[FOM] R: Cantor and "absolute infinity"
drago at unina.it
Sun Feb 19 17:26:31 EST 2006
Avron wrote :
>... Did he have any characterization of this "set",
> except as the set of all sets which are subsets of A
> (a clear application of unrestricted comprehension)?
> ...Cantor [needed] to give
> a convincing argument why P(N) is a set
> and not an absolutely infinite collection .... I cant see what
> this argument could have been. In fact, I cant see what
> criterion Cantor could have had for telling what collections
> are "absolutely infinite" (and according to which
> P(N) is not "absolutely infinite") except to ban collections/sets
> the assumption of which leads to contradictions...
Then Friedman replied:
This issue is easily resolved if you look at set theory as an extension of
finite set theory.
Specifically, HF = the *infinite set* of all hereditarily finite sets, is
the standard model of (pure) finite set theory.
I ask to Friedman:
how you are sure that this extension is unique? In the case of multiple
extensions, why you choose?
In my opinion, set theory is the first scientific theory which introduced
philosophy in science; just after it, special relativity introduced the
human observer as an essential component of his theory; after two decades
quantum mechanics was built by manifestly declaring to rely upon the
"Copenhagen spirit", a gentle way to avoid to call it "philosophy".
Indeed, in Mathematics the word "set" is universally defined as a "primitive
notion". Hilbert temporarily solved this puzzle by relying his systems upon
notions implicitly defined , in the hope to assure the consistency of them.
That was not the case.
Hence, the question remains totally valid: what is a "set"? And moreover,
what means that "x BELONGS to a set"?
There exists a plain answer: "set" is a word which cannot be defined inside
Mathematics; rather it can be defined by only some philosophical notions (I
trying: totality, wholeness, comprehension, etc. ), i.e. inside the realm
with Mathematics, philosophy.
I can support my thesis in historical terms. In 1772 a memoir to the Berlin
Academy of Science by Lagrange launched the program to unify Mathematics
(geometry in particular) under calculus (conceived as theory of
infinitesimals). Instead in 19th Century Mathematics
surprisingly proliferated in a lot of appartently irreconciliable theories;
even geometry proliferated in different theories. After a serious crisis
(not accepted by mathematicians) Cantor invented set theory which gaves to
the mathematician a great certainty because it offered an universal and
uniform treatement of almost the whole, enormous body of mathematical
theories; i.e. a result which never has been obtaind before; a promotion to
>From what set theory do obtain its so great power of unification, hanging
whatsoever mathematical theory? It is apparent that inside Mathematics the
answer has to be: from no more than its notions of "set" and "belongs". But
these two words constitute a very little basis with respect to the enormity
mathematical theories involved by the above question.
Hence, it is trivial to deduce that only the philosophical meanings of these
two words can achieve such an universal capability.
All that for supporting the idea that likely as infinitesimals were
philosophically biased, but at present we well know that they constituted an
useful tool for discovering a great part of both mathematics and theoretical
physics, so set theory can have suggested good mathematics, though by not
completely mathematical means.
Hence, no bans; but a pluralist attitude, which is overt to all possible
the foundations of Mathematics, and at the same time is ready to even more
accurately circumscribe them.
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