[FOM] Logically determined

[Paul Budnik]

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