[FRIAM] anonymity/deniability/ambiguity

[Frank Wimberly]

The badly rendered part:

{\displaystyle \left|{\sqrt {2}}-{\frac {a}{b}}\right|={\frac
{|2b^{2}-a^{2}|}{b^{2}\left({\sqrt {2}}+{\frac {a}{b}}\right)}}\geq {\frac
{1}{b^{2}\left({\sqrt {2}}+{\frac {a}{b}}\right)}}\geq {\frac
{1}{3b^{2}}},}[image:
{\displaystyle \left|{\sqrt {2}}-{\frac {a}{b}}\right|={\frac
{|2b^{2}-a^{2}|}{b^{2}\left({\sqrt {2}}+{\frac {a}{b}}\right)}}\geq {\frac
{1}{b^{2}\left({\sqrt {2}}+{\frac {a}{b}}\right)}}\geq {\frac
{1}{3b^{2}}},}]



On Thu, May 21, 2020 at 11:30 AM Frank Wimberly <wimberly3 at gmail.com> wrote:

> Clinicians often call that "being oppositional".
>
> You say that I've known authorities.  I was just talking to John Baez
> about my advisor Errett Bishop, often called the inventor of constructive
> mathematics.  Here is a constructive proof, with no use of the excluded
> middle, of the irrationality of sqrt(2) that I found in Wikipedia.
> Apologies to those who don't care:
>
> In a constructive approach, one distinguishes between on the one hand not
> being rational, and on the other hand being irrational (i.e., being
> quantifiably apart from every rational), the latter being a stronger
> property. Given positive integers *a* and *b*, because the valuation
> <https://en.wikipedia.org/wiki/Singly_and_doubly_even#Definitions> (i.e.,
> highest power of 2 dividing a number) of 2*b*2 is odd, while the
> valuation of *a*2 is even, they must be distinct integers; thus |2*b*2 −
> *a*2| ≥ 1. Then[17]
> <https://en.wikipedia.org/wiki/Square_root_of_2#cite_note-17>
> {\displaystyle \left|{\sqrt {2}}-{\frac {a}{b}}\right|={\frac
> {|2b^{2}-a^{2}|}{b^{2}\left({\sqrt {2}}+{\frac {a}{b}}\right)}}\geq {\frac
> {1}{b^{2}\left({\sqrt {2}}+{\frac {a}{b}}\right)}}\geq {\frac {1}{3b^{2}}},}[image:
> {\displaystyle \left|{\sqrt {2}}-{\frac {a}{b}}\right|={\frac
> {|2b^{2}-a^{2}|}{b^{2}\left({\sqrt {2}}+{\frac {a}{b}}\right)}}\geq {\frac
> {1}{b^{2}\left({\sqrt {2}}+{\frac {a}{b}}\right)}}\geq {\frac
> {1}{3b^{2}}},}]
>
> the latter inequality being true because it is assumed that *a*/*b* ≤ 3 −
> √2 (otherwise the quantitative apartness can be trivially established).
> This gives a lower bound of 1/3*b*2 for the difference |√2 − *a*/*b*|,
> yielding a direct proof of irrationality not relying on the law of
> excluded middle <https://en.wikipedia.org/wiki/Law_of_excluded_middle>;
> see Errett Bishop <https://en.wikipedia.org/wiki/Errett_Bishop> (1985,
> p. 18). This proof constructively exhibits a discrepancy between √2 and
> any rational.
>
> On Thu, May 21, 2020 at 10:50 AM Steve Smith <sasmyth at swcp.com> wrote:
>
>>
>> On 5/21/20 10:32 AM, uǝlƃ ☣ wrote:
>> > Don't be fooled. "The problem with communication is the illusion that
>> it exists." Or ie I believe in a stronger form of privacy than you believe
>> in.
>> I KNOW! I know just what you mean!
>>
>> <note to Frank...  one of the species of animal in this group is "the
>> Contrarian", but you probably already guessed that>
>>
>>
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>
>
> --
> Frank Wimberly
> 140 Calle Ojo Feliz
> Santa Fe, NM 87505
> 505 670-9918
>


-- 
Frank Wimberly
140 Calle Ojo Feliz
Santa Fe, NM 87505
505 670-9918
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