
Cantor's argument and the Potential Infinite.
Posted:
Nov 16, 2012 2:49 AM


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 biinjective 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 ultrafinistim 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 ultrafinitism 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

