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.
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Being known is stronger requirement than being computable. Something is known only when it has been computed. That is why there is inherent uncertainty or ambiguity in nonterminating decimal because even if we every digit is computable we may not be able to compute all of them. In the new real number system every nonterminating decimal can be approximated by its nth Caucy term with margin of error 10^(-10).
In this sense only terminating decimals are known without uncertainty; they are closed under addition and multiplication. Every segment (terminating and a term of its Cauchy sequence) of a computable decimal is known. Such segments are also closed under addition and multiplication (this is our standard way of adding and multiplying decimals).
E. E. Escultura
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