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[HM] Infinity and the "Noble Lie"
Posted:
Dec 11, 2005 12:11 PM
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On Wednesday 07 December 2005 22:06, Joe Shipman
<joeshipman@aol.com> wrote:
It is certainly true that the Axiom of Infinity is a tenet of our
"ZFC religion", and we are all encouraged to use it when formalizing --
but at the same times many mathematical philosophers (either
mathematicians on their day off, or philosophers writing about
mathematics) claim to disbelieve in actual infinities.
AZ:
Bourbaki formulate this "tenet of the "ZFC religion" as follows:
AXIOM OF INFINITY. There exists an infinite set.
There is nowhere any word as to whether this 'infinite' is actual
or potential.
Some high professionals in modern axiomatic set theory (AST)
suppose that the "actual vs potential problem belongs to a philosophy
and is not a part of mathematics". Some other well-known AST-people
state that they can prove the uncountable INFINITY of the INFINITE
continuum without usage not only term 'actual infinity' but even term
'infinity' itself, i.e., not using algorithmically any property of just
INFINITE sets.
But it's well known that if the infinity is potential than all
Cantor's "Study on Transfinitum" as well as all theories of transfinite
ordinal and cardinal 'numbers' of modern AST become empty fictions
which, according to Kronecker, belong even not to a philosophy, but to
an unnatural, doubtful theology (a well known Cantor's intention <to
build a transfinite stairs up to a Heaven> in order to cognize the Lord
essence. (J.W.Dauben, V.N.Katasonov)).
Further Joe Shipman writes:
It seems to me that there is a bit of the "noble lie" here -- because
these finitists (and also the agnostics about infinity) are benefiting
from the use of the Axiom of Infinity by the entire society of
mathematicians, even when they don't use it in their own work, because
the Axiom of Infinity has been so useful in the development of
mathematics as a whole. And of course those skeptics who DO nonetheless
use the axiom are in an even less defensible position.
Does anyone perceive an ethical issue here?
AZ:
Yes, there is a crying ethical issue here.
In modern mathematics and modern AST there are two inexplicit
axioms.
ARISTOTLE's AXIOM. All infinities are potential.
CANTOR's AXIOM. All infinite sets are actual.
From Aristotle's time, (Classical) mathematics as a whole (results
of which can be ultimately testified by calculations or practice) was
(and is now) based on Aristotle's axiom and therefore there was no need
ever to formulate it explicitly. Moreover, an explicit formulation of
Aristotle's axiom and its addition to, say, Peano's axiom system will
change nothing in classical arithmetic.
After Cantor actualized all infinities, his axiom became a
necessary condition of any 'diagonal' proofs (of course, if these proofs
are mathematical): "a necessary condition" in the direct mathematical
sense that without that condition basic Cantor's statements as to
different 'transfinite' cardinalities are simply not provable.
But modern AST never and nowhere formulated this necessary
condition explicitly.
From the point of view of Classical Mathematics a non-deliberate
concealment of the necessary condition of a proof is a rude mathematical
error testifying to a fatal scientific ignorance, but a deliberate
concealment of that fact (as modern AST does it) is not a bit of the
'noble lie', it is a large crying lie, testifying, according to Brouwer,
to the fact that "Cantor's set theory as a whole is a pathological
incident in the History of Mathematics and a moral crime against future
generations (of meta-mathematicians and AST-people)".
So, there is a crying ethical issue generated by the
interpretation and the usage of the Axiom of Infinity within the
framework of the modern AST.
More minutely these questions (and a lot of others) are analyzed
in recent papers:
A.A.Zenkin, Logic of Actual Infinity and G.Cantor's Diagonal Proof
of the Uncountability of the Continuum. - The Review of Modern Logic,
Vol. 9, Number 3&4, 27-82 (2004).
A.A.Zenkin, Scientific Intuition of Genii Against Mytho-'Logic' of
Cantor's Transfinite 'Paradise'. - Philosophia Scientia, 9 (2), 145 -
163 (2005).
Some strings from the papers' conclusions.
The traditional Cantor's proof of the uncountability of real
numbers contains two HIDDEN necessary conditions. The explication of
these two conditions makes Cantor's proof invalid.
The first hidden necessary condition is Cantor's axiom (above).
The explication of this necessary condition makes Cantor's statement
'Continuum is uncountable' unprovable.
The second hidden necessary condition is as follows.
THE HIDDEN CANTOR-HODGES' POSTULATE. Within the framework of
Cantor's diagonal proof, from the RRA-assumption "X is countable" it
follows that only such the indexings of real numbers in the sequence x1,
x2, x3, . . . (E) are admissible which use ALL elements of the set
N={1,2,3,4,5, :}. Any other indexing which employs FEWER THAN ALL
natural numbers (e.g., any proper infinite subsets of the countable set
N) is categorically forbidden.
I would like to emphasize here that the hidden Cantor-Hodges'
postulate is the second necessary condition of "Cantor's diagonal
argument", since all 'bad' indexings of real numbers of the set X,
according to W.Hodges, really "fail to reach Cantor's conclusion" (see
his famous paper 'An Editor Recalls Some Hopeless Papers' in the
BSL-1998, Vol. 4, no. 1, 1-17.).
Moreover, the Cantor-Hodges' postulate, outlaws the transitivity
relation of equivalent sets within the framework of Cantor's proof, and
the postulate itself is simply a teleological assertion (only 'good'
indexings are permitted, since only these allow us "to reach a desired
Cantor's conclusion") which can have no relation to, by Feferman,
'really working' mathematics.
It is obvious that the logical failure of Cantor's theorem on the
uncountability of continuum changes essentially traditional logical and
methodological paradigms of mathematics-XX and philosophy of infinity,
and, at last, opens a way to solve main problems connected with I, II,
and III Great Crises in foundations of mathematics. [Kleene 1957],
[Zenkin 2004-1999, 1997].
Alexander Zenkin
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