Re: Existence of reals and observation of them

On Dec 10, 8:11 pm, cbr...@xxxxxxxxxxxxxxxxx wrote:
As Frege pointed out, the translation is possibly suspect; and more
so, we don't know the exact context of Cantor's statements (the
contributor is well known for taking quotes out of context).

A friend sent me a somewhat larger excerpt in German, and the
translation seems alright. I attach it at the end of this post.

What Cantor (apparently) proposes is essentially what we do when we
assume the existence of a Hamel basis for the reals as a vector space
over Q: each real is uniquely defined by a sum of a finite subset of
some uncountable set of basis vectors (i.e. every definition is
finite, but the set of symbols is uncountable).

Maybe, but how could he think there might be a humanly manageable
uncountable set of symbols?

Keep in mind that the paradox itself relies on the assumption that the
reals can be well-ordered; so we're already in the realm of the
utterly non-constructive. Konig tried (and failed) to show that such
non-constructive assumptions were logically inconsistent; but instead
he demonstrated something else.

Note that Konigs argument stands even if we grant that there exists an
uncountable system for definition. We can still ask "but what happens
if we only use a countable language (e.g., ZFC) to define some
countable /subset/ of the reals, and then well-order the remaining
'undefined' reals to define 'the least undefinable real'?".

Yes, the paradox stands if we allow for a language to be self-
referential in the sense that it is able to define a number by
reference to all of its possible definitions of numbers. But Cantor
could have thought that only an uncountable language could follow so
closely our thought.

This is odd, anyway, because the problem arises in natural language,
which admits only finite definitions (as Cantor underlines) by means
of a finite set of symbols.

It looks like Cantor thought an uncountable set of definitions is
available in natural language and no countable language would permit
König-style definitions.

It really isn't a matter of the consequences of Cantor's theory of
cardinality. In the long run, Konigs paradox (like Richard's and
Berry's) has to do with exposing the general mathematical concept of
"definable", formally speaking.

Yes, definability paradoxes have definitely to do with some problem
with the informal concept of definition. Maybe this concept is
'indefinitely extensible' like the concept of set according to
Dummett, Shapiro, Wright...

But I meant Cantor could simply not realize that a finite or countable
set of symbols can only yield a countable set of finite chains.

Strange, anyway.

----------------
Cantor's letter can be found in "Georg Cantor
Briefe", H. Meschkowski, W. Nilson (editors), Springer, Berlin
(1991),p.446.

I have no copy but perhaps someone here has one and can tell us what
Hilbert replied. German text:

[QUOTE] Ich hätte mit Ihnen unter Anderem gern über die Königschen und
Poincaréschen Versuche in der Mengenlehre gesprochen, die meiner
Meinung nach auf irrigen Grundsätzen beruhen und nur geeignet sind,
Verwirrung anzurichten.
König will zwei Arten von reellen Zahlen unterscheiden; solche, die
"endliche Definitionen" zulassen und solche, die "unendliche
Definitionen" erfordern.
Eine jede Definition ist aber ihrem Wesen nach eine endliche, d.h.
sie erklärt den zu bestimmenden Begriff durch eine endliche Anzahl
bereits bekannter Begriffe B1, B2 , B3, ..., Bn.
"Unendliche Definitionen" (die nicht in endlicher Zeit verlaufen)
sind Undinge. Wäre Königs Satz, daß alle "endlich definirbaren"
reellen Zahlen einen Inbegriff von der Mächtigkeit aleph_0 ausmachen,
richtig, so hieße dies, das ganze Zahlencontinuum sei abzählbar, was
doch sicherlich falsch ist.
Es fragt sich nun, welcher Irrthum liegt dem angeblichen Beweise
seines falschen Satzes zu Grunde?
Der Irrthum (welcher sich auch in der Note eines Herrn Richard im
letzten Hefte der Acta mathematica findet, welche Note Herr Poincaré
in dem letzten Hefte der Revue de Métaphysique et de Morale mit
Emphase herausstreicht) ist, wie mir scheint, dieser:
Es wird vorausgesetzt, dass das System {B} der Begriffe B, welche
eventuell zur Definition von reellen Zahlenindividuen herangezogen
werden müssen, ein endliches oder höchstens abzählbar unendliches sei.
Diese Voraussetzung muß ein Irrthum sein, da sich sonst der falsch
Satz ergeben würde: "das Zahlencontinuum hat die Mächtigkeit aleph_0".
Irre ich mich, oder habe ich Recht.? [END OF QUOTE]



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