First order logic, set theory and God

[Amarok]

The way the proof avoids Russell's paradox boils down to how concepts are defined and classified.


It is crucial to understand that with respect to composition, we have a hierarchy of complexity.

Let us define the words - componential as the attribute of being a component and composite as the attribute of having components.

1- At the lowest level is the componential non-composite entity A which is a component but does not have any component, such that x∉A∈B, for some B.

2- Next, are componential composite entity A which both have components and are components, such that E∈A∈B, for some E and B,

3- Finally, you have the non-componential composite entity A which have components but is never a component of anything, such that B∈A∉x, for some B.


Now according to the above definitions, V can be defined as the whole of reality which is that non-componential composite entity whose components are precisely all existing entities who are necessarily componential and can either be composite or non-composite.

Now according to Russell's paradox, the pertinent question to ask ourselves is if V as the set of all existing componential entities, a componential entity itself and therefore a member of that set?

Well no, since by the very definition of V, it has already been established that it is non-componential. But then one might ask " how can we know that V is non-componential and hence not a component of anything ? " Well because if we assumed V to be componential, then by definition it cannot be the set of all existing componential entities as one will commit the error of Russell's paradox where V becomes a member of the set of all componential entities. Therefore V∈V.

Alternatively we ask, can V as an non-componential entity be the set of all componential entities ? Yes, because non-componential and componential are two distinct categories and thus V can become the set of all existing componential entitities without becoming a member of that set. On that basis, we have completely disqualified V as a universal sets of all sets(and therefore Russell's paradox is not an issue anymore) by restricting V as a set of all sets that satisfies the property that every entity A  of a set is componential.  


As for your argument about the principle of suffiency - your notion V being an infinite system requiring no cause is a baseless assumption, and I would even further argue, that such an assumption can be nullified by the very fact that V is composite. Since we can admit that V does not even exist until all its components exist, then it begs the question as to when will V ever be formed if it would take an infinite amount of time for it components to come into existence ? Never. Thus to entertain V as an infinite system is tantamount to the denial of it's very existence.

Well, V is an abstract set, so is not something that needs to come into existence (you haven't included time in your formal system). And, once again, you have not given an axiom that allows for the construction of V.
Now, your argument fails in the case of an infinite regress: it always exists as an infinite regress, so does not have to 'come into existence'. In fact, if we have an infinite regress, let S be the system of all entities in that infinite regress. Then S cannot be caused because any cause would have to come before that regress, which is impossible.

So, once again, your P1 is brought into question in the case of infinite collections.

This doesn't show that ALL infinite collections have no cause, but it does show that all infinite collections that contain an infinite regress are without causes.

Again, a much more reasonable version of P1 is to assume that all finite systems are caused, but leave open the issue of infinite systems. This is a minimal extension of what we can hope to know or verify and so is a more reasonable assumption to make than the stronger version you choose.

Furthermore, because of the question of infinite regresses (which are certainly a possibility), the version of P1 you adopt is likely to be false.

With this modification, you have the issue of determining whether your collection V is finite or infinite. Only in the former case are you guaranteed a cause.

And again poly you lay down the law against people who repeat the oft unsupported nonsense apologists trot out about an infinite regress . It's almost as bad as their unsupported assertions about nothing .