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Topic: mathematization of quantifiers, Euler's formula, and two paradoxes
Replies: 8   Last Post: Jan 25, 2014 6:20 PM

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fom

Posts: 1,969
Registered: 12/4/12
mathematization of quantifiers, Euler's formula, and two paradoxes
Posted: Jan 23, 2014 12:53 AM
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There is an outstanding question in
the foundations of mathematics
concerning the linear or successive
arrangement of large cardinal axioms
in set theory.

This post offers a thesis in answer
to that question.

Let me observe that I have spent some
time considering arithmetic this
year. My axioms would be useless
to the general practitioner. However,
they are based on a view that emphasizes
cardinality over ordinality. And, this
has motivated the interpretations given
here.

In summary, the principal relation is
that of "proper division". A notion
of "monadic division" is expressed using
a non-well-founded relation. Monadic
divisors are the multiplicative
identity and the primes. One may consider
the monadic divisors to be similar to
Quine atoms, although I have not really
investigated whether or not this can be
implemented in NFU.

This methodology has permitted me to
view Peano arithemetic as primarily
expressing the ordinal structure essential
to the metamathematical use of numbers
as nominal labels. What follows will
be an analysis of how the duality between
these two views of natural numbers may
be seen to coincide with the duality
of quantifiers.

It is important to remember that the
nineteenth century had Boole and Pierce
"mathematizing" logical quantifiers
before Frege and Russell attempted to
"logicize" mathematics. Boole's work is
well-known. He introduced an interpretation
of addition and multiplication which we
now associate with union and intersection
in set-theoretic Boolean algebras such as
in a power set.

But, Pierce's work -- to the best of my
knowledge -- introduced the idea of using
a Sigma for existential quantification and
a Pi for universal quantification. This
notation is still used today in the classification
hierarchies when indicating the first
quantifier in a string of quantifiers.

With regard to a logical domain, these
quantifiers stipulate membership differently.

In the case of Pi, there is a "course-of-values"
specification:

Name_1 and Name_2 and ...

In the case of Sigma, there is a disjunction
of variable assignments for a one-variable
language:

( x = Name_1 ) or ( x = Name_2 ) or ...

Now, before going further, one may consider
the Goedel incompleteness theorems in their
original context with respect to Hilbert's
program. Hilbert had already formulated an
explanation of geometry indicating that geometry
would be consistent relative to arithmetic.
And, conceding the indemonstrability of a
completed infinity, Hilbert hoped to demonstrate
the consistency of mathematics through a
metamathematical arithmetization. Goedel's
incompleteness theorems demonstrate a fundamental
relationship between logic and arithmetic
that thwarts Hilbert's original hopes.

Now consider the Euler product formula
described in the link:

http://en.wikipedia.org/wiki/Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function#The_Euler_product_formula

Let me express this formula with the
expression,

Sigma_(n=1...oo) [1/n^s] = Pi_(n=1..oo) [1/(1-p_n^-s)]

I have added the metamathematical indexing
of the primes which does not occur in
the link.

Consider the statement of this formula
under iterations where s=0,


1/1 =/= 1/(1-1)

1 =/= oo [=omega]



1/1 + 1/1 =/= 1/(1-1) + 1/(1-1)

1 + 1 =/= oo * oo [=omega^2]



1/1 + 1/1 + 1/1 =/= 1/(1-1) + 1/(1-1) + 1/(1-1)

1 + 1 + 1 =/= oo * oo * oo [=omega^3]


:
:

omega = omega * oo [=omega^omega]


Let me empasize that my statements are not
about the mathematical correctness of the
symbols used.

The fact is that arithmetic grounded the
interpretation of quantifiers. This particular
formula expresses certain notions when s=0
that are familiar to logicians.

First, there is the sorties paradox:


1 grain of wheat does not make a heap.
If 1 grain of wheat does not make a heap then 2 do not.
If 2 grains of wheat do not make a heap then 3 do not.
:
:
If 9,999 grains of wheat do not make a heap then 10,000 do not.
----------------------------------------------------------------
10,000 grains of wheat do not make a heap.



That is, no finite iteration will lead to
equality when the actual mathematics of
the formula is taken into account.

The second, of course, is Zeno's paradox.

In order for the Sigma expression on the
left to be equal to the Pi expression on
the right, the completed infinity must be
assumed.

This is precisely the outcome of foundational
investigations such as "Principia Mathematica".

So, why is this account to be considered
as significant?

First, quantifiers had been reinterpreted
with respect to arithemtical operations.
This is a specific formulation -- a single,
definite instance -- that relates the
mathematical reinterpretation to the appropriate
paradoxes.

