Re: A consideration concerning the diagonal argument of G. Cantor
- From: Julio Di Egidio <julio@xxxxxxxxxxxxx>
- Date: Sun, 18 May 2008 23:35:49 EDT
Ross A. Finlayson wrote:
Virgil wrote:
In article<dcd91fc3-00a4-4449-8008-845a1dc7efca@xxxxxxxxxxxxxxxx
legroups.com>,
WM <mueckenh@xxxxxxxxxxxxxxxxx> wrote:unless it is reached in
o
oo
ooo
...
Actual infinity is not reached in the vertical
ZF, they do not existthe horizontal.
Regards, WM
Unless such strings of characters are sets within
within ZF, and are irrelevant to ZF.
There is no such thing as "actual infinity" at all, unless we mean something like the element 'oo in N*'.
There is not even notion of "actual potential infinity", unless we mean something like the limit 'n -> oo, n in N(*)'.
OTOH, the notion of POTENTIAL is revealing a fundamental building block for _any_ logical framework -- then come finite/non-finite etc. etc.
Unions of fisons:
DEF: The UNION of _all_ fisons over _any_ *given* domain is "the maximal fison" (the covering fison for the domain).
DEF: The EXTENDED UNION of _all_ given fisons over _any_ domain is "the omega-fison" (still the covering fison for the domain).
Within the framework where ZF is, ZF's universe
contains itself.
ZF is incongruent, and so is its universe.
Those unary sequences as expansions trivially
represent natural
integers. That there is no maximal element in the
naturals yet omega
appended is that, has that for no set of those not
including it is it
that set.
ZF contains itself, ZF's universe is the Russell set
and contains
itself. Regular it is not.
Russell's paradoxical set is the empty set, but ZF is even more empty than the empty set itself!
Hopefully, ZF is not *our* universe...
-LV
.
Ross
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