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Formalizing objective mathematics

Paul Budnik
Mountain Math Software
paul@mtnmath.com

Copyright © 2010 Mountain Math Software
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This web page is intended for those with some knowledge of the foundations of mathematics. It should provide the information needed to understand this proposal. Related information is available through linked pages to clarify the argument and make it accessible to a wider audience.

1  Introduction

Objective mathematics attempts to distinguishes between statements that are objectively true or false and those that are only true, false or undecidable relative to a particular formal system. This distinction is based on the assumption of an always finite but perhaps potentially infinite universe1. This is a return to the earlier conception of mathematical infinity as a potential that can never be realized. This is not to ignore the importance and value of the algebra of the infinite that has grown out of Cantor's approach to mathematical infinity. It does suggest a reinterpretation of those results in terms of the countable models implied by the Löwenheim-Skolem theorem. It also suggests approaches for expanding the foundations of mathematics that include using the computer as a fundamental research tool. These approaches may be more successful at gaining wide spread acceptance than large cardinal axioms which are far removed from anything physically realizable.

1.1  The mathematics of recursive processes

A core idea is that only the mathematics of finite structures and properties of recursive processes is objective. This does not include uncountable sets, but it does include much of mathematics including some statements that require quantification over the reals[2]. For example, the question of whether a recursive process defines a notation for a recursive ordinal requires quantification over the reals to state but is objective.
Loosely speaking objective properties of recursive processes are those logically determined by a recursively enumerable sequence of events. This cannot be precisely formulated, but one can precisely state which set definitions in a formal system meet this criteria (see Section 5).
The idea of objective mathematics is closely connected to generalized recursion theory. The latter starts with recursive relations and expands these with quantifiers over the integers and reals. As long as the relations between the quantified variable are recursive, the events that logically determine the result are recursively enumerable.

1.2  The uncountable

It is with the uncountable that contemporary set theory becomes incompatible with infinity as a potential that can never be realized. Proving something is true for all entities that meet some property does not require that a collection of all objects that satisfy that property exists. Real numbers exist as potentially infinite sequences that are either recursively enumerable or defined by a non computable, but still logically determined, mathematical expression. The idea of the collection of all reals is closely connected with Cantor's proof that the reals are not countable. For that proof to work reals must exist as completed infinite decimal expansions or some logically equivalent structure. This requires infinity as an actuality and not just a potential.
This paper is a first attempt to formally define which statements in Zermelo Frankel set theory (ZF)[4] are objective. The goal is not to offer a weaker alternative. ZF has an objective interpretation in which all objective questions it decides are correctly decided. The purpose is to offer a new interpretation of the theory that seems more consistent with physical reality as we know it. This interpretation is relevant to extending mathematics. Objective questions have a truth value independent of any formal system. If they are undecidable in existing axiom systems, one might search for new axioms to decide them. In contrast there is no basis on which relative questions, like the continuum hypothesis, can be objectively decided.

1.3  Expanding the foundations of mathematics

Defining objective mathematics may help to shift the focus for expanding mathematics away from large cardinal axioms. Perhaps in part because they are not objective, it has not been possible to reach consensus about using these to expand mathematics. An objective alternative is to expand the hierarchy of recursive and countable ordinals by using computers to deal with the combinatorial explosion that results[3]. (To learn more about this approach see the ordinal calculator page.)
Throughout the history of mathematics, the nature of the infinite has been a point of contention. There have been other attempts to make related distinctions. Most notable is intuitionism stated by Brouwer[1]. These approaches can involve (and intuitionism does involve) weaker formal systems that allow fewer questions to be decidable and with more difficult proofs. Mathematicians consider Brouwer's approach interesting and even important but few want to be constrained by its limitations. A long term aim of the approach of this paper is to define a formal system that is widely accepted and is stronger than ZF in deciding objective mathematics.
2  Background
3  Mathematical Objects
4  Axioms of ZFC

