Re: "Kantor dust"

["Bhupinder Singh Anand" <re@alixcomsi.com>]

On Sunday, February 08, 2009 8:33 PM, Paul Taylor wrote in Categories:

" ... people should be clear about WHICH OF MANY "CONSTRUCTIVE" THEORIES OF
MATHEMATICS they mean as the context of their comments. ... If you want to
construct Cantor space from the reals by the "missing third" construction, I
would think that it makes little difference whether you start from Cauchy or
Dedekind."

Actually, there seem to be two issues involved here.

The minor of the two: that the "missing third" construction yields a
'limit', namely the "Kantor dust";

The other: whether we can agree on any 'constructive' concept at all, such
as, say, the standard interpretation of first-order Peano Arithmetic.

As to the first: Consider an equilateral triangle ABC of height h and side
s. Divide the base AC in half and construct two isosceles triangles of
height h.d and base s/2, where 1>d>0. Iterate the construction on each
constructed triangle ad infinitum.

If the "Kantor dust" is a legitimate (fractal) set, then this construction,
too, should yield a limiting configuration.

Since the height of the constructed triangles at the n'th construction is
h(d^n), and 1>d>0, it would seem that the base AC of the original
equilateral triangle will always be the limiting configuration of the
opposing sides.

This is indeed so if 1/2>d>0, since the total length of the opposing sides
is a Cauchy sequence whose limiting value is, indeed, the length s of the
base AC.

However, if d=1/2, the total length of all the sides opposing their base on
AC is always 2s!

Moreover, if d>1/2, the total length of all the sides opposing their base on
AC is a monotonically increasing value.

(To give it a practical flavour, let s be one light-year, and consider how
long it would take a light signal to travel from A to C along the sides
opposing the base in each of the above cases; and whether it makes any sense
to assert that all the cases must have a limiting configuration.)

So, whatever it is that the "missing third" construction is supposed to
yield, it certainly cannot have any relation to the terms "third",
"construction", "limit" and "set" under any interpretation of these terms
that we normally conceive of in mathematics.

The issue - as Thoralf Skolem emphasised in "Some remarks on axiomatized set
theory", delivered in an address before the Fifth Congress of Scandinavian
Mathematicians in Helsinki, 4-7 August 1922, with respect to the the Axiom
of Choice - is that set-theory admits statements that are essentially
non-verifiable under any conceivable interpretation; statements, moreover,
which do not express any definable content and cannot, therefore, be
expected to communicate any meaningful information unambiguously under
interpretation:

"So long as we are on purely axiomatic ground there is, of course, nothing
special to be remarked concerning the principle of choice (though, as a
matter of fact, new sets are not generated univocally by applications of
this axiom); but if many mathematicians---indeed, I believe, most of
them---do not want to accept the principle of choice, it is because they do
not have an axiomatic conception of set theory at all. They think of sets as
given by specification of arbitrary collections; but then they also demand
that every set be definable. We can, after all, ask: What does it mean for a
set to exist if it can perhaps never be defined? It seems clear that this
existence can be only a manner of speaking, which can lead only to purely
formal propositions---perhaps made up of very beautiful words---about
objects called sets. But most mathematicians want mathematics to deal,
ultimately, with performable computing operations and not to consist of
formal propositions about objects called this or that."

That the problem lies deeper - in fact at the very core of our fundamental
assumptions - is seen if we note that our logical thinking is universally
founded upon the validity of Aristotle's particularisation.

This holds that an assertion such as, 'There exists an x such that F(x)
holds'---usually denoted symbolically by  '(Ex)F(x)'---can be validly
inferred in the classical logic of predicates from the assertion, 'It is not
the case that, for any given x, F(x) does not hold'---usually denoted
symbolically by '~(Ax)~F(x)'.

Now, in his 1927 address, Hilbert reviewed in detail his axiomatisation of
classical Aristotlean predicate logic as a formal first-order
epsilon-predicate calculus, in which he used a primitive choice-function
symbol, 'epsilon', for defining the quantifiers 'for all' and 'exists'.

In an earlier address "On The Infinite", delivered in Munster on 4th June
1925 at a meeting of the Westphalian Mathematical Society, Hilbert had shown
that the formalisation proposed by him would adequately express Aristotle's
logic of predicates if the epsilon-function was interpreted to yield
Aristotlean particularisation.

This, as Hilbert remarked in his 1927 address, was what he had set out to
achieve as part of his 'proof theory':

"... The fundamental idea of my proof theory is none other than than to
describe the activity of our understanding, to make a protocol of the rules
according to which our thinking actually proceeds."

What came to be known later as Hilbert's Program---which was built upon
Hilbert's 'proof theory'---can be viewed as, essentially, the subsequent
attempt to show that the formalisation was also necessary for communicating
Aristotle's logic of predicates effectively and unambiguously under any
interpretation of the formalisation.

This goal is implicit in Hilbert's remarks:

"Mathematics in a certain sense develops into a tribunal of arbitration, a
supreme court that will decide questions of principle---and on such a
concrete basis that universal agreement must be attainable and all
assertions can be verified."

"... a theory by its very nature is such that we do not need to fall back
upon intuition or meaning in the midst of some argument."

The difficulty in attaining this goal constructively along the lines desired
by Hilbert---in the sense of the above quotes---lies in the fact that, as
Rudolf Carnap emphasised in a 1962 paper, "On the use of Hilbert's
epsilon-operator in scientific theories", the Axiom of Choice is formally
derivable as a theorem in a set theory ZF_epsilon, which is, essentially,
Zermelo-Fraenkel set theory where the quantifiers are defined in terms of
Hilbert's epsilon-function.

