July
05, 2017 Wednesday
Bedtime
Story
Understanding the Term "Essentially Incomplete" of a Formal System of Zeroth-Order Logic
Last
night we saw how Gödel had proved the incompleteness of Principia Mathematica
as a formal system of propositional calculus or zeroth-order logic.
But
as I had left last night’s story with the words that he did not stop at this.
He
went a notch further.
Gödel
further degraded the formal system of Principia by emphasizing that not only
was it incomplete, but that it was essentially incomplete.
We,
if you recall, had briefly discussed in one of our bedtime stories this subtle
difference between being incomplete and being essentially incomplete.
What
it implied was that even if in the Principia the formula G was added as an
axiom, the enhanced and expanded system would still be inadequate to allow
derivation of all true theorems.
All
such kind of addition of axiom would achieve is to generate yet another true
but undecidable formula of the new formal system.
It
is understandable that the new system thus created would be more complex than
the original Principia simply because in addition to the old one, it would be
having one more new axiom.
This
will make both the number theory and the notion of demonstrability in the new
system to be more complex.
Let
us denote this new complex number theory relationship with dem’ (x, z).
This
new undecidable formula is contrived in the same manner as the way in which Gödel
constructed the true but undecidable formula in the Principia itself.
Using
this technique of Gödel, one can always generate new undecidable formulas
(which can be labeled dem’’ (x, z), dem’’’ (x, z), dem’’’’ (x, z) and so on,
irrespective of how many new formulas are added as axioms to it.
More
importantly, exhibition of this phenomenon is not due to some quirks or
peculiarities in the foundations of Principia Mathematica or some flaw in
formalization by Whitehead and Russell.
Undecidable
formulas will always arise in any system, as long as it totally formal and as
long as it is founded on axioms that define elementary properties of natural
numbers, including addition and multiplication.
This
came as a great shock and setback to those who had complete faith in the powers
of axiomatic deductive reasoning.
Gödel
prevailed on them to recognize this fundamental limitation of mathematics based
on axioms and sequential deduction.
There
would always remain a vast ocean of mathematical truths that would stay
stubbornly improvable if one were to stick to the basic axioms and the assigned
rules of inference.
With
this we are done with the fourth argument and point (4) of Gödel reasoning.
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.
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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:
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