5  The Objective Parts of ZF

Objectivity is a a property of set definitions. Its domain is expressions within ZF (or any formalization of mathematics) that define new sets (or other mathematical objects). A set is said to be objective if it can be defined by an objective statement.
The axiom of the empty set and the axiom of infinity are objective. The axiom of unordered pairs and the axiom of union are objective when they define new sets using only objective sets. The power set axiom applied to an infinite set is not objective and it is unnecessary for finite sets.
A limited version of the axiom of replacement is objective. In this version the formulas that define functions (the An in this paper) are limited to recursive relations on the bound variables and objective constants. Both universal and existential quantifiers are limited to ranging over the integers or subsets of the integers. Without the power set axiom, the subsets of the integers do not form a set. However the property of being a subset of the integers
S(x) ≡ ∀y y ∈ x → y ∈ ω
can be used to restrict a bound variable.
Quantifying over subsets of the integers suggests searching through an uncountable number of sets. However, by only allowing a recursive relation between bound variables and objective constants, one can enumerate all the events that determine the outcome. A computer program that implements a recursive relationship on a finite number of subsets of the integers must do a finite number of finite tests so the result can be produced in a finite time. A nondeterministic computer program2 can enumerate all of these results. For example the formula
∀r S(r) → ∃n (n ∈ ω  → a(r,n))
is determined by what a recursive process does for every finite initial segment of every subset of the integers3. One might think of this approach as a few steps removed from constructivism. One does not need to produce a constructive proof that a set exists. One does need to prove that every event that determines the members of the set is constructible.

6  Formalization of the Objective Parts of ZF

Following are axioms that define the objective parts of ZF as outlined in the previous section. The purpose is not to offer a weaker alternative to ZF but to distinguish the objective and relative parts of that system.

6.1  Axioms unchanged from ZF

As long as the universe of all sets is restricted to objective sets the following axioms are unchanged from ZF.
  1. Axiom of extensionality
    ∀x ∀y    (∀z    z ∈ x ≡ z ∈ y) ≡ (x=y)
  2. Axiom of the empty set
    ∃x ∀y    ¬(y ∈ x)
  3. Axiom of unordered pairs
    ∀x ∀y   ∃z  ∀w   w ∈ z ≡ ( w = x ∨w = y )
  4. Axiom of union
    ∀x ∃y   ∀z   z ∈ y ≡ (∃t   z ∈ t ∧t ∈ x)
  5. Axiom of infinity
    ∃x  ∅ ∈ x ∧[∀y  (y ∈ x) →(y ∪{y} ∈ x)]

6.2  Objective axiom of replacement

In the following An is any recursive relation in the language of ZF in which constants are objectively defined and quantifiers are restricted to range over the integers (ω) or be restricted to subsets of the integers. Aside from these restrictions on An, the objective active of replacement is the same as it is in ZF.

B(u,v) ≡ [ ∀y (y ∈ v ≡ ∃x [ x ∈ u ∧An(x,y)])]

[ ∀x ∃ ! y An(x,y) ] → ∀u ∃v (B(u,v))
7  An Objective Interpretation of ZFC
8  A Creative Philosophy of Mathematics

References

[1]
L. E. J. Brouwer. Intuitionism and Formalism. Bull. Amer. Math. Soc., 20:81-96, 1913.
[2]
Paul Budnik. What is Mathematics About? In Paul Ernest, Brian Greer, and Bharath Sriraman, editors, Critical Issues in Mathematics Education, pages 283-292. Information Age Publishing, Charlotte, North Carolina, 2009.
[3]
Paul Budnik. A Computational Approach to the Ordinal Numbers. Mountain Math Software, Los Gatos, CA, 2010.
[4]
Paul J. Cohen. Set Theory and the Continuum Hypothesis. W. A. Benjamin Inc., New York, Amsterdam, 1966.

Footnotes:

1A potentially infinite universe is one containing a finite amount of information at any point in time but with unbounded growth over time in its information content.
2In this context nondeterministic refers to a a computer that simulates many other computer programs by emulating each of them and switching in time between them in such a way that every program is fully executed. The emulation of a program stops only if the emulated program halts. The programs being emulated must be finite or recursively enumerable. In this context nondeterministic does not mean unpredictable.
3Initial segments of subsets of the integers are ordered and thus defined by the size of the integers.


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