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

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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.
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Thus there are infinitely many
``decimals'' (which is your word for decimal expansions of realnumbers) and infinitely many ``such algorithms.''
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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.
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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.
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This is part of the inherent uncertainty of infinite set, that you cannot identify every element.

Cheers.

E. E. Escultura
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