First order logic, set theory and God

[polymath257]

Below is a copy-paste of my own thread that was posted in another forum. I'm reposting it here in hopes to spark an intelligent discourse on what I believe to be the most refined proof of God's existence.

Here it goes.

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Written in first order logic, Hatcher's proof of God is based on three axioms that he calls "empirically grounded" and an apriori assumption that "something exists."

The axioms are that:

P1. The principle of sufficient reason: All phenomena are either self-caused (i.e. A->A) or other-caused (B->A; B is not equal to A) but not both. Put another way, this principle says that the question "why?" is always meaningful. Everything happens for a reason.

P2. The potency principle: If A -> B then for all C element of B, A -> C. In other words if A is the cause of B then A is the cause of every part of B. There are several notions of causality in philosophy. Hatcher's notion of causality is total causality; i.e. it is not the straw that breaks the camel's back but the 1000 straws before it, the camel, gravity, and so forth, that give rise to the camel breaking its back.

P3. The principle of limitation: For all A, where A is an element of B, B -> A does not hold. This says a system (which Hatcher represents as a set) cannot be the cause of its own components. Hatcher justifies this by explaining any system has (1) form (the parts) and (2) function (the relationship between the parts). A car (the system) cannot be the cause of its own steering wheel (a part), because the car does not even logically exist until the steering wheel exists. Thus the car's existence cannot precede the steering wheel's existence.

Hatcher shows that the logical outcome of these 3 axioms together with the above noted assumption are the existence of a "unique, universal, uncaused cause."

Throughout this work, Hatcher strives to make his assumptions (his axioms) and modus operandi (first order logic) explicit. Unlike many proofs of God (beginning with the proof advanced by Aristotle) Hatcher's proof does not appeal to the absurdity of an infinite regression of causes. Hatcher argues that because his proof is formulated in first order logic one must invalidate one or more of his three empirically grounded axioms to refute it. At the same time he shows that doing so is difficult as it commits oneself to beliefs not commonly accepted in the scientific community such as the existence of non-causal systems (something not observed to date).

In the following I have provided a more complete formulation of Hatcher's proof.

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The following is a excerpt from pages 82 through 86 of "Love, Power, and Justice: The Dynamics of Authentic Morality" by William S. Hatcher.

Copyright 1998 by William S. Hatcher

Chapter 3, section 4

The Existence of God

Our purpose in undertaking this study is to establish the existence of God on a totally objective basis, as a necessary logical feature of the overall structure of reality itself.

+ ( R ) ;Reality = the totality of existence, everything there is.
+ (P);Phenomenon = some portion of ( R )  
+ Causality = relationship between (P) A and (P) B, which holds whenever A is a cause of B (symbolized A → B).

This means that A contains a sufficient reason for the existence of B. => everything B that exists must either be preceded by a cause A different from B (A → B and A ≠ B), or else contain within itself a sufficient reason for its existence (B → B).

<=>In the former case, we say that B is caused or other-caused and in the latter uncaused or self-caused.

* The principle that every existing (P) must either be caused or uncaused (and not both) is the principle of sufficient reason.

<=>Another basic relation between (P) is the relation of part to whole: we write A ∊ B whenever the entity A is a component of the system (composite phenomenon) B.

Notice that A may also be composite, but must be an entity (not just an arbitrary system) in order to be a component of another system B (whether the latter is an entity or not).

Two systems (whether entities or not) may also be related by one being a subsystem of the other. We write A ⊂ B whenever A is a subsystem of B. This means precisely that every component E ∊ A is also a component E ∊ B.
=> For example: a single leaf would be a component of a tree, but all the leaves together would constitute a subsystem of the tree.

If E is either a component or subsystem of B, then E is a part of B.

From the strictly logical point of view, the defining or characteristic feature of an entity A is that A can be a component of some system B, A ∊ B.

<=> entities are components while systems have components (they are composite phenomena). Moreover, some systems also are components.

Thus, with respect to composition, we have three distinct categories of phenomena:

- (P) may be noncomposite (have no components), in which case it is necessarily an entity.
- (P) may be a composite entity, in which case it both has components and is a component.
- (P) may be composite without being an entity, in which case it has components but can never be a component.

Causality and composition are related to each other by the obvious potency principle, which says that if A → B, then A must also be a cause of E, where E is any component or any subsystem of B.

<=> to be a cause of B is to be a cause of every part of B -- its components and its subsystems. This means that our notion of causality is that of complete cause (philosophy recognizes several different notions of "cause").

Finally, the existence of a whole system obviously cannot precede the existence of its components (rather, the constitution of a whole obviously supposes and depends upon the prior or simultaneous existence of its components).

We thus have the principle of limitation= every composite phenomenon A, A cannot be a cause of any of its components.

=> It follows immediately from these principles that no composite phenomenon can be self-caused, for suppose A → A where A is composite. Then, by the potency principle A → E, where E is any component of A. But this contradicts the limitation principle.

In fact, from these valid principles of causality and composition, we can logically deduce the existence of a unique, noncomposite, self-caused, universal cause G. This entity, whose existence we prove, is God (by logical definition). This God is not some abstract figment of our imagination but the actual, ultimate cause of all existing phenomena and entities, the origin of all being.

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[==== Since the proof is easy, we give it here in full. However, the reader who already accepts and understands the existence of a universal uncaused cause (i.e., God) can safely skip the details of the proof without diminishing his or her understanding of the subsequent sections of the course.

Let V=collection (universe) of all existing entities.

Since V is composite it cannot be self-caused (see above) and so must have a cause G (different from V itself).

Thus, G → V; G ≠ V

Moreover, every existing (P) A is either an entity, and thus a component of V, or else a system all of whose components are in V -- in which case A is a subsystem of V.

Thus, G is either a component or a subsystem of V. But, in either case, G → G by the potency principle.

Thus, G is self-caused and hence noncomposite (no composite can be self-caused as shown above).

Finally, since G → V and every (P) A is a part of V then by the potency principle, G = universal cause (the cause of every existing phenomenon, including itself).

Finally, we show that G = the only uncaused phenomenon, for suppose there is another such phenomenon G'.

Then G → G' (since G is a universal cause). But since G' is self-caused it cannot be other-caused by the principle of sufficient reason.
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Thus, G = G' and the uniqueness of G is established.===]

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With that being said, I would be more than curious to see if anyone could spot a noticeable error in Hatcher's logical deduction

Which version of set theory are you using for your 'component' relation? In particular, what allows you to form the system V? How do you avoid Russell's paradox in your set theory? We *know* that naive set theory is inconsistent and no 'set' of the form V can exist in standard ZF set theory.

At the very least, you don't have a first order system when you allow the formation of arbitrary collections (that being a second order construction), *unless* you specifically give the rules of set formation.

This doesn't even address whether the axioms used  for the causality relation have anything at all to do with reality (and others have commented that issue).