Piecing together structures: question

[phoenixthoth]

Is it possible to form or "glue together" somehow a structure out of other (not necessarily identical to one another) structures such that each of the other structures is elementarily embeddable within the formed structure?

With ultraproduct formulation and Los's theorem, it appears that only a large number of structures, not all, are elementarily embeddable within the ultraproduct of those structures.

 

[phoenixthoth]

I think I found what I was looking for, what's called a reduced product.

Along the way, I found a structure such that all structures regardless of symbol set are elementarily embeddable within it.

Now for any given structure, what sort of things about that structure should I investigate that would be interesting?

 

[phoenixthoth]

I was wondering if anyone would be interested and willing to read what I have so far..
I would appreciate any feedback.
The highlight is a structure U with the property that all structures, regardless of underlying symbol set, are elementarily embeddable within U.
The strange title of the essay stems from Max Tegmark's mathematical universe hypothesis in which it is posited that reality is isomorphic to a structure.
I hope the attachment thingy works..I've never used it before.

 

[bpet]

I was wondering if anyone would be interested and willing to read what I have so far..
I would appreciate any feedback.
The highlight is a structure U with the property that all structures, regardless of underlying symbol set, are elementarily embeddable within U.
The strange title of the essay stems from Max Tegmark's mathematical universe hypothesis in which it is posited that reality is isomorphic to a structure.
I hope the attachment thingy works..I've never used it before.

The paper is very interesting. As a reader not familiar with NFU sets, one small detail I got stuck on was the concept of $$\in$$ as a relation. In the definition of stratified formulas on page 4 it's referred to as such, yet the Russell's paradox argument of the previous theorem seems to imply that it isn't. I guess, if $$\in$$ isn't a relation, what is it?

 

[phoenixthoth]

That is a very good and if I dare say subtle point.

Perhaps "is an element of" is a relation symbol in the metalanguage to describe NFU that just doesn't correspond to a set in NFU. Would that make sense?

Thinking about a model of NFU. "Is an element of" is a relation symbol in the symbol set of that model that doesn't correspond to a constant symbol (ie, an NFU set) in that model.

?

 

[phoenixthoth]

The abstract of the attached document is still essentially what I want it to say, though let me add some more thoughts.

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 possesses 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.