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Russell's paradox is a famous theorem in set theory. It asserts that "the collection of all sets is not a set itself". In the other words "the set of all sets doesn't exist" in the world which ZFC axiomatic system describes. Note that sets are the only legitimated objects in ZFC system. So in the ZFC point of view the collection of all sets is not an object in the realm of existence.

Russell's proof for this theorem uses the self-reference of the notion of "the set of all sets". Also I think this is a formal form of the famous discussion for non-existence of God which is a self-reference notion itself. For example according to the usual definition of God, he is an eternal immortal being with unlimited power to do everything. But he can annihilate itself by his unlimited power too so he is mortal, a contradiction. This fact shows one should restrict the properties of God in the definition in order to avoid self references and contradictions.

Question 1: Did Russell mention the above correspondence between the problem of existence of the set of all sets in set theory universe and the problem of existence of God in the real universe in his works explicitly or implicitly?

Question 2: What are the philosophical impacts of Russell's paradox as a theorem of set theory on the problem of existence of God in theology?

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Theologians of course try to restrict god's power in such paradoxes, by saying that his power is thorough possibilities not impossibles!(Although not treating such modal assertions exactly!!) But they never think that such a restriction results even terrible problems for god; Probably worse than his limited power. Although,i myself think that the problem of existence of god is far more complicated than most people might think.I'm not talking about whether hi exists or not; I'm talking about the concept of "Existence" for something beyond matter... Such a conception is maybe based on an unskillful extension of the concept of "Existence" in the level of matter!...

  • Good point Amir! The notion of "existence" is very important and complicated generally and particularly in this problem which is about a transcendental being like God. Anyway I beleive we should define the notions of "God" and "faith" beyond logic which allows God to have contradictory behaviors. As a summery I think the faith begins where the proof fails. – user4801 Nov 25 '13 at 5:55
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This is Wittgensteins answer to both questions in the Tractatus:

3.031 It used to be said that God could create everything, except what was contrary to the laws of logic. The truth is, we could not say of an “unlogical” world how it would look.

This speaks for itself; but one should also note the following proposition:

3.032 To present in language anything which “contradicts logic” is as impossible as in geometry to present by its co-ordinates a figure which contradicts the laws of space

This also speaks for itself - but to expand: Russells paradox is not a paradox, has absolutely nothing to do with the divine; and represents only the limitations of what was known about the construction of set theory in Russells time; it represents an obstacle by which theory must be modified otherwise it anulls itself. There may be more than one viable modification.

One particular direction is to adopt a paraconsistent logic as done by Newton Da Costa for the foundations of set theory (rather than the first-order logic of Russells to a first approximation). Here a Universal set U and a Russell set R is given; and one can prove theorems like the union of every set in R is actually U.

What this example also affirms is that we do not know logic as a complete mechanism of thought.

Wittgenstein ends the Tractatus by the following:

7 What we cannot speak about we must pass over in silence.

This echos Jesus Ben Sirachs admonition:

what is too sublime for you, do not seek; do not reach into things that are hidden from you. What is committed to you, pay heed to; what is hidden is not your concern.

This is expanded by the Summa Theologica of Aquinas, in the first article:

I answer that, it was necessary for man's salvation that there should be a knowledge revealed by God, besides philosophical science built up by human reason.

Wittgenstein final proposition is simply saying that philosophy that admits of nothing other than logic mode is too small to speak of a divine realm. It can gesture towards it only in silence, admit its possibility only in silence. It can only speak of 'what is committed to you' - the world, your self, others - to the extent possible; what is 'hidden' is the 'private language' of revelation that is only known to those that have that veil of mystery lifted; and they know it with both certainty and uncertainty, the admixture uncertain and varying as it befits our finite human mode.

For philosophy to accept revelation, however, turns it into a different discipline - theology - at least in the West;in Islam, this remains the main mode of philosophical pursuit.

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Russell's proof for this theorem uses the self-reference of the notion of "the set of all sets". Also I think this is a formal form of the famous discussion for non-existence of God which is a self-reference notion itself. For example according to the usual definition of God, he is an eternal immortal being with unlimited power to do everything. But he can annihilate itself by his unlimited power too so he is mortal, a contradiction. This fact shows one should restrict the properties of God in the definition in order to avoid self references and contradictions.

Generally speaking omnipotence has not included doing everything. It tended to apply to things that are at the very least internally coherent and not self defeating.

If you think about the age old question "Can God create a stone so big that even he could not lift it" That is a internally incoherent statement. You can make the case that statements like these are statements without any meaning. Like baby jabber noises that your mouth can make without any meaning associated with them.

Almost like asking... If God cannot make a gobbldyduke so big that even the great minatos in the beforethereafter cannot lift it how can God or gobbldyduke exist? All noises but no meaning.

OR if you really want to you can believe that God can do the internally incoherent and therefore can make a stone so big that he cannot lift it and then lift. Internally incoherent you say?... Yes but I believe he can do the internally incoherent.

Should not call into question his omnipotence either way.

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Question 1: Did Russell mention the above correspondence between the problem of existence of the set of all sets in set theory universe and the problem of existence of God in the real universe in his works explicitly or implicitly?

Answer 1: I do not know but I very much doubt that he did. By the way, keep in mind that it is not the case that Russell's paradox applies to ZFC. Russell's paradox applies to naive set theory. See Zermelo–Fraenkel set theory for instance. It is debatable whether it is permissible to say, "God in the real universe" because God is transcendental (not immanent) and so is not co-existent with any part of the real world so to speak - taking theologians at their own word. And in what way is the existence of God a problem? There are opposing beliefs with respect to the existence of God, yes, but is it correct to say that the existence of God is a problem?

Question 2: What are the philosophical impacts of Russell's paradox as a theorem of set theory on the problem of existence of God in theology?

Answer 2: Short answer - none. Long answer - I wonder why you would think that there is connection between a certain mathematical paradox and the existence of God. Is it because you think that somehow set theory is foundational to mathematics and that somehow mathematics is foundational to the real world? If you look for systemic foundations you are always sure to end up back where you started. Consider Ouroboros before you venture too far.

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