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Topic: Cantor's argument and the Potential Infinite.
Replies: 17   Last Post: Nov 17, 2012 10:59 PM

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Zaljohar@gmail.com

Posts: 2,665
Registered: 6/29/07
Cantor's argument and the Potential Infinite.
Posted: Nov 16, 2012 2:49 AM
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I'll here present my version of potential infinity, which intends to
capture that concept, and prove that Cantor's diagonal argument is
applicable to that context also. So it doesn't necessitate a completed
actual infinity interpretation. However I'll also show another kind of
potential infinity scenario, which I call the strict potential
infinity, under grounds of which Cantor's diagonal argument cease
working, and I'll discuss why that strict form of potential infinity
is defective.

Generally speaking the argument of potential infinity says that NO
infinite set exists in the sense of a complete actual infinite set, so
the set N of all naturals is never completed, it is in a continual
state of becoming, and all completed sets of naturals are finite. So
if we denote {x| x is a natural} to be an object that stands for the
"potential" of infinitude of naturals, then we'll have

For all x. x is a completed set of naturals -> x is a finite proper
subset of {x| x is a natural}.

Notice here that {x| x is a natural} do not mean an actual completed
set of all naturals, it is just an object that uniquely stands for the
predicate "natural". It is neither finite nor infinite since those
would be terms defined only for completed sets, and {x| x is a
natural} is not a completed set, it is viewed to be in continual
becoming; Lets call such objects Potential sets.

Under those grounds it is said that Cantor's argument of
uncountability of the reals vanishes.

But this is NOT true.

We still can characterize Cardinality in this setting.

Two potential sets are said to have equal cardinality iff there is a
potential injection from one to the other at each direction.

Example: the potential sets N and E
There is a potential injective F map from N to E that is {(n,x)| x=2n
& n is a natural & x is even}
Also in the other direction you have a potential injective map G that
is
{(x,n)| n=x & x is even & n is natural}

The idea is that one cannot demonstrate any element of N that is not
in the potential domain of F. Since that domain is clearly N itself.

But can we have a similar potential bi-injective mapping between N and
R?

The answer is NO. Cantor's diagonal argument is also applicable
here!!!

Say there can be a Potential injection from R to N, lets call it I
Lets take the converse of I, denoted it as I^-1, which will be an
injection from the range of I to R. Now define a diagonal in a
potential manner by changing the i_th member of the digit sequence
representing the real in R that the i_th natural in the domain of I^-1
is coupled to, where the ordering is the ordinary natural order which
of course can be potentially defined. Now take the Potential
collection of all changed elements, and we'll have a potential
diagonal that is not in the potential range of I^-1, i.e. not in R. A
contradiction.

So Cantor's diagonal is applicable to potential infinity context!

The next scenario that I'll discuss is the STRICT potential infinity
scenario:

Here in this scenario, there is NO representation of any object that
can stand uniquely for a predicate that is potentially infinite, so
the predicate "natural number" is of course a potentially infinite
predicate since every finite set of naturals is not a completed set of
all naturals but yet this scenario simply stipulates that there is no
object that can stand uniquely for such predicate. So sets (which are
objects) only stand uniquely for finite predicates, there is no actual
infinite set, and there is also no potential infinite set like that
described in first scenario. There are only "PREDICATES" that
qualifies to be potentially infinite, however those are further
stipulated to be only described by formulas which are parameter free,
which of course known to be countable in number. So at the end we
clearly have no grounds for any proof of uncountability.

The problem with this scenario is that it is too restrictive, a super-
task for example cannot be represented by it, it is actually not
faithful to the concept of potential infinity itself, since informally
a potentially infinite predicate yields a potentially infinite
collection of objects that stands for that predicate, which serve as a
potential extension of that predicate. Now to go and shun that object
from existence like that makes one wonder about the potential those
predicates are all about, it is a potential in vain, from one aspect
those predicates range over objects in a continual manner, and from
the other aspect we don't see that continual extension, it simply
vanished, just like that? its like continually blowing into
nowhere???

Actually to me a more faithful argument would be to call the above
scenario "finitism", this would suit it better, which is though
restrictive in the above manner, but yet it is faithful to its
original stance, albeit not fully so to speak.

The real faithful scenario is actually ultra-finistim which simply
says that there are no infinite extensions, nor there is anything in
continual being. Everything is finite and ends up by some large
finite, and that's it. So this doesn't only shun potential infinite
collections, it also shuns MOST of finite numbers from existence, and
only accept the few handy ones that we can experience with and can
communicate, those that our machines and us can reach with the
strongest abbreviation notions we can have (which is of course also
finite).

Of course under that scenario which is claiming to be a reality
scenario, I say under this scenario just mentioning the matter of
infinite whether potentially or complete is deemed as a fantasy, and
any thought about it relates to speech about fantasies whether that
argument was consistent in form or not, it is not significant since it
is not about the real world we are living in, that's how matters are
seen from this perspective.

However the subject of whether ultra-finitism is true or not, is
actually another subject that is not about potential infinity. What I
wanted to say is that concepts like the Actual infinite or even the
potential infinite that I've presented at the head post are more
faithful concepts to their informal background, than the argument of
strict potential infinity that from one angle attracts those who wish
to speak about the infinite in a potential manner, but yet on the
other hand stipulate a restriction that is not faithful to what it
began with in the first place.

So in nutshell even under potential infinity background, still
Cantor's diagonals can be constructed and works to show that the
potential set R of reals is still having potentially more elements
than the potential set N of naturals.

Zuhair









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