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From: wij <wyniijj5@gmail.com>
Newsgroups: comp.theory
Subject: Re: Another proof: The Halting Problem Is Undecidable.
Date: Thu, 10 Oct 2024 23:26:25 +0800
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On Thu, 2024-10-10 at 22:43 +0800, wij wrote:
> Axiom: Part is smaller than the whole.
>=20
> Theorem: A system (physical device, computer,...) cannot compute/emulate =
a=20
> =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 bigger system ... (like =
a computer cannot simulate a system (whatever)
> =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0 that contains it).
>=20
> The concept is general. If applied to HP, H cannot compute the property (=
except
> trivial) of D, simply because D contains (maybe logically) H as a part.
>=20
> Thus, can the part equal to the whole? Definitely not, by definition. The=
=20
> public are also fooled by Cantor's magic of infinite set: The set of even=
=20
> number, say X, is actually a distinct set isomophic to the natural number=
 (not
> a part of it). The element in X is not 'even' in X.... Basically, there a=
re many
> set of natural number, not just one. The set of 'natural number' has to b=
e
> explicitly specified to avoid ambiguity in discussion.
>=20
> Snippet from https://sourceforge.net/projects/cscall/files/MisFiles/RealN=
umber-zh.txt/download
> and translated by Google Translator:
>=20
> Appendix4: 2D-number can express plane. In 2D-number, as long as the dist=
ance
> =C2=A0=C2=A0=C2=A0 postulate (1. Distance between points is invariant by =
movement 2.The
> =C2=A0=C2=A0=C2=A0 ratio of distance between points is invariant by scala=
r multiplication) are
> =C2=A0=C2=A0=C2=A0 satisfied, Euclidean geometry system can be establishe=
d. What is meant to
> =C2=A0=C2=A0=C2=A0 say is that: Such a 'mass-point universe' is construct=
ed based on our
> =C2=A0=C2=A0=C2=A0 preset property. We are ultimately exploring the seman=
tics of our own
> =C2=A0=C2=A0=C2=A0 knowledge. And, as long as the logic holds, the respec=
tive reality should
> =C2=A0=C2=A0=C2=A0 be expected. Inversely, exploring 'real number' by phy=
sics is basicly valid.
> =C2=A0=C2=A0=C2=A0 In the digital era, universe (semantics) is a natural =
computer.
> Appendix 5: ....
> Appendix 6: From Appendix 4, it can be roughly concluded that: a system
> =C2=A0=C2=A0=C2=A0 (physical device, computer, k,...etc.) cannot calculat=
e (or simulate) the
> =C2=A0=C2=A0=C2=A0 characteristics of the system containing it. Basically=
, it is the concept of
> =C2=A0=C2=A0=C2=A0 "parts are smaller than the whole" (This is the defini=
tion). In addition,
> =C2=A0=C2=A0=C2=A0 shutdown problems can also be explained by this concep=
t. Cantor's infinite
> =C2=A0=C2=A0=C2=A0 set theory may lead to the fallacy that "parts are equ=
al to the whole",
> =C2=A0=C2=A0=C2=A0 such as "the number of even numbers is the same as the=
 number of natural
> =C2=A0=C2=A0=C2=A0 numbers". But, like the 0.999... problem, there is mor=
e than one 'set of
> =C2=A0=C2=A0=C2=A0 natural numbers'. N<0,+2> =3D {0,2,4,6,..} can also be=
 regarded as a set of
> =C2=A0=C2=A0=C2=A0 natural numbers, in which 2,6,10,.. are odd numbers in=
 N<0,+2>.
>=20
>=20

This "0.999...!=3D1" proof can also demonstrate the idea "Part is smaller t=
he=20
 whole" from various kinds of instances:

 Expression B<1-A <=3D> B+A<1 always holds (Assume B=3D0.999.., A=3Dthe par=
t to add to B):

 Ex: 0.9+0.09 < 1
     0.99+0.009 < 1
     0.999+0.0009 < 1
     ... (so on infinitely)
     0.999... < 1

 If "0.999...=3D1", the expression "B<1-A <=3D> A+B<1" does not hold.