An open problem raised in Tegmark's "ensemble TOE" paper (published in 
'98, if I recall) is to answer the question that is the subject of this 
thread.

I believe recent work I have done has the potential to be a step towards 
that answer.

First, a link to the very rough draft (please forgive formatting 
errors), and second what I think the deficiency is with this document:

http://www.universalsight.org/math/9-26-08/01-03-structure_of_all_structures.pdf

Abstract:
In this document, the author presents a structure with the
property that all structures are elementarily embeddable within it. One 
essen-
tial tool is a version of New Foundations set theory, ?first developed 
by Quine,
as presented by Holmes in [1]. The motivation is given by The Mathematical
Universe article by Max Tegmark, [2], in which it is hypothesized that 
physical
existence is mathematical existence and, consequently, it is 
hypothesized that
the structure with the aforementioned property could be central in the Math-
ematical Universe Hypothesis as being at least keenly connected to the 
literal
universe, if not literally being the universe.
The author assumes some knowledge of mathematical logic such as, for ex-
ample, the inductive de?finition of a fi?rst-order well-formed formula.


The intended audience is primarily Max Tegmark, honestly, but more 
generally, any physicist interested in Tegmark's self-proclaimed 
"bananas articles" like the MUH paper, and who have already been exposed 
to the basics of mathematical logic.

Prior to drafting this document, I contacted Prof. Tegmark regarding the 
core ideas in the draft. I described what I was attempting and if I 
recall I sent him the abstract. As I will describe shortly, this is 
incomplete, so I didn't send him this pdf yet. I hope he wouldn't mind 
my inclusion of his response, which I think many here might find highly 
debatable (and worthy of discussion), sent by email:

<quote>
It sounds to me from what you're saying that A would be the Level IV 
multiverse, i.e., all of physical reality.
</quote>

Now for the deficiency I see with my document. -If- there aren't any 
other errors, then something wrong with my ultimate structure is that it 
is the ultimate structure with respect to just one symbol set. I need a 
structure that is ultimate with respect to -all- symbol sets. The basic 
idea I had which I have not yet tried to formalize is encoding all 
symbol sets into an ultimate symbol set which in human mathematics is a 
countable set; so something like the set of natural numbers will encode 
all possible symbols. One simple way to do this would be to say all 
numbers congruent to 0 mod 3 are encodings of constant symbols, all 
numbers congruent to 1 mod 3 are encodings of n-ary relation symbols, 
and all numbers congruent to 2 mod 3 are encodings of n-ary function 
symbols.
So, note that I did not finish what I set out to do in my abstract: "the 
author presents a structure with the property that all structures are 
elementarily embeddable within it." I believe what I have done is this: 
a structure over a fixed symbol set S with the property that all 
S-structures are elementarily embeddable within it.

Now on to the subject of time.

If Tegmark is correct and an ultimate structure literally is all of 
physical reality, what strikes me is that this ultimate structure 
appears quite static. What then is the source of our perceptions of 
transition, ie, time? This ultimate structure I presume (safely, I 
believe) is constant yet we perceive things to change. Why and how? IOW, 
what is the mechanism that converts the static ultimate structure into a 
fluid appearance of transition? These questions are still valid even if 
the ultimate structure I have in mind is wrong; Tegmark still 
hypothesizes that some math structure is all of physical reality.

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