Re: Another proof: The Halting Problem Is Undecidable.

On 12/10/2024 02:53, wij wrote:

Archimedes likely believes that all (real) numbers, including pi, sqrt(2), are
p/q representable. Is that what you suggest?

By the time of Archimedes it had been known for several hundred
years that "sqrt(2)" is irrational.

Established, at the latest, by the Pythagoreans [~500 BCE, nearly 300y
before Archimedes].  Possibly known earlier. ...

[The status of "pi" remained unknown
for a further ~2K years.]

First proved by Lambert, in 1761;  1973y after Archimedes died. ...

      So no, Archimedes did not believe that, not
least when he laid some of the foundations of calculus.

Euclid's "Elements" book 10 is about irrational numbers, and
Archimedes referred to that book in his writings, so he was certainly
aware that root(2) is irrational. ...

That is a fabrication (there are many, but... accepted, as a fabrication)

... So everything above can be verified by anyone able to google or
wiki.  Accusing me of lying is not a good way to get the mathematical help
that you clearly need, esp when the accusation is so clearly false.

Archimedean axiom is an *assertion* that infinitesimal does not exist without
knowing the consequence (violating Wij's Theorem which is provable from the rules
stronger than 'assertion').

If "Wij's Theorem" is inconsistent with the axioms of real numbers,
then it is not a theorem of real numbers.  Try one of the other systems of
numbers, which you would probably find more to your taste, given the other
things you say in this group.

Are you kidding? "x>0 iff x/n >0, where n∈ℤ⁺" is inconsistent?

It was /your/ claim that "Wij's Theorem" is "violated" by the
Archimedean axiom.  /If/ you are right, /then/ your theorem is not true
in the standard reals, as used by all numerate scientists and engineers
and as understood by all mathematicians [even if they /also/ use NSA or
surreals or other systems].  FWIW, /I/ think your theorem is correct
in standard analysis, but you seem to object to that.

   With your real, yes. My real is based on the abacus that can be physically modeled. Tell me, how can
it be inconsistent?

Perhaps you should first explain how you represent infinity and
infinitesimals on a standard abacus?  "Your" reals can, of course, be
inconsistent if you insist on axioms that are inconsistent with them.
[If you persist in insulting those who are trying to help you,
then you will not get any further reply from me.  I don't intend to play
"Fetch" with you.]
--
Andy Walker, Nottingham.
    Andy's music pages: www.cuboid.me.uk/andy/Music
    Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Necke