Date: Jan 24, 2014 6:18 PM Author: fom Subject: Re: An attempt at a finite consistency thesis On 1/24/2014 4:51 AM, fom wrote:

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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

near-fields on 9 elements. The algebra of

these near-fields 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(1-t), [b, s, c]] = [[a, rt(1-s), b], t, [a, rs(1-t), 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 fixed-point argument in

an interval. Given that the Dedekind-Cantor

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 well-founded

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/2003-January/006087.html

and have further noted that it is more closely

related to Noetherian (well-founded) 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 non-fundamental 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 non-fundamental

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 non-Archimedean 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 square-free primes to model my

original set-theoretic 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.

:-)