"Limits To Science: God, Godel, Gravity"
http://www.science20.com/hammock_physicist/limits_science_god_godel_gravity
Here is my comment:
An important question is whether or not a TOE will be finite in
length. I am taking 'TOE' to be, as a working definition, a complete
description of reality or a complete description of everything that
exists. Reality is infinitely vast at least for the reason that it
contains all the integers, not to mention the vastness of the physical
multiverse. So a TOE can be an infinite document. But like the digits
of pi, perhaps this infinitely long document can be computed to
arbitrary precision in a finitely long program, set of instructions.
Then one *might* consider this program which generates a TOE to
arbitrary precision to be "the" TOE, a compression of an infinitely
long document into a finitely long document, thus showing that reality
at its core does not possess the trait of Kolmogorov randomness. Being
that reality contains the uncomputable, it *seems* unlikely that
everything can be finitely describable.
However, I believe that there is a TOE (complete description of
reality) whose *form* can be written down. This TOE has a "shape" to
it, but without specifying any more details than that. It's an
existence proof of a plausible form a TOE could be in. It is roughly
based on Tegmark's article entitled the Mathematical Universe
Hypothesis (available on arxiv.org) which can be broken down to rely
on the axiom that reality is independent of humans which is possibly
controversial.
The argument is made that a TOE can be in the form of a logical
structure which is a tuple consisting of an underlying set, a set of
distinguished constants (like zero), functions (like successor), and
relations (like less than) on this underlying set. Making the
additional assumption that if there is a structure such that *all*
logical structures can be "embedded" within it, then this type of
universality endows such a structure with the same structure as
reality. Thus this sort of ultimate structure would be in an intuitive
sense like ultimate reality. Thus a description of this ultimate
structure would be a description of reality.
To do this, I employ a different-than-usual set theory called NFU
which stands for new foundations with urelements as explained by
Randal Holmes' textbook on NFU. The NFU has been shown to be
consistent which cannot be said of the more famous ZF or ZFC set
theories. The NFU also has a universal set (a set containing all sets)
and a "stratified comprehension theorem" which essentially states that
any object of the form {x : F} where F is any "stratified" formula is
a set in NFU. An example of a *non*-stratified formula F is the
infamous formula used in Russell's paradox: x is not an element of x.
Thus the object considered in Russell's argument isn't a set and from
this argument, no Russell-type contradictions can be derived from the
universal set axiom + stratified comprehension.
Within NFU, it is possible to see that the object which contains *all*
structures is a set. Then one can form the "reduced product" of all
structures, using this set as the index set. One feature of a reduced
product is that it is a logical structure and another feature is that
every structure used to form the product (in this case, every
structure) is embedded within the reduced product.
The reduced product of all structures is the ultimate structure as
described a few paragraphs above.
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