Math Forum Discussions

george

Posts: 800
Registered: 8/5/08

Re: Objections against Cantor
Posted: Nov 19, 2012 7:13 PM

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On Nov 19, 5:47 am, Zuhair <zaljo...@gmail.com> wrote:
> (3) Argument of Finitism: Only finite sets and finite processes exist,
> nothing else. So Cantor's argument is flying high up in imaginative
> thinking far away from the grounds of reality.

It CAN'T be VERY far.
Even if you think that only finite things exist, i.e., even if you
think
that every set has a finite natural number as a cardinality, the
problem
becomes that there are AN INFINITE number OF THOSE.
If some things are cardinalities, and some are not, or if
cardinalities
are encoded as sets (in 1st-order ZFC we typically encode
every cardinal as an initial ordinal), then you need some
justification
for claiming that there is NOT a set of all and only the cardinals.
The point is, you can't credibly claim that there "exist only" finite
things when THE NUMBER OF such finite things IS PROVABLY infinite.

> The problem with that is that there is no clear justification of why
> should the notion of "finiteness" be given so much credit over
> "infiniteness"

But it just plain ISN'T! EVEN if we say that EVERY THING is finite,
we STILL
wind up with an INFinite number OF THINGS!!

> if we say that everything in our world is finite and
> deem infinity as being at best a logically consistent imaginative
> ideal,

THAT IS NOT credible because these INFINITELY many different
cardinalities are EACH AND EVERY AND ALL *concretely*IN* our world.
Our concrete world ITSELF contains INFINITELY many CONCRETE things.
So YOU CAN'T credibly or consistently banish "infinity" to the realm
of "imaginative ideal". The universe of discourse as a whole, GIVEN
THAT EVERYTHING IN IT IS FINITE AND CONCRETE, IS NOT imagined or
ideal.
IT TOO MUST be concrete.

> then the same can be exactly said about finitism also, since it
> accepts large finite sets that we may not happen to even touch in any
> finite way, like numbers that we cannot describe using all of our
> abbreviation capacity,

Obviously, there are NO such numbers. For any number not described by
a given abbreviation capacity, THERE IS A DIFFERENT abbreviation/
notation THAT DOES describe it.

> Actually the infinite seems much simpler and easily touched by human
> imagination than most of large finites.

Well, yes, conceded, my argument did presume that the other side had
already insisted on the concrete relevance of EVERY finite
cardinality, no
matter how large. If you are going to disavow some of those, then
nobody
has anything to say except "why?". That they can't be referred to or
imagined
is frankly obviously false. Given world enough and time, each and
every one of them could in fact be written out in unary. The fact
that individual human brains are too finite to cope with that is
SURELY IRrelevant.
Seriously, do you REALLY want to claim that Ack(6,6) DOES NOT EXIST??