[FOM] Logically determined
Paul Budnik
paul at mtnmath.com
Mon May 31 17:43:06 EDT 2010
2+2=4 is true in all universes with cardinality of at least 4 assuming
one is using those symbols to refer to standard integers. In contrast
parallel lines may or may not meet in a given universe depending on the
geometry of that universe. This illustrates what I mean by logically
determined. Once a geometry is specified, the fate of parallel lines may
become logically determined.
The consistency of a formal system is logically determined by its
axioms. The axioms are either consistent or inconsistent. Gödel proved
that one may not be able to decide this question from those axioms. Thus
the intuitive idea of logically determined must mean something more than
deducible from axioms. Gödel proved the incompleteness of logic as well
as mathematics.
Gödel also pointed the way to a deeper understanding of what we mean by
logically determined. The axioms of a formal system determine not only
its consistency but also its omega consistency. An example of an omega
inconsistent system is one that proves a particular Turing Machine (TM)
halts that does not halt. The axioms of the system say something about a
recursively enumerable sequence of events that is not true.
A large fragment of mathematics is composed of statements that say
something about a recursively enumerable sequence of events that would
seem to be objectively true or false. For example the arithmetical
hierarchy can be constructed from iterating the question does a TM have
an infinite number of outputs, does it have an infinite number of
outputs an infinite subset of which are the Gödel numbers of TMs that
themselves have an infinite number of outputs, etc. One can iterate
this up to any recursive ordinal and construct the hyperarithmetical
hierarchy. Even some questions that require quantification over the
reals can be interpreted as referring to a recursively enumerable
sequence of events. One example would be asking if a TM defines a
notation system for a subset of the recursive ordinals.
I suggest that logically determined statements are those that make an
objective statement about a recursively enumerable sequence of events.
The catch is defining which statements (and thus which relationships
between events) are objective. Since logic is provably incomplete this
cannot be precisely defined. It is a philosophical idea not a
mathematical assertion. It is a suggested refinement to the notion of a
Platonic ideal reality. It is based on contemporary mathematics and
computer science and on the old idea that infinity refers to an
unlimited and thus unrealizable potential.
This approach puts some statements in ZF about uncountable sets in the
category of Euclidean geometry. They are not objectively true or false
but have an objective interpretation relative to specific formalizations
of mathematics. The provably definable objects in formal systems
constructed by finite beings have definitions that are recursively
enumerable as are the provable relationships between those objects.
Mathematics, as a cultural institution, is a great many things. We
arrive at a deeper understanding through many divergent and even at
times mutually inconsistent paths. However there seems to be an
underlying ideal of irrefutable truth in an uncertain world that
mathematics strives for. Thus I suggest that ideally mathematics is
about formalizing which conclusions are logically determined by
assumptions. This includes both choosing interesting assumptions to
consider and expanding logic to decide a wider variety of interesting
questions.
Paul Budnik
www.mtnmath.com
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