Re: parallels
- From: "John Edser" <edser@xxxxxxxxxxxxxx>
- Date: Fri, 19 Jan 2007 13:50:23 -0500 (EST)
Aatu Koskensilta aatu.koskensilta@xxxxxxxxx wrote:-
JE:-was
I strongly suggest you replace your consistent rhetoric (previously I
accused of "babbling" and now I am accused of "metaphorical rambling")with
NON EVASION.
I have not pretended to address your rambling, or if you like the term
better, babbling, with any arguments.
JE:-
They are both your own rhetorical terms where neither are appropriate and
you know it. The only ethical thing for you to do is to apologize and
respond rationally to the arguments presented. In the sciences you discuss
and test, you do NOT just dictate. By what divine right do you simply
dictate that I was only "rambling" and "babbling"? This happens to be a
science list so we do not include arguments by divine right here, even by
divine mathematicians :-)
Indeed, there is no point for me
to do so, since my interests lie solely in the incompleteness theorems
and their invocations by your fine self and others.
JE:-
Are you arguing that what Gödel discovered has no valid application outside
of mathematics what so ever? Please do not attempt to evade this question
and please do not answer ambiguously.
Going on about
"irreversible linkage", "tautologicity" and what have you, in relation
to the incompleteness theorems is pointless, unless you somehow connect
your ideas to the actual mathematical content of the theorems. Otherwise
you might as well quote the fundamental theorem of arithmetic or
Dirichlet's theorem.
JE:-
You claim to be a logician yet you seem not to have the faintest idea as to
what logic DOES: separates a sequitur from a non sequitur THAT IS ALL.
http://en.wikipedia.org/wiki/Non_sequitur_%28logic%29
Any valid sequitur can have TWO forms: reversible and irreversible.
PLEASE indicate CLEARLY if you agree or disagree.
Language and computer programming can only allow a reversible sequitur as a
nested subset of an entirely irreversible sequitur. The sentence: "the cat
sat on the mat" does not have the same meaning as "the mat sat on the cat"
simply because subject and predicate cannot be reversed within any rational
language (or computer program). If you reverse them then you reverse their
critical set nesting. If I just define a cat to be mat, then a mat is also
just a cat where this entirely reversible sequitur (represented by just a
set intersection) is known as a TAUTOLOGY. Tautologies can be expanded
indefinitely becoming amazingly complex (as they are within mathematics).
However, they can have no rational meaning unless they become predicated to
something else in an entirely non reversible way: e.g. the matcat thing is
black but not all black things are matcat things. Only now can a matcat
claim to make rational sense. The predicate "black" constitutes a critical
induction from which the matcat thing must be able to be deduced. In the
sciences this allows inductions to be tested to refutation via ANY deduction
from them. All of reasoning remains based on just this one SIMPLE
proposition: IF the induction is true then ANY valid deduction from it MUST
also be true BUT NOT THE REVERSE.
I remain flabbergasted that you can see no connection between the above and
what Gödel discovered which Fanzen puts in (non reversible English) to
mean:-
"The mathematician Godel proved that a system of axioms can never be
based on itself: statements from outside the system must be used in order to
prove its consistency."
Please very carefully explain to all of us here why any rationalist cannot
just retort: Well, very OBVIOUSLY only because ALL of the axioms of
mathematics were and remain, TAUTOLOGIES. IF this is NOT true then please
provide just a single axiom which is not a tautology.
Your continued pretence that tautological sequiturs have no connection
what-so-ever to what Gödel discovered remains ludicrous. It appears to me
you are attempting to evade this issue simply because you do not wish
mathematics to be proven to be just a tautology (especially by Gödel). This
is why I said the thing I respect most about him was his complete honesty.
It takes real guts to prove that what the subject matter you have spent your
life on was not consistent. It takes even more guts to say WHY. It appears
you remain gutless.
I have read the article with exasperation. AT NO TIME does TorkelFranzén
even mention the concept of a TAUTOLOGY let alone include
how it necessarily contrasts to a NON tautology within any
RATIONAL system.
