      mijcar Quantum Meta-Fields  Home | Wave-Particle Theory | Quantum Meta-Fields | Real World Mathematics -- An Alternative View | Arithmetic and the Real World - An Imperfect Fit | An Inexact Reality | Windows 10 What is a Quantum Meta-Field?

Quantum Meta-Field over the real numbers:

A set R* is a Quantum Meta-Field over the real numbers iff

- Any member M of R* is a set that meets the following conditions:

- - M is a subset of the real numbers and for any m in M, there is a "next" m* (that is, there is no member of M that is between m and m*).

Postulate: R* is well-defined and of order c.

Theorems that are obvious:

- Any finite subset of the real numbers is a member of R*.

- Any subset of the integers is a member of R*.

- For any given real e>0, there is a set E that is a member of M and which has the properties such that any real number is within e of a member of E.

- For any given real e>0 and any finite subset S of the real numbers, there is a finite member F of M such that for any member s of S, there is a member of F that is within e of s.

- For any given e>0 and any finite math problem (i.e., finite numbers and finite operations) there are an infinite number of finite subsets P of M such that the numbers of the problem are members of P and that the solution if within e of some member of P.

- For any finite real world system with a finite number of computations, there is an infinite number of finite members of M that will solve that problem within any desired degree of accuracy.

Questions: Can one set e>= 0? Yes, in most cases (I say that only to be cautious -- I have not yet conceived of any situation where the answer is "no").

However, there is no real world situation in which e would equal 0.  Reply and/or comment to: (c) Michael J. Carroll [March, 2018]