Re: Usenet-controversial


Garrett Wollman wrote:

> For any given irrational x, we can choose rational values q and r such
> that q < x < r.

For any given rational x, we can also do this.

"x having a different value" means that there exists
> some q and r for which this inequality is true, given the system of
> math we are familiar with, but not in this hypothetical alternate
> mathematics, under the assumption that the rationals and the relations
> all mean the same thing in both systems.

> This is clearly not the case for some irrationals: if x is "the
> positive square root of two", it cannot have any other value (assuming
> the arithmetic operators on Q are unchanged).

It seems to me you have things exactly backwards; sqrt(2) having a
different value, following your definition of what that means, might
mean that there is a number x, such that x^2 = 2, but it is not the
case that 7/5 < x < 3/2. But this in fact is true; x could be -sqrt(2).
Is that not an example of the square root of two taking a different
value? If not, why not? What about the 7-adic integer which
approximates to 3 mod 7, 10 mod 49, -235 mod 2401, and so forth to
futher values easily calculated by Newton's method. Certainly, these
don't satisfy 7/5 < x < 3/2, because they don't lie in an ordered
field. Does that count?

(However, as I
> mentioned, the fact that every term in certain Taylor series is
> rational suggests that they are, in which case a legitimate question
> may be "why?".)

Every transcendental number is the value of a function f(1) defined by
a Maclaurin series with rational coefficients which converges in a
neighborhood of 1 and converges to the value in question at 1.

.

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