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fom
Posts:
1,969
Registered:
12/4/12


Re: An attempt at a finite consistency thesis
Posted:
Jan 24, 2014 6:18 PM


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.
:)



