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Topic: An attempt at a finite consistency thesis
Replies: 2   Last Post: Jan 24, 2014 9:14 PM

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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
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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
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.


:-)







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