formal systems (as in

Tegmark), then by Godel there should be unprovable propositions (ie. worlds

that are never instantiated). This seems in direct contradiction to the

actual existence of everything conveivable, does it not?

`Are you sure Tegmark identify worlds with propositions of FS? Anyway,`

what logicians (and modal logicians in particular) are used to do is to identify

worlds with maximal consistent sets of propositions (or sentences). Then you

can extract from Godel that any FS can be instantiated in alternative worlds.

For example if you take a typical FS like Peano Arithmetic, the proposition that

PA is consistent is undecidable. This means that there is at least two

maximal consistent sets of propositions extending the set of theorems of PA:

one with the proposition that PA is consistent and one with the proposition

that PA is inconsistent. In that sense the non provable propositions are

instantiated in worlds. In general worlds are not effective (computable) objects:

we cannot mechanically (even allowing infinite resources) generate a world.

What we need to do is to put a measure on those maximal consistent extensions.

(maximal = can no more be extended without making the world inconsistent (containing

a contradictory proposition).

what logicians (and modal logicians in particular) are used to do is to identify

worlds with maximal consistent sets of propositions (or sentences). Then you

can extract from Godel that any FS can be instantiated in alternative worlds.

For example if you take a typical FS like Peano Arithmetic, the proposition that

PA is consistent is undecidable. This means that there is at least two

maximal consistent sets of propositions extending the set of theorems of PA:

one with the proposition that PA is consistent and one with the proposition

that PA is inconsistent. In that sense the non provable propositions are

instantiated in worlds. In general worlds are not effective (computable) objects:

we cannot mechanically (even allowing infinite resources) generate a world.

What we need to do is to put a measure on those maximal consistent extensions.

(maximal = can no more be extended without making the world inconsistent (containing

a contradictory proposition).

bruno

http://iridia.ulb.ac.be/~marchal/