Date: Nov 16, 2012 2:49 AM Author: Zaljohar@gmail.com Subject: Cantor's argument and the Potential Infinite. 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