Recently, the Prof. Harvey Friedman published a paper in which he proves the existence of God starting from the consistency of the mathematics. Does someone know if this proof has been refuted or it can be considered the last word in the ontological evidences? Thanks.

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    Just to point out, a conditional proof with the consistency of mathematics as its antecedent isn't a categorical proof, since it is an open question whether mathematics is consistent. Moreover, a face-value reading of Godel might suggest that no proof to the effect that mathematics is consistent is possible. So I don't see any particular reason to worry about a conditional with an antecedent that is impossible to prove. – Paul Ross Nov 7 '13 at 14:31
  • Quick glance. Does this paper go like: if god then consistent math? If so, then it simply isn't an (attempted) ontological proof. Is this claimed somewhere (other than in the question v1 above)? – user3164 Nov 7 '13 at 17:23
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    @Ricardo Let me be explicit. Where is it stated that this is supposed to be a "proof for the God's existence"? – user3164 Nov 7 '13 at 18:05
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    @Riccardo.Alestra Well, I don't see it. What I do see: "We have been able to prove, using an Angel, that mathematics is free of contradiction (I.e., ZFC is consistent)." "POWERFUL AXIOM: THERE EXISTS AN ANGEL!" This paper/ppt does not claim to contain a proof for the existence of gods/angels. If you agree, I suggest changing the question. – user3164 Nov 7 '13 at 18:24

After a quick reading of the paper, this is what I have been able to make of it.

As far as attempts to link mathematics and theology go, it at least presents some nuanced ideas (that no object accessible to us is perfect, for instance); but it does not seem to me that this paper either proves the existence of god, or even makes any sort of strong argument in the case of god. What it does do is show how the notion of a 'positive' predicate on properties, introduced by Gödel in his ontological proof of god, very nearly represents a powerful concept which can be used to underwrite the consistency of ZFC.

p. 2:

[Gödel's formalization of earlier ontological arguments for the existence of God] relied heavily on modal logic, but what we found particularly striking was the use of "positive properties" [...] It is clear that Gödel was using "positive properties" as, mathematically speaking, an ultrafilter on properties. In fact, at least implicitly, he was using "positive properties" as an ultrafilter on extensions of properties. I.e., whether a property is positive depends only on what objects it holds of. [...] It occurred to us that perhaps this highly intriguing ultrafilter, viewed as an ultrafilter on classes of objects, can be used to prove the consistency of mathematics.

There was a problem, however: the notion of a perfect God as an object of the theory prevents the property of 'positiveness' from being useful in this way. The problem is precisely that as a mathematical structure, the 'positive' predicate singled out a single object, i.e. what Gödel would identify with God, which more or less defines the positive property in that a property is positive if and only if God has that property.

p. 2–3:

At the time, we just didn't see how to get such ambitious mathematical mileage out of this "positivity ultrafilter", at least in any simple basic conceptual way. This goal seemed particularly remote since the positivity ultrafilter, as discussed by Gödel and implicitly by others, is what is called a "trivial ultrafilter". I.e., an ultrafilter consisting merely of the classes containing some given special point. In particular, following Gödel, and going back at least to Leibniz [...] a class of objects is positive if and only if it contains God. And trivial ultrafilters are, as the name suggests, mathematically trivial. So it would appear that one cannot expect to do anything substantial, mathematically, with the positivity ultrafilter.

And then the authors struck on an idea: if they wanted this positivity ultrafilter to have the mathematical power they wanted, what they had to do was remove god from the theory.

It occurred to us that if we take God out of the class of all objects, treating God as exceptional, but keeping the positivity ultrafilter, pruned to be over the class of objects excluding God, then the positivity ultrafilter is no longer trivial. In fact, it is a nontrivial ultrafilter over the class of all objects without God.

They take pains to describe the situation in terms not of God not existing, but just as an object which is not accessible to us through the structure of the theory, although 'God' is thought to be the source of the theory (in that he is posited to have made the notion of "positivity" accessible to us). It is in Section 3 that they treat the positivity property and describe (as I will choose to phrase it at this juncture) how to understand a model of goodness which does not include God.

p. 16:

In our framework, we can view the attribute of Positive as a facility that God has created and given the world access to, perhaps in order to direct us into "positive behaviors". In our context, we are not incorporating any direct access to God, but we do have access to various of God's creations that he has chosen to give us access to.

p. 22–23:

Unfortunately, if we admit a perfect object v [an object which only has positive properties, i.e. which only belongs to classes which are positive], then [...] the class consisting of exactly v must be positive; i.e., v ∈ {v} ⇒ {v} is positive. But {v} has exactly one element [which implies that the 'positive' property is trivial, i.e. not powerful enough to underwrite the consistency of any substantial piece of mathematics...] we must deliberately exclude any perfect object from the totality of objects, E.g., redefine the objects to consist of the imperfect objects.

A very natural viewpoint is this: no object is perfect, but objects can "approximate perfection" without limit. I.e., any level of "approximate perfection" can be achieved by imperfect objects. In particular, all objects have flaws. This is an interesting statement in its own right, formalized as follows.

(∀v1)(∃A1)(v1 ∈ A1 ∧ ¬POS(A1)).

(As religious treatments of mathematics go, this one comes from a very protestant mindset: all objects are imperfect, but can be improved upon. Of course, this sentiment is also shared by secular progressive movements.)

Section 5 treats so-called "divine objects". The property of being a divine object turns out to be being a member of all classes which are both positive and definable. The latter is described in Section 4, and amounts just to being recursively constructible in a finite number of steps: the authors interpret this as being within our ability to access as finite beings.

Of course, there is nothing in their theory which actually singles out "positivity" as corresponding to what we think of as the moral quality. In fact, the quality that they call "positivity" is much more convincing as a property of Bigness — any collection of objects is either Big or Small but not both, the union of two Small collections is Small (the intersection of two Big collections is Big), the empty set is Small, and every Big collection contains at least two elements — in fact, infinitely many, via the theory of non-trivial ultrafilters. The so-called "divine objects" correspond to those objects which belong to every Big class which can be finitely constructed. As the complement of any finitely constructable set is also finitely constructible, we might say that a class is Really Quite Small if it is both Small and finitely constructible; then the divine object axiom corresponds to the proposition that there is at least one object which doesn't belong to any Really Quite Small class. One could wax on about how Smallness here corresponds to "Not Positive"-ness, and that it means that the divine object axiom posits a creature which is above all of our finite, petty evils — but if you bear in mind that no object is perfect, the moralistic interpretation is that the moral failings of that object are enormous beyond even our ability to conceive. Granted, this would certainly solve the problem of evil, but perhaps that's not what the authors were aiming for.

In Section 7, they use their theory of divine beings (or Really Big Collections, if you prefer my interpretation) to prove that their system is almost as strong consistency-wise as ZF plus the assumption that there is a "measurable cardinal" (see p. 28–29 for the initial discussion); and in Section 6 they prove in particular if their system is consistent, it is enough to ensure the consistency of ZFC.

In conclusion, the paper does not prove the existence of God; for all of its talk about what God has provided and given us access to, it can just as easily be interpreted as saying that ZFC is consistent if there is no God, nothing is perfect, and there are tremendous morally ambiguous forces in the world; and in any case makes a lot more sense as a theory of Big Collections Of Things which aren't really involved in morality plays.

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