July
04, 2017 Tuesday
Bedtime
Story
Formula G is Asserting the Truth yet it is Not Demonstrable Within the System
Last
night we were having a look at the formula G at its meta-mathematical level.
G
at the level of number-theory level states that there is no number x that
shares the ‘dem’ relationship to the number sub (n, 17, n).
But
to us, as human apes who go by “meaning”, the validity of the meta-mathematical
meaning should suffice to establish the truthfulness of G.
However,
we have shown quite comprehensively that within the Principia G is undecidable,
and hence has no proof.
And
yet, meta-mathematically, that is exactly what G contends!
That
means G is asserting a truth.
Fascinatingly,
we have succeeded in establishing a truth of number-theory not from the axioms
and using the rules of inference of the Principia, but by taking a
meta-mathematical argument.
This
is the end of the third argument of Gödel.
Now
we will move on to the fourth argument on incompleteness and study it in
detail.
(4) Let us understand the meaning of the term
“completeness” in relation to propositional calculus or zeroth-order logic.
A
formal system is said to be “complete” if its every true theorem or the
statement can be derived from the axioms of the systems using the allowed rules
of inference.
If
all the true statements of the system cannot be derived in this manner, then
the system would be deemed “incomplete”.
Now
please go back and consider the discussion we had in point (3).
We
had agreed finally that the formula G of the Principia is the true formula (and
not ~G), and yet it is not possible to derive it within the system.
Then
it follows that the system of Principia is an incomplete one, assuming that it
is consistent.
It
has to be pointed out that this conclusion could have been reached without
going through the argument stated in point (3).
Meaning,
it is not relevant for the completeness argument to know whether it is the
formula G that is true or the formula ~G.
Suffice
to say that within the Principia, G is not decidable.
It
is sufficient to know that one of them expresses the truth and that both are
not derivable within the system.
It
may be that psychologically it feels better to know which one is the offender,
though it is not a prerequisite for our current argument.
Gödel
did not stop at this.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.in/
Good night mon ami and my fellow cousin ape.
Advertisements
Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
No comments:
Post a Comment