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From: sci.math on 15 Jul 2010 00:03
On May 24 2009, 12:20 pm, Martin Musatov <marty.musa...(a)gmail.com<
wrote:
< Martin Musatov wrote:
< < WM wrote:
< < < On 24 Mai, 09:17, lwal...(a)lausd.net wrote:
< < < < On May 23, 2:14 am, WM <mueck...(a)rz.fh-augsburg.de< wrote:
<
< < < < < On 23 Mai, 03:45, "Dik T. Winter" <Dik.Win...(a)cwi.nl< wrote:
< < < < < < I have still not found a definition of "potential
infinity" that is valid
< < < < < < within ZF.
< < < < < The definition of potential infinity would avoid that
obvious
< < < < < contradiction. I do not know whether it would make ZF free
of
< < < < < contradictions. But that is not my concern. I need that
axiom only for
< < < < < arithmetics where the natural numbers already are known.
<
< < < More precisely, I don't need any axiom at all. Natural numbers
are
< < < there whether or not someone takes the trouble to "formalize"
them.
< < < But it will help you (Dik) to understand potential infinity.
<
< < < < Once again, this thread is quickly approaching that thousand
post
< < < < mark yet again, which means that Google users such as WM and
< < < < myself will be leaving the thread soon.
<
< < < So it is.
<
< < < < I noticed how earlier, a standard set theorist (I forgot which
one)
< < < < made a crack about how since WM is an ultrafinitist and
Google's
< < < < maximum thread length is a thousand,
<
< < < This is so since the thread "A consideration concerning the
diagonal
< < < argument of G. Cantor", started by albrecht here in sci.logic in
Jan.
< < < reached 9244 posts in Oct. 2008.
<
< < < < this means WM must
< < < < believe that 1001 is the largest number. Debates about Google
and
< < < < other ways to access Usenet aside, I doubt that there's a link
< < < < between ultrafinitism and Google use. (Phil Carmody might have
< < < < found a link between being a so-called "crank"/"idiot" and
Google,
< < < < but not all "cranks" are ultrafinitists. In particular, I'm
not an
< < < < ultrafinitist -- I freely admit that the number 1002 exists,
and WM's
< < < < upper bound far exceeds this number.)
<
< < < Please note: There is no upper bound. There is no largest
natural
< < < number.
<
< < < < Since I'll be leaving the thread soon, let me at least comment
on
< < < < WM's latest attempt to work with Potential Infinity:
<
< < < < < Axiom Of Potential Infinity: For every natural number there
is a set
< < < < < that contains this number together with all smaller natural
numbers,
< < < < < and for every set of natural numbers there is a natural
number that is
< < < < < larger than every number of the set.
<
< < < < I see nothing wrong with this axiom, or trying to replace the
usual
< < < < Axiom of Infinity (which WM calls the Axiom of _Actual_
Infinity in
< < < < order to distinguish it from his new axiom) of ZF with this
axiom, to
< < < < obtain the new theory ZF-(Actual) Infinity+WM's Potential
Infinity. It
< < < < has no affect on the Axiom of Extensionality, unlike WM's
previous
< < < < attempts to define Potential Infinity, so the standard set
theorists
< < < < can't use Extensionality as an excuse to ignore this axiom.
<
< < < < We might even try to write the axiom in the language of ZF. We
try:
<
< < < < AneN (Ex (AmeN (m<=n -< mex))) & Ay (EneN (AmeN (mey -< m<n)))
<
< < < < But there's one problem -- the set N, of course, is not
supposed to
< < < < exist in WM's theory, so how can we even mention N in this
axiom,
< < < < when we can only talk about that which _exists_?
<
< < < N exists just as a potentially infinite set, AneN means for all
< < < natural numbers that you can think of, specify, name, identify,
< < < write, ... briefly: For all that are there.
<
< < < < Perhaps, instead of a set N, we can define the one-place
predicate
< < < < N(x), intended to agree with the standard definition of
natural number
< < < < (so that N(x) <-< x is a natural number). Then the axiom
becomes:
<
< < < < An (N(n) -< Ex (Am ((N(m) & m<=n) -< mex))) & Ay (En (N(n) &
Am ((N(m)
< < < < & mey) -< m<n)))
<
< < < < (As an aside, what's interesting about WM's axiom is that if
we
< < < < replace the word "natural" with "ordinal," then the resulting
< < < < formula is actually a theorem of ZF, and serves as a
definition
< < < < of successor and limit ordinals.)
<
< < < Is it? "For every set of ordinals, there is an ordinal larger
than the
< < < ordinals of that set."
< < < This means that you can pick every set of ordinals. Doesn't it
imply
< < < the existence of the set of all ordinals? Or do you presuppose
that
< < < you can pick every set of ordinals because the set of all
ordinals is
< < < anyhow excluded?
<
< < < However: If so, then you see that set theory does not improve
the
< < < situation.
< < < In potential infinity the set N is never understood as a set
that
< < < cannot be increased.
< < < In set theory, N is a set that cannot be increased, but on the
next
< < < level the set of all ordinals, as a set that cannot be
increased, must
< < < be excluded.
<
< < < This always reminds me on the problem of initial creation. The
world
< < < could not create itself. So God is introduced. But the question
who
< < < created God is forbidden.
< < <Jesus is Lord. Amen!
< < < If we use the above complete set N (that cannot be increased by
any
< < < element), then we are forced to introduce larger ordinals. But
then we
< < < are not allowed to use the set of all these ordinals. So we
haven't
< < < won anything but have left the solid grounds of science and have
< < < invented a highly questionable hypothesis. That is why I find
that
< < < mathematics has evolved to what better would be called math.
<
< < < Regards, MM
<
< There is only one thing the preceding chaos proves and proved
< definitively and that is the fact that P==NP. "Only a foold would
< insist on an impossibility to which there is no gain." Martin
Michael
< Musatov %P-Complete:

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