Re: Contradicrtion-free mathemattics (The new nonstandard analysis
> >A decimal is known by its digits. Therefore, it exists or is known or
> >well-defined if every
> >digit is known or computable. Being computable means there is an algorithm
> >or rule or
> >scheme for computing each digit or determining it uniquely from the basic
> >integers 0, 1, . . ., 9. Since computation is a finite process, the set of
> >such algorithms is finite.
>
> This is clearly false. For each positive integer n there is a
> completely explicit, concrete algoprithm for producing the decimal
> expanion of the square root of n. Thus there are infinitely many
> ``decimals'' (which is your word for decimal expansions of real
> numbers) and infinitely many ``such algorithms.''
>
> Note that this does not depend on classical logic in any way. Both
> Bishop and Brouwer would agree that the set of algorithms which produce
> decimal expansions is infinite.
It is even false for the set of natural numbers, as there is an
algorithm for each natural and more than any finite number of naturals.
--------------
Although I have answered you questions in my responses to matthias@xxxxxxxxxxx and others I'll make some clarification.
A decimal is well-defined if every digit is known or computable, i.e., there is a rule or algorithm for determining every digit. If you write those rules you will find them finite. But I agree that you can add one more to the list. In other words, the number of such rules is finite but unbounded. So is the space of decimals as I have constructed them on the basis of the three axioms.
E. E. Esculturra
.
Relevant Pages
- Re: Penroses Computing Pi Description?
... > <snip penrose> ... > digit as output. ... > "computing Pi" on its own, ... is the algorithm the computation of one particular digit or is ...
(sci.logic) - Re: arithmetic in ZF
... Torkel manifestly obviously IS NOT *saying* that, ... > Or an algorithm can be given in terms ... > for computing a function from ... > naturals to naturals can be simulated ...
(sci.logic) - Re: Contradicrtion-free mathemattics (The new nonstandard analysis
... >>digit is known or computable. ... > decimal expansions is infinite. ... algorithm for each natural and more than any finite number of naturals. ...
(sci.math) - Re: Contradicrtion-free mathemattics (The new nonstandard analysis
... >>digit is known or computable. ... >>scheme for computing each digit or determining it uniquely from the basic ... algorithm for each natural and more than any finite number of naturals. ...
(sci.math) - Re: Top 500 is out
... Computing pi by a more sensible algorithm will use a couple of dozen ... gives details, describes them as "almost linear time", and specifically ... We give algorithms for the computation of the d-th digit of certain ...
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