[FOM] The boundary of objective mathematics
paul at mtnmath.com
Fri Mar 13 13:36:02 EDT 2009
Monroe Eskew wrote:
> Your position seems inconsistent. You say that infinite objects do
> not exist, yet you permit statements whose truth value is determined
> by infinite objects.
No. I allow statements that are determined by a recursively enumerable
sequence of events. The events never ALL happen but EACH of them can
happen in a potentially infinite universe. Thus a statement about all of
them has on objective meaning in that it refers only to events each of
which is determined by a mechanistic process.
> There is no finitistic proof of the consistency
> of ZFC. But you say it is an objective statement because it is
> determined by checking a recursive statement for each natural number.
> (i.e. Check for each n whether n codes a proof of 0=1 from ZFC.) The
> set of theorems of ZFC is recursively enumerable (r.e.) but not
> recursive, hence is an infinite set for which the membership relation
> cannot be reduced to a finitistic property.
My position is finitism with respect to what exists. It is not finitism
with respect to which questions are objectively meaningful. I consider
questions about an infinite number of recursively enumerable events to
be human creations that can have immense practical value, but do not
correspond to anything that exists physically or in some Platonic universe.
> You should also note that any set of n-tuples of natural numbers is
> r.e. if and only if it is \Sigma_1. Thus, many collections of natural
> number n-tuples definable by a number theoretic formula with
> alternating quantifiers are not r.e. If you want to say that
> quantification over natural numbers is always meaningful, then this
> goes far beyond r.e. sets.
I like to think of infinite sets as properties of recursive processes
and these go way beyond r. e. sets. In particular I consider the
following sequence of questions to be objectively meaningful.
Does a TM have an infinite number of outputs?
Does it have an infinite number of outputs an infinite subset of which
are the Godel numbers of TMs that themselves have an infinite number of
Each new questions asks if the TM has an infinite number of outputs that
satisfies the previous question?
In this way one can construct the arithmetic hierarchy. All the events
that determine each question are r. e., but there is a complex logical
relationship between them that determines the final answer.
This also answers the question for your second post on this subject:
> Why not iterate this, allowing allowing statements to be determined by
> statements determined by finite events, statements determined by
> statements determined by statements determined by finite events, and
> so on?
I do. My phrasing may have been confusing on this point.
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