Re: Contradicrtion-free mathemattics (The new nonstandard analysis
Your definition of ``known'' is inconsistent in the sense that the sum
of two computable numbers may not be a ``known'' number.
The following example is called a ``weak counteraxample'' by
intuitionists.
Let x be the number 0.99999999999999999..... which is clearly known.
Let y be the following decimal number. The nth digit of y is 0 unless
the number n is a Godel number for a proof that there are infinitely
many twin primes. The only reason I chose that particular statement is
that it is unknown whether or not it is provable that there are
infinitely many twin primes. If n is such a number, then the nth digit
of y is 1.
Every digit of y is easily computable. To compute the nth digit, just
check whether the number n codes a proof of the twin primes theorem.
Thus y is a known number.
Now if it is provable that there are infinitely many twin primes then
y > 0 (and y is not infinitesimal or anything like that) and so the
first digit of x + y will be 1. Otherwise, the first digit of x + y
will be 0, because y will be 0 and x + y will be exactly x. I am doing
addition here in the system R* that you have defined. The important
point about the addition table is that there is a carry when you add 1
and 9.
This means that we can tell by looking at the first digit of x + y
whether the twin primes theorem is provable. Since we don't actually
know whther this theorem is provable, we don't ``know'' the first digit
of x + y. The ``known'' numbers are not closed under addition.
.
