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.
--------------

This is simply my requirement for a decimal to exist or be well-defined, that every digit is known or computable or "determinable" uniquely.

--------
My mathematical spaces consists of decimals (base 10) well-defined by the three axioms. Triadic numbers, binary numbers, base 5 numbers, etc., are simply different mathematical spaces well-defined by their respective suitable axioms.
------------

Thus there are infinitely many
``decimals'' (which is your word for decimal expansions of realnumbers) and infinitely many ``such algorithms.''
-----------

When you LIST DOWN the rules for finding those digits they will turn out to be finite. In other words, you cannot write infinite rules.
----------

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.
------------------
This is part of the inherent uncertainty of infinite set, that you cannot identify every element.

Cheers.

E. E. Escultura
.

Relevant Pages

  • Re: Paper & pencil password algorithm
    ... Computing something like mul, C) would be twice as much ... Testing if a digit is multiple of three is easy to remember. ... Look at the first empty slot from the left, ... *addKey: Add the key digitwise to the last row. ...
    (sci.crypt)
  • 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: 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: Contradicrtion-free mathemattics (The new nonstandard analysis
    ... >digit is known or computable. ... >scheme for computing each digit or determining it uniquely from the basic ... ``decimals'' (which is your word for decimal expansions of real ... decimal expansions is infinite. ...
    (sci.math)
  • Re: An uncountable countable set
    ... They are satisfactorily indexed by the infinite set of finite natural ... You just proved that there are infinitely digit positions which are not ... And each of the infinitely many naturals is only finitely large. ...
    (sci.math)