Das Kalenderblatt 090820

The only possible conclusion [given the Lowenheim-Skolem Theorem]
seems to be that notions such as countablility and uncountability are
inherently relative. [...] What we ourselves take to be P(omega) never
appears to be anything but uncountable. The relativist, convinced that
our own point of view is in turn limited, urges us to acknowledge the
possibility that what we ourselves take to be P(omega) is not - as
viewed from some even higher vantage-point from which it may yet be
countable. But how are we to make sense of this? Certainly not by
trying to view P(omega) from two different points of view at once;
that would be incoherent. Nor by trying to view it simply from this
point of view; that would make the possibility unintelligible. But if
it were possible to view it from an absolute standpoint, then
relativism itself would lose its rationale and there could be no
objection to saying that P(omega) contained all of omega's subsets and
that it was unconditionally uncountable. So if we do deny the absolute
uncountability of P(omega), then what exactly are we denying and
where, so to speak, are we denying it? (The mere fact that there are
legitimate concepts of countability and uncountability which do
involve relativisation to certain domains is beside the point. The
relativist wants to insist that there are no absolute concepts of
countability and uncountability - that it makes no sense to describe P
(omega) as unconditionally uncountable). We, in mounting a general
investigation into what sets are like, can only aspire to know whether
or not P(omega) is countable, as it were /here/. It is not. But we can
have no grasp on any distinction between what is true here and what is
true simpliciter. So P(omega) is uncountable simpliciter.
Yet the very use of the word "here" appears to vindicate the
relativist. There remains a real predicament. That which cannot
legitimately be stated (relativism) appears, for all that, to impress
itself upon us as soon as we step outside mathematical practice and
reflect on what is revealed therein. This predicament is directly
analagous to that which Wittgenstein faces in the Tractatus. There is
no particular point of view in the world which can be spoken of as
here: our point of view is a limit of the world. That is, there is no
particular set in the hierarchy of sets which can be spoken of as the
intended range of the quantifiers: they are intended to range over the
whole hierarchy (though not even this can properly be said). And here
an intriguing possibility arises. If we are prepared to extend the
analogy with the Tractatus, then it will become apparent that, despite
the fact that relativism defies any coherent statement, the debate
between the relativist and the non-relativist is in a very deep sense
irresoluble. For what the relativist means is quite correct; only it
cannot be said, [1]

Manches doch! Die Relativitätstheorie schimmert überall durch. Nichts
(und nicht einmal das) ist absolut. Generalsuperintendentist D. Dr.
sc. Hütte knurrt [2]: "Das Zwölfprophetenbuch im AT berichtet von 12
Propheten, das steht unwiderleglich fest." Zwei Absätze weiter fährt
er fort (Hervorhebung von mir): Wir haben also die *Wahl*, uns auf die
in ihrer trostlosen Überschaubarkeit Finitophobien stimulierende
Anzahl von nur 12 Propheten zu beschränken oder in sublimem Trotz an
eine übererzählbare Unzahl von Propheten zu glauben.

[1] A. W. Moore: "Set Theory, Skolem's Paradox and the Tractatus",
Analysis 1985, 45.
http://analysis.oxfordjournals.org/cgi/pdf_extract/45/1/13

[2] Manuskript, verschollen. Aber eine analoge Relativierung ist aus
der modernen Beweistheorie bekannt.

Gruß, WM
.

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