For example, the orthomodular lattice extends into a separable Hilbert space because the set of closed subspaces of this Hilbert space is lattice isomorphic to the orthomodular lattice.
This question is important because if such founding structures exist, then it is possible to derive the structure of physical reality from one founding structure or a small set of founding structures.
This again is important because it becomes apparent that the lower levels of physical reality are inaccessible to investigation via experiments.
One of the problems with the example is that the Hilbert space applies a number system for the specification of its inner product. That number system must be a division ring.
If the answer is positive, then physical reality needs only to incorporate these founding structures to also incorporate many parts of mathematics.
https://docs.com/hans-van-leunen
First, Hilbert spaces (in QM for example) are state spaces, not physical reality, so they do not have the same properties, they are just intermediate mathematical modeling tools for abstraction.
When you try to give a mathematical model to the real physical space, you will inevitably end up with some sort of chart concept to link your physical world model to a know mathematical structure and become able to define notions like derivability and integrability on the physical world (using transition maps, push-forwards and pull-backs). This is basically a way to encapsulate and control the arbitrary part of your theory.
By doing this your physical reality space will not be based on a number field like R or C but rather on some function space, that is, chart-transition maps, which indeed will be a division ring. This leads to the fact that real physical world fields are not vector spaces in the common sense but become moduli, that is a generalisation of a vector space over a ring, not over a number field. This is the construction that we have in differential geometry to define bundle theory and their so-called sections, that is fields.
If (Quantum and/or Relative) field theories, either in our current set of knowledge or in any kind of extended formalism, are considered the foundation of physical world, then the answer to your question should be yes, because this whole construction is indeed derived from pure set theory, without any arbitrary element. The construction does happen to cover many parts of mathematics.
You can find a very detailed course of several lectures describing this construction for both relativistic and quantum theories by looking up for Frederic Schuller, which is also an excellent teacher ;)
Hilbert spaces require number systems that are division rings for the specifications of its inner products. This means that the choice is restricted to the real numbers, the complex numbers and the quaternions.
Each infinite dimensional quaternionic separable Hilbert space owns a non-separable companion Hilbert space that embeds the separable Hilbert space. This combination suits as a proper model in which the Hilbert spaces act as a repository for athe dynamic geometric data of all elementary particles. Elementary particles are elementary modules that constitute all other modules and modular systems.
By defining reference operators that use the rational elements of number systems as their eigenspace, these reference operators supply parameter spaces to functions and the function values define a new category of defined operators that supply sampled continuums as their eigenspaces. If a similar trick is done to the non-separable Hilbert space, then function theory is merged with Hilbert space operator technology. The purely mathematical Hilbert Book Model applies this approach.
https://doc.co/kQjP6b
Microsoft ended doc.com. So you better read
TheStructureOfPhysicalReality; http://dx.doi.org/10.13140/RG.2.2.10664.26885
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