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[HM] Infinity and the "Noble Lie"
Posted:
Dec 11, 2005 12:11 PM


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 wellknown ASTpeople 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 nondeliberate 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 metamathematicians and ASTpeople)".
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, 2782 (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 CANTORHODGES' POSTULATE. Within the framework of Cantor's diagonal proof, from the RRAassumption "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 CantorHodges' 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 BSL1998, Vol. 4, no. 1, 117.).
Moreover, the CantorHodges' 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 mathematicsXX 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 20041999, 1997].
Alexander Zenkin



