Working with the set of real numbers
Rustom Mody
rustompmody at gmail.com
Wed Feb 12 15:11:37 CET 2014
On Wednesday, February 12, 2014 3:37:04 PM UTC+5:30, Ben Finney wrote:
> Chris Angelico writes:
> > On Wed, Feb 12, 2014 at 7:56 PM, Ben Finney wrote:
> > > So, if I understand you right, you want to say that you've not found
> > > a computer that works with the *complete* set of real numbers. Yes?
> > Correct. [...] My point is that computers *do not* work with real
> > numbers, but only ever with some subset thereof [...]
> You've done it again: by saying that "computers *do not* work with real
> numbers", that if I find a real number - e.g. the number 4 - your
> position is that, since it's a real number, computers don't work with
> that number.
There is a convention in logic called the implicit universal quantifier
convention: when a bald unqualified reference is in a statement it means
it is universally quantified. eg
"A triangle is a polygon with 3 sides"
really means
"ALL polygons with 3 sides are triangles" ie the ALL is implied
Now when for-all is inverted by de Morgan it becomes "for-some not..."
So "computers work with real numbers" really means "computers work with
all real numbers" and that is not true
> That's why I think you need to be clear that your point isn't "computers
> don't work with real numbers", but rather "computers work only with a
> limited subset of real numbers".
Yes both these statements are true by above.
In fact computers cannot work with real numbers because the real number
set is undecidable/uncomputable. In particular, trivial operations like
equality on reals -- IN GENERAL -- is undecidable.
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