Re: Contradicrtion-free mathemattics (The new nonstandard analysis

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.

.

Relevant Pages

  • 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 ... you cannot write infinite rules. ... This is part of the inherent uncertainty of infinite set, that you cannot identify every element. ...
    (sci.math)
  • Re: The Modified Halting Problem, Take ??? .
    ... Turing tapes" will not be a set or even a proper class in NAFL, ... each tape itself is an infinite entity. ... They're usually described as potentially infinite or finitely unbounded. ... and there is no last digit of Pi ever computed which one would think ...
    (sci.logic)
  • Re: An uncountable countable set
    ... >>> string up to digit number n. ... You state that an infinite string could be ... but contains only finite digit positions. ...
    (sci.math)
  • Re: abundance of irrationals!)
    ... >> WE have a number of axiom systems, ... such set meets Cantor's definition of an infinite set. ... digit position in the nth listed number. ... >> with internal contradictions, but what we are talking about has been ...
    (sci.math)
  • Re: abundance of irrationals!)
    ... >> It is not a matter of understanding, but a matter of contradiction. ... > in which infinite sets are REQUIRED by the axioms. ... The axiom requires n+1 for given n. ... > last digit of one of your listed numbers, ...
    (sci.math)