Re: Another proof: The Halting Problem Is Undecidable.

On Sun, 2024-10-13 at 22:01 +0100, Ben Bacarisse wrote:

wij <wyniijj5@gmail.com> writes:
 

If 0.999..=1, you have to explain your arithmetic system.

 
Almost.  First you have to explain the notation.  That's easy (but
relatively advanced) as 0.999... denotes the limit of a sequence of
partial sums (sometimes called a series limit).  The arithmetic system
(the reals, where all such sums converge) comes after saying what the
... denotes.
 
When *you* say that 0.999... =/= 1 you always avoid saying what the
notation (specifically the ...) means in formal terms.
 

What I would say now is probably not different from
https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber2-en.txt/download

"..." conventionally means "so on,..,etc.", likely infinitely. I use it like that.

As indicated in the above file:
"Theorem 5: The set of elements composed of finite discrete symbols and the set
         of elements composed of infinite discrete symbols cannot form 1-1
         correspond."

(did your formal system say that? I guess no)

That means whatever a formal system can do is limited. Almost everything
formally stated is impossible.

Take Archimedes' Axiom for example: It did not mention anything about infinity.
How can it *formally* deny the existence of infinity/infinitesimal? And, the
Archimedes' Axiom argument is likely suffering from circular reasoning as an
axiom. It needs to say what the R it is addressing. (The Axiom is new, it lacks
the source, and likely a 'popular' rumor to me if it is also 'formal'. E.g.
I just saw another one https://proofwiki.org/wiki/Axiom_of_Archimedes
∀x∈ℝ: ∃n∈ℕ, n>x
The problem is what exactly had Archimedes said? ).