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.
--~--~---------~--~----~------------~-------~--~----~
You received this message because you are subscribed to the Google Groups "Everything List" group.
To post to this group, send email to everything-list.domain.name.hidden
To unsubscribe from this group, send email to everything-list+unsubscribe.domain.name.hidden
For more options, visit this group at
http://groups.google.com/group/everything-list?hl=en
-~----------~----~----~----~------~----~------~--~---
Received on Fri Sep 26 2008 - 11:55:28 PDT