More flying rabbits and dragons

From: Alastair Malcolm <amalcolm.domain.name.hidden>
Date: Sat, 16 Oct 1999 09:03:17 +0100

I would be interested to know if anyone can find any problems with the following new solution to the much discussed flying rabbit problem (discussed below as the dragon universe problem). Here the measure is initially based on the symbols comprising the formal systems which include the mathematical structures specifying universes (as in Tegmark's paper).

The starting point is all possible symbol strings containing all possible symbols. Now, bearing in mind that any symbol can have any interpretation, we can start to select those strings which contain symbols that can be interpreted in their entirety as well-formed formulae (wff's). Of these we can select those which comprise mutually consistent axioms (both directly, and after taking into account derived theorems via standard rules of inference), connected by an 'AND' (that is, a string of the form 'Axiom1 AND Axiom2 AND ...'). We can also strike out any cases of duplicate axioms.

We now have a list of symbol strings that are interpretable as axiom sets of all possible consistent theories (theorem sets). Since all axioms are representable by an (anthropically unbiassed) subset of all possible combinations of symbols, then we should not lose generality by scaling up our measure from symbols to axioms. Now, since the axioms for a TOE completely specify our universe (and perhaps others), it is clear we cannot just add extra axioms to specify a dragon event, because that would lead to an inconsistency (theorems derivable from the TOE would specify a non-dragon event at that time and place). However it is possible to have a set of axioms that would specify the path of our visible universe, but with a visible dragon event - a dragon universe theory; this would have to be represented by a much longer axiom string than the TOE set.

Now, as an extreme approximation (all that is necessary here), we can say that our TOE requires n axioms, the dragon universe 2n axioms. In comparing all possible theories containing a finite number of axioms, then it is reasonable to suppose that occurences of TOE's will vastly outnumber dragon universe theories, once functionless axioms are taken into consideration: consider all strings up to m axioms in length, for large finite m; there will be n more surplus axioms for the TOE than for the dragon universe theory (m-n as opposed to m-2n), and so bearing in mind that each axiom can contain an infinite variety and number of different symbols, and that each axiom (functionless or not, participating in the specification of another universe or not), contributes to the measure, then there will be far more combinations of m axioms that include our TOE than is the case for dragon universe theories, with the result that we are far more likely to be in a simpler (TOE-based) universe - dragon universes are not more probable.

(Two further points here: firstly, the outnumbering of dragon universes is maintained as m increases to infinity - simply take a finite random sample; secondly, if one wishes to take into consideration all possible paranormal events, one also needs to take into consideration the much larger number of equal-axiom-number theories that specify our universe, but with invisible paranormal events, events in other universes and so on - these will vastly outnumber visible-paranormal-event theories.)

Alastair
Received on Sat Oct 16 1999 - 01:36:00 PDT