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fom
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mathematization of quantifiers, Euler's formula, and two paradoxes
Posted:
Jan 23, 2014 12:53 AM


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 nonwellfounded 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 wellknown. He introduced an interpretation of addition and multiplication which we now associate with union and intersection in settheoretic 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 "courseofvalues" specification:
Name_1 and Name_2 and ...
In the case of Sigma, there is a disjunction of variable assignments for a onevariable 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/(1p_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/(11)
1 =/= oo [=omega]
1/1 + 1/1 =/= 1/(11) + 1/(11)
1 + 1 =/= oo * oo [=omega^2]
1/1 + 1/1 + 1/1 =/= 1/(11) + 1/(11) + 1/(11)
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 nonstandard, 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
(n1)! = 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.



