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Re: An attempt at a finite consistency thesis
Posted:
Jan 24, 2014 9:14 PM


On 1/24/2014 3:18 PM, fom wrote: > On 1/24/2014 4:51 AM, fom wrote: >> > > <snip> > > > I found a post this morning that makes it > a somewhat easy to explain the analysis > given at the top of the thread. In the > link, > > https://groups.google.com/forum/#!original/sci.math/0plz5618K4c/eoCyGg9Sa7IJ > > > > 'federation2005' describes axioms for > ternary algebras for affine spaces over > a field. The axioms fail for 2 element > fields and 3 element fields. > > In the case of a 2 element field, he > observes that vector spaces over 2 > element fields are Boolean rings. > > Now, by recognizing that the 16 basic > Boolean functions relate to one another > as a finite affine geometry and by > using the Miracle Octad Generator to > situate them, they became naturally > organized as a finite vector space. > > This is important because Boolean rings > satsify the usual algebraic axioms of > ring theory. So, the analysis introduces > a reduction to algebraic forms. > > As for the 3 element field, the projective > planes of order 9 are built using the > nearfields on 9 elements. The algebra of > these nearfields are similar to the structure > of the complex numbers in that they have > real and imaginary components. The real > components are comprised of the 3 element > field, {1, 0, 1}. 'federation2005' discusses > this field in his post. > > The axiom which presents problems for > 'federation2005' with respect to the > 3 element field is A3 because the needed > associativity law for the ternary algebras > cannot be proven for that field. > > A1: [a, 0, b] = a > > A2: [a, 1, b] = b > > A3: [a, rt(1t), [b, s, c]] = [[a, rt(1s), b], t, [a, rs(1t), c]]. > > In describing his axioms, 'federation2005' describes > the ternary product of the algebra with the > parenthetical remark, > > "(to be thought of as the affine operation (1  r)a + rb)" > > Anyone familiar with analysis know that this is > an analytic expression for convexity. > > Now, Bolzano's proof of the intermediate value > theorem relies on a fixedpoint argument in > an interval. Given that the DedekindCantor > program for an arithmetic of limits asserts > an object identity for points, the resolution > of the intermediate value theorem relies > upon a 3 element system. This is where the > logic and the algebra meet. > > Now, on my account of arithmetic using relations, > the asserted identity associated with wellfounded > induction had been given through the axioms, > > > ExEy( Az( x mdiv z ) /\ S(x,x,y) ) > > > AxAy( x = y <> Au( Av( u mdiv v ) > Ev( S(u,x,v) <> S(u,y,v) ) ) ) ) > > > > I have elsewhere noted Dana Scott's remarks on > this view of natural numbers, > > http://www.cs.nyu.edu/pipermail/fom/2003January/006087.html > > and have further noted that it is more closely > related to Noetherian (wellfounded) induction > and the descending chain condition, > > http://en.wikipedia.org/wiki/Noetherian_induction > > http://en.wikipedia.org/wiki/Noetherian_topological_space > > In the present circumstance, the import of this > axiom lies in the position that the metamathematical > use of ordinal numbers must be taken into account. > So, in keeping with an existence assertion that > requires 2 objects to be asserted simultaneously, > one has also that these natural numbers index the > first two primes, > > p_1 = 2 > > p_2 = 3 > > In effect, then, the entire construction involves a > reduction to algebraic analysis with respect to the > two finite fields that cannot satsify the ternary > algebra axioms described by 'federation2005'. > > Now, this ought not be surprising. > > Although Chang and Keisler refer to model theory > as "logic + universal algebra", it is my > understanding that Wilfred Hodges has refered > to it more along the lines of "algebraic geometry". > I suspect that such a view reflects the influence > of stability thoery. I have no knowledge of > sability theory beyond the fact that it introduces > geometric notions into model theory. > > In the context of a reduction to algebraic > notions, it is then much easier to explain > the expression of the Heegner numbers. > > The foundations of mathematics of the late > nineteenth century had been heavily focused > on definitions for natural numbers. Although > Frege ended up using 0 as a base for his > definition, he gave extensive arguments > concerining the nature of units. > > The Heegner numbers are characterized in the > context of the class number conjectures of > Gauss, > > http://en.wikipedia.org/wiki/Class_number_problem#Status > > As can be seen, there are 9 fundamental discriminants > that correspond to the Heegner numbers. In addition, > the class number 1 has 4 nonfundamental discriminants > > These are two of the cardinal numbers that have been > associated with the invariants of this construction. > > In their positive formulation, > > http://en.wikipedia.org/wiki/Heegner_number > > they are comprised of either the multiplicative > identity or are prime. Hence, they are represented > in the arithemtical constructions that have > been proposed in support of this analysis. > > The distinction between fundamental and nonfundamental > discriminants can be found by following the > relevant links: > > http://en.wikipedia.org/wiki/Imaginary_quadratic_field#Discriminant > > http://en.wikipedia.org/wiki/Fundamental_discriminant > > http://en.wikipedia.org/wiki/Absolute_value_%28algebra%29 > > I fully confess that while I have the knowledge > to understand these links, it has been so long > since I studied the material, I have no express > knowledge of these matters. > > With respect to the absolute value link (pertaining > to Archimedean and nonArchimedean embeddings) let > me observe that one can follow the definitions from > norms on algebraic forms to their reliance on > absolute values in Serge Lang's "Algebra". I know > this from searching for the source of trivial metrics > and trivial norms because of the choice axioms and > the bases for vector spaces. > > To conclude this posting, let me provide the definition > for the Moebius function whose codomain is the > 3 element field. It is defined with respect to > unique prime factorizations involving square free > prime numbers, > > http://en.wikipedia.org/wiki/M%C3%B6bius_function#Definition > > > As discussed in the link, > > > http://mathforum.org/kb/plaintext.jspa?messageID=7945648 > > > I used a system of squarefree primes to model my > original settheoretic axioms. Because I had been > using circular reference, I had been concerned about > paradoxical outcomes. So, I had searched for a > model of the axioms based on circular definitions. > > If one associates the square free primes with the > sequence of exponents in prime factorizations > as I did in the link, > > > http://mathforum.org/kb/plaintext.jspa?messageID=9344636 > > > then one could interpret the Moebius function in > the following sense. > > In the following representations for continued > fractions, let the parity distinction between > "odd" length sequences and "even" length sequences > be accommodated with the use of 0. What is being > expressed in what follows is a convergence to > the golden ratio and its inverse. These are > significant, once again, to the symmetries > associated with the Mathieu groups and the > 24 loci of the MOG array. > > Here are some links: > > > http://en.wikipedia.org/wiki/Mathematical_constants_%28sorted_by_continued_fraction_representation%29 > > > http://en.wikipedia.org/wiki/Binary_icosahedral_group#Elements > > http://en.wikipedia.org/wiki/Icosian > > > And, here is the interpretation described > above with respect to parities, > > [0;] > [1;] > > [0;1] > [1;1] > > [0;11] > [1;11] > > [0;111] > [1;111] > > [0;1111] > [1;1111] > > and so on. > > > :) > > > >
I'd hope I'd have a good comment for you. There is one.
http://en.wikipedia.org/wiki/Mathematical_constants_%28sorted_by_continued_fraction_representation%29
I expect you to make factual sense as you do.
Here, the, the Moebius function is a rather general transform in what it preserves, then having unsigned data and with that. There is what is preserved and what is not.
Then you describe general linear constructions of terms as counting.



