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.
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 universe^{1}. 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.
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.
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.
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 Background3 Mathematical
Objects4 Axioms of
ZFC
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 A_{n} 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 program^{2} 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
integers^{3}. 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.
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.
In the following A_{n} 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 A_{n}, the objective active of replacement
is the same as it is in ZF.
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.
Paul J. Cohen. Set Theory and the Continuum
Hypothesis. W. A. Benjamin Inc., New York, Amsterdam,
1966.
Footnotes:
^{1}A 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.
^{2}In 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.
^{3}Initial segments of subsets of the
integers are ordered and thus defined by the size of the
integers.
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