On Tue, Sep 27, 2011 at 1:02 PM, meekerdb <meeke...@verizon.net> wrote:

>      Not in general, unless one is only going to allow only Boolean logics
>> to exist. There have been proven to exist logics that have truth values that
>> range over any set of numbers, not just {0,1}. Recall the requirement for a
>> mathematical structure to exist: Self-consistency.
> Okay, there may be other subjects, besides number theory and arithmetical
> truth where other forms of logic are more appropriate.  For unambiguous
> propositions about numbers, do you agree with the law of the excluded
> middle?
> Jason
> I think this an assumption or another axiom.  Consider the conjecture that
> every even number can be written as the sum of two primes.  Suppose there is
> no proof of this from Peano's axioms, but we can't know that there is no
> proof; only that we can't find one.  Intuitively we think the conjecture
> must be true or false, but this is based on the idea that if we tested all
> the evens we'd find it either true or false of each one.  Yet infinite
> testing is impossible.  So if the conjecture is true but unprovable, then
> it's undecidable.

Propositions can be undecidable in the context of a given set of axioms, but
there are stronger systems in which the proposition is decidable.

In any case, whether or not some proposition is decidable (can be
demonstrated as true or demonstrated as false in a series of logical steps
leading to the axioms in question) does not suggest that a mathematical
proposition is true or false dependently of us.  Conversely, I think it is
one of the strongest arguments against the idea that math is man-made.  Any
system of axioms we develop is imperfect in the sense that it cannot answer
all questions concerning the numbers.

Those who think that the objects of study in mathematics are human
inventions are living in the early 20th century.


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