Re: Contradicrtion-free mathemattics (The new nonstandard analysis

In article <1138586403.969919.71570@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
matthias@xxxxxxxxxxx wrote:

> E. E. Escultura
>
> >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.
.

Relevant Pages

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  • Re: Well Ordering the Reals
    ... > Okay, I don;t think I understood what you were saying. ... > naturals, you might as well call it something, I suppose. ... In TO's system of "whole numbers", there is a most significant digit and ... > infinite unending string of bits, even if most are generally ignored. ...
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  • Re: Well Ordering the Reals
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  • Re: INFINITY Revisited
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  • Re: abundance of irrationals!)
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