Second, the analysis is dual with respect
to an ordinal sequence of succession and
a cardinal system of aliquot parts. What
is asymmetric is the use of the ordinal
sequence as a metamathematical index. But,
this is a nominal use and not an arithmetical
use.

In general, mathematicians conflate this
metamathematical use. There is no criticism
in this account of that practice. To the
contrary, if one takes primes as "urelements"
in the sense of monadic divisors as discussed
above, then the unique prime factorization
theorem depends upon the metamathematical
use of the ordinal notation as exponents.

The assertion here is that this particular
formula expresses the difference between
ordinal succession, cardinal aliquot parts,
and nominal metamathematical usage while
simultaneously relating the symbols by
which quantification had been mathematically
interpreted.

On this thesis, then, the observed linearity
or successive ordering of the large cardinal
axioms is a consequence of the duality
expressed in this specific instance asserted
to correspond with the interpretations of
Boole and Pierce.

Suppose that one next considers the case
that s=1,

http://en.wikipedia.org/wiki/Proof_of_the_Euler_product_formula_for_the_Riemann_zeta_function#The_case

As explained in the Wikipedia link, the
Riemann zeta function diverges at this
value for s. In effect, this is an
analytical proof of the infinitude of
prime numbers. That is also explained
in the link.

What is of specific interest in this case
is the role of the numbers (p_k - 1) in


... (1 - 1/5)(1 - 1/3)(1 - 1/2) * zeta(1) = 1

... (4/5)(2/3)(1/2) * zeta(1) = 1


Because the axioms used in the arithmetic that
I have been using are defined with respect to
circular reference, their semantics is best
understood with respect to the revision theory
described by Gupta in the link,

http://plato.stanford.edu/entries/definitions/#CirDef

If you examine Gupta's description, he explains
its interpretation as generated sequence.

V, delta(V), delta(delta(V), ...

Please compare this with the discussion of a fixed point for
pointed directed complete partial orders,

http://en.wikipedia.org/wiki/Directed_complete_partial_order#Properties

Whereas the wikipedia article speaks in terms of an
object and a set, note that the order relation is
a transitive, reflexive order. In Moschavakis monograph
on inductively defined structures, what is important is
that a monotone operator on a power set preserves the
subset relations. My formulations are all based on
a strict transitive order, a grammatical equivalence
asserting a Leibnizian principle of identity of indiscernibles,
and an asserted identity based on existence assertions.
This is non-standard, However, it is not unreasonable.

So that one might understand that the general idea
here is not without precedent, I managed to find
this paper discussing some of Feferman's work with
inductively defined structures.

http://www1.maths.leeds.ac.uk/~rathjen/fest.pdf

What is without precedent is my analysis of
arithmetic that suggests the thesis I have offered
here.

Now, recently I became aware of Wilson's theorem in an
unexpected way. The reason I emphasized the role of
the numbers (p_k - 1) in


... (1 - 1/5)(1 - 1/3)(1 - 1/2) * zeta(1) = 1

... (4/5)(2/3)(1/2) * zeta(1) = 1


is because a notion of arithmetic which divorces
ordinal sequencing from cardinal aliquot parts
must have some sense of how the two form are
bound to one another.

Very often mathematicians use 0 to identify
distinguished objects. And, with respect to
inductive defintions, the notion of fixed points
come into play. But, this is not possible
with the basic idea of a base point and succession.

If one looks at Wilson's theorem,

http://en.wikipedia.org/wiki/Wilson%27s_theorem

one has that an arbitrary natural number n
is a prime if and only if

(n-1)! = -1 mod(n)

Now, consider first that the arithmetical
theory in question contains no 0. So,
modular arithmetic is not intrinsic to its
formulation.

Next, consider the usual theory which may
given an interpretation using string
monoids. That interpretation is based
on taking an "empty string" as the identity
element.

If one imagines a metamathematical index
beginning with '0', then one may imagine
its metametamathematial "empty string" as
corresponding ot '-1'

The point here involves an analogy with
set theory and limit ordinals. The
empty set is a limit ordinal by a vacuous
interpretation of the definition. And,
with the axiom of choice, the entirety
of transfinite arithemetic consists of
unit succession generating limit ordinals.

The suggestion here is that the
primes have a similar relationship
in the naturals that are expressed
in the modular arithmetic of Wilson's
theorem.

There is not much more to be said.

This cannot be proven. But, it can
be offered as a thesis if one remembers
that the original relationship between
mathematics and logic had interpreted
logical quantifiers arithmetically.
The analysis here suggests that a
single identity evaluated at 0 and
at 1 conforms with the duality expressed
between quantifiers and the duality
expressed between ordinal interpretation
and cardinal interpretation. In
addition, the basic paradoxes associated
with quantifier interpretation are
given an account.





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