Discussion:
Even in Dedekind arithmetic, one does nothing with fake real numbers, only with subsets ...
John Gabriel
2018-02-12 15:00:32 UTC
Even in Dedekind arithmetic, one does nothing with fake real numbers, only with subsets which are composed of rational numbers.

Dedekind, like Cauchy was a fraud.

A number is the measure of a magnitude.

Sets don't measure anything. In fact, set theory is bogus because a set is undefined.

"For it cannot occasion surprise that different persons should arrive at the same logical conclusions when they have already agreed upon the fundamental laws (axioms), as well as the methods by which other laws are to be deduced therefrom." - Einstein

Einstein was ignorant in more than one way. Mathematics is not about consensus. It is either right or wrong. Modern orangutan academics have invented rules for their mythmatics, but in mathematics there are NO rules, only hard, cold facts.

Dedekind and Cauchy fail:

Python
2018-02-12 18:45:58 UTC
Post by John Gabriel
Dedekind and Cauchy fail
You did. They didn't.

This is a plain, cold fact.
John Gabriel
2018-02-12 20:46:42 UTC
Post by Python
Post by John Gabriel
Dedekind and Cauchy fail
You did. They didn't.
This is a plain, cold fact.
You will address me as Mr. Gabriel, you vile piece of shit. Chuckle.

I guess there is nothing you can refute there so the next best thing is for you to throw merde. Tsk, tsk.
Jan
2018-02-12 23:52:07 UTC
Post by John Gabriel
Even in Dedekind arithmetic, one does nothing with fake real numbers, only with subsets which are composed of rational numbers.
In Dedekind's system numbers are defined as certain sets, yes. As long as
they satisfy the desired properties, such things are as good as any other
possible alternate definition.

Philosophically many may find this sort of thing unsatisfying and a "kludge"
but mathematically this is sound (or at least as sound as the axioms of
set theory which nobody knows for sure).

--
Jan
John Gabriel
2018-02-13 01:04:51 UTC
Post by Jan
Post by John Gabriel
Even in Dedekind arithmetic, one does nothing with fake real numbers, only with subsets which are composed of rational numbers.
In Dedekind's system numbers are defined as certain sets, yes. As long as
they satisfy the desired properties, such things are as good as any other
possible alternate definition.
No Jan, they're not as good as any other definition. Sets don't measure anything in the way that rational numbers do. You can't do anything, even with Dedekind arithmetic which is only possible with rational numbers.

We understand a given magnitude (whether distance, mass, volume, etc) by a number which describes it. Neither a Dedekind Cut nor a Cauchy sequence do this.
Post by Jan
Philosophically many may find this sort of thing unsatisfying and a "kludge"
but mathematically this is sound (or at least as sound as the axioms of
set theory which nobody knows for sure).
How is it sound when it has been proven to be invalid?

http://youtu.be/LSWIFXP2r14

The entire concept is absolute hogwash because one can't relate non-numbers to indistinguishable and unreifiable points on a mythical "real" number line.

To reify a point on the number line, all of the following must be possible:

i. It must be possible to mark off the magnitude on the number line.
ii. There must be a number which describes its measure.

You can reify any rational number on the number line, but it is not possible to reify the magnitudes pi, e or sqrt(2) because there is no number describing any of these. These are all symbols for incommensurable magnitudes. They are decidedly not numbers.

Even before one can consider Dedekind Cuts, the real number line must be established. There is no real number line - it's a myth.
Post by Jan
--
Jan
Ross A. Finlayson
2018-02-13 10:02:49 UTC
Post by John Gabriel
Even in Dedekind arithmetic, one does nothing with fake real numbers, only with subsets which are composed of rational numbers.
Dedekind, like Cauchy was a fraud.
A number is the measure of a magnitude.
Sets don't measure anything. In fact, set theory is bogus because a set is undefined.
"For it cannot occasion surprise that different persons should arrive at the same logical conclusions when they have already agreed upon the fundamental laws (axioms), as well as the methods by which other laws are to be deduced therefrom." - Einstein
Einstein was ignorant in more than one way. Mathematics is not about consensus. It is either right or wrong. Modern orangutan academics have invented rules for their mythmatics, but in mathematics there are NO rules, only hard, cold facts.
http://youtu.be/LSWIFXP2r14
The JG-bot, is yet a wrecked affliction
spawn-bot of though almost aritied just
the dubious poor lacking ignorant poor
side of an ancient argument long ago
relegated to the dustbin of long exceeded
argument about the continuum and infinity.

Or, the JG-bot's like a castrati,
but it can't sing.
Zelos Malum
2018-02-15 14:17:36 UTC
Post by John Gabriel
Even in Dedekind arithmetic, one does nothing with fake real numbers, only with subsets which are composed of rational numbers.
Hey dipshit! Dedekinds definition deals with sets of rational numbers. That is the entire fucking point!
Post by John Gabriel
Dedekind, like Cauchy was a fraud.
No, He said you could construct using these subsets a structure that has all the properties of real numbers.
Post by John Gabriel
Sets don't measure anything. In fact, set theory is bogus because a set is undefined.
Who cares baout "measure" here, whatever that means in mathematics?

Sets are the objects in the universe of discourse in set theory. That is all it is.
Post by John Gabriel
Einstein was ignorant in more than one way. Mathematics is not about consensus. It is either right or wrong. Modern orangutan academics have invented rules for their mythmatics, but in mathematics there are NO rules, only hard, cold facts.
Facts belong to science, mathematics is about axioms and theorems.