Of course not. There is no apparent connection between these things and
the actual mathematical content of the incompleteness theorem.
JE:-
Well, here we CAN detail just one of our differences: no APPARENT
connection. This does not prove that there is NO CONNECTION. Why have you
closed your mind to any argument which even attempts to connect them?
The closest we seem to get to this seems to be:for
"Gödel's proof makes essential use of what is called the diagonal lemma
T. This is a general result about T stating that for every formula B(x)with
one free variable x - meaning that B(x) asserts something about the
unspecified number x - a formula A (known as a fixpoint for B(x)) can be
constructed such that
T |- A<=>B(#A)"
Perhaps you might to like explain why "<=>" does not mean "entirely
reversible" proving just a tautologous logical linkage?
Here Torkel is just explaining a technical theorem pertaining certain
formal theories T. According to the diagonal lemma, as he explains, for
any formula B(x) we can find a formula A such that it is provable in T
that A is true just in case the Gödel number of A has the property
expressed by B(x). What this has to do with "entirely reversible" or
"tautologous logical linkage" cannot be determined unless you provide
some way of relating these notions to the standard logical notions of
formal provability, equivalence, Gödel numberings and so forth.
JE:-
Yet again, you have simply evaded my question.
"Perhaps you might to like explain why "<=>" does not mean "entirely
reversible" proving just a tautologous logical linkage?"
Please ANSWER the ******* question.
Franzen clearly noted what he considers to be a CORRECT statement inWORDS
(i.e. as a non reversible rational proposition with a subject and abased
predicate) as to what Gödel had discovered:
"The mathematician Godel proved that a system of axioms can never be
on itself: statements from outside the system must be used in order toprove
its consistency."
This is indeed a correct statement: a consistent system satisfying
certain technical conditions can only be proved consistent by using
principles not contained in the system. However, what this has to do
with "a non reversible rational proposition with a subject and a
predicate" remains completely obscure.
JE:-
"Obscure" to you but not to me (or ANY rationalist). Please attempt to open
your mind to arguments that you appear to reject only because you dislike
them.
Please note: Franzen has acknowledged above that what Gödel discovered does
have valid applications outside of mathematics (which has to be true to
remain consistent to what he discovered).
Please list these valid applications for sbe reader's to view.
Your argument that "though there are certain marginal set theories witha
universal set, such a set is not a part of ordinary mathematics or set
theory" was not correct.
Sure it was, as you will find out for yourself if you choose to study
set theory.
JE:-
Every reference I have on the logic of mathematics defines a universal set
by SIMPLE NECESSITY. This gave rise to Russell's Paradox (which was just an
infinite regress "solution" i.e. was just no solution at al) and
subsequently, to Gödel's proof that mathematics cannot be consistent.
Gödel added nothing POSITIVE to Russell and Whitehead's proposed "solution".
It remains easy to see: the universal set is the defined PREDICATE, i.e.the
INDUCTIVE ASSUMPTION from which everything else exists as just adeduction
(including all tautologies nested within it).
As usually understood a universal set V is a set such that for every
object a, a is a member of V. The existence of such a set is, on the
face of it, a purely mathematical supposition -
JE:-
Dear oh dear...Russell and Whitehead demonstrated that to be consistent V
must be a set of an even more universal set, which itself has to be subset
of an even more universal set, "ad finitum" . In other words every set is a
universal set but some are more universal than others! Now WHERE did I hear
something very-much-like-that before?
http://en.wikipedia.org/wiki/Infinite_regress
http://en.wikipedia.org/wiki/Russell's_paradox
Why doesn't the following: "Russell's Paradox", "infinite regress" even
leave your lips? Do they remain sealed against these just as they remain
sealed against the term most dreaded by pure mathematicians, "tautology"?
...a false one, as sets are
usually conceived in mathematics, as it happens.
JE:-
Incorrect. The universal set WAS a conception of physics but remains just a
misconception of mathematics. The science of physics induced the universe
concept and NOT the non science of mathematicians.
John Edser
Independent Researcher
edser@xxxxxxxxxxxxxx
.
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