The significance of this lies in the accepted interpretation of Paul Cohen's
argument in his 1963-64 papers; they are primarily taken as definitively
establishing that the Axiom of Choice is independent of a set theory such as
ZF.

Now, Cohen's argument---in common with the arguments of many important
theorems in standard texts on the foundations of mathematics and
logic---appeals to Aristotle's particularisation when interpreting the
existential axioms of ZF (or statements about ZF ordinals) in the
application of the seemingly paradoxical (downwards) Lowenheim-Skolem
Theorem for legitimising putative models of a language (such as the standard
model 'M' of ZF, and its forced derivative 'N', in Cohen's argument).

Thus, Cohen's argument should really be taken to establish that, not only is
Aristotle's particularisation 'stronger' than the Axiom of Choice, but that
there is no sound interpretation of ZF that can appeal to Aristotle's
particularisation.

Moreover, the larger significance of Hilbert's formalisation of Aristotle's
particularisation is that---in formal languages that prefer the more
familiar '[A]' - as in '[(Ax)]' - as a primitive symbol to Hilbert's more
logical choice function 'epsilon'---it implicitly gives formal legitimacy to
Alfred Tarski's standard definitions of the satisfaction, and truth, of the
formulas of a formal language under an interpretation, since these
definitions faithfully mirror the particular interpretation of Hilbert's
formalisation that appeals to Aristotle's particularisation.

The reason: Under Tarski's definitions, the formally defined logical
constant '[E]' in an occurrence such as '[(Ex)]'---which is formally defined
in terms of the primitive (undefined) logical constant '[Ax]' as '[~(Ax)~
...]'---always appeals to an interpretation such as 'There is some x such
that ...' in any formal first-order mathematical language.

In other words, Tarski's definitions ensure that, if the first-order
predicate calculus of a first-order mathematical language admits
quantification, then any putative model of the language must interpret
existential quantification as Aristotle's particularisation.

Selecting such a strong interpretation---i.e., one which favours Aristotle's
particularisation---for the standard interpretation, say S, of the formal
Peano Arithmetic PA has significant consequences.

For instance, if we accept the logical validity of such interpretation, then
S is sound (i.e., every PA-theorem interprets as true under S.

Further, if S is sound, then PA is omega-consistent (i.e., we cannot have a
PA-formula [F(x)] such that [F(n)] is PA-provable for any given PA-numeral
[n], and [~(Ax)F(x)] is also PA-provable).

Now, in his seminal 1931 paper, Godel showed that if a Peano Arithmetic such
as his formal system P is omega-consistent, then it is incomplete, in the
sense that he could constructively define a P-formula [R(x)] such that
neither [(Ax)R(x)] nor [~(Ax)R(x)] are P-provable.

However, he also showed in this paper that if P is consistent and [(Ax)R(x)]
is assumed P-provable, then [~(Ax)R(x)] is P-provable.

By Godel's definition of P-provability, it follows that there is a finite
sequence [F_1], ..., [F_n] of P-formulas such that [F_1] is [Ax)R(x)], [F_n]
is [~(Ax)R(x)], and, for 2=< i =< n, [F_i] is either a P-axiom or a logical
consequence of the preceding formulas in the sequence by the rules of
inference of P.

Now, a proof sequence of P necessarily interprets as a sound deduction
sequence under any sound interpretation of P. It follows that we cannot have
a sound interpretation of P under which [(Ax)R(x)] interprets as true and
[~(Ax)R(x)] as false.

Since both [(Ax)R(x)] and [~(Ax)R(x)] are closed P-formulas, it follows that
the P-formula [(Ax)R(x) => ~(Ax)R(x)] interprets as true under every sound
interpretation of P.

By Godel's completeness theorem, [(Ax)R(x) => ~(Ax)R(x)] is, therefore,
P-provable; whence [~(Ax)R(x)] is P-provable.

Since Godel also showed that, if P is consistent, then [R(n)] is P-provable
for any given P-numeral [n], it follows that P is not omega-consistent.

Since Godel's argument holds in PA, we further have that the standard
interpretation S of PA is not sound; moreover, no sound interpretation of PA
can appeal to Aristotle's particularisation!

Thus the difficulty in agreeing upon the concept of a 'constructive' theory
is deep-rooted in our dependence on Aristotle's logic of predicates which,
whilst allowing us the luxury of expressing the most subjectively conceived
of our abstract concepts not only in languages of common discourse such as
English, but also in mathematical languages such as ZF, is inadequate for
ensuring that that which we express in the most basic of our mathmatical
languages, namely Peano Arithmetic, can be communicated effectively and
unambiguously.

That a sound interpretation of Peano Arithmetic exists - moreover, one that
allows us to communicate effectively and unambiguously - is indicated by the
fact that we unhesitatingly entrust our lives each moment of each day to
mechanical and electronic artefacts whose reliability is essentially founded
on the ability of PA to admit unambiguous and effective communication.

Accordingly, in a recently arXived paper (link below), I consider a
weakened, finitary, interpretation B (of an omega-inconsistent PA) which
avoids appealing to Aristotlean particularisation in the interpretation of
the existential quantifier, and which is actually implicit in Turing's 1936
analysis of computable functions.

This is the interpretation B of PA obtained if, in Tarski's inductive
definitions---of the satisfaction and truth of the formulas of PA under the
'standard' interpretation S of PA---we apply Occam's razor and weaken the
definition of subjective Tarskian satisfiability by replacing it with an
algorithmically verifiable definition of objective Turing-satisfiability.

Not only is the interpretation sound, but it implies that PA is categorical;
we can thus, in principle, communicate perfectly with technologically
advanced extra-terrestrial intelligences.

http://arxiv.org/PS_cache/arxiv/pdf/0902/0902.1064v3.pdf

Regards,

Bhup