Tarski wrote:My initial impression on the axioms is as follows:
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Axiom 3: The property of being God-like is positive
This axiom is not self evident even if one uses some tentative familiar notion of "positive". Why should it be a "good" thing to necessarily exist? Perhaps existing contingently is more desirable. Who is to say what is positive?
This is an example in my opinion which illustrates that symbols, math do not map onto words cleanly and necessarily. Even you acknowledge this by asking what does he mean by "positive". It never occurred to me that what Godel meant by positive might be "good" or a "good" thing to necessarily exist. I think what would make more sense is that positive means "existing" as opposed to negative "not existing". But my point is that what exists in symbol format does not neatly map onto words. This is also why I think you aren't as good at word format logic as you think you are.
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Axiom 4: If a property is positive, then it is necessarily positive
Once again, there is too much packed into the axiom. Axioms should be simple and hopefully self evident. This axiom assumes that whatever positiveness is, it cannot be relative to the context or a matter of opinion. A moral relativist would have no truck with this axiom.
Oddly enough I think I do understand Godel's argument. If a property exists, then it exists in all worlds or it exists conceptually necessarily. So for example, ultimate knowledge may not exist in our physical world and may be impossible to ever exist given limits to our perceptions and abilities, but it exists conceptually and necessarily irrespective of our abilities to perceive.
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Axiom 5: Necessary existence is positive
I can not see why I should accept this axiom either. Does it seem self evident? Once again, the notion of positiveness needs to be clarified.
Without plausibility arguments for the axioms and a solid agreed upon notion of positive that agrees with its use in the axioms, the formal validity of the argument is not very impressive. Truth has not been established since we cannot see that the argument is sound.
On the other hand, we are talking about Godel, so to be fair, one should search the literature to find out to what extent Godel did try to clarify positivity (he did say something) and to what extent he was able to argue the plausibility of the axioms.
Frankly I don't think Godel's argument is all that complex, I really do think it is simply about ultimate knowledge/platonic knowledge and it has nothing to do with theology and the Gods of theology. So in my opinion this axiom is simply saying that in a conceptual world, ultimate platonic knowledge necessarily exists.
The version we are reading is actually not Godel's formulation but is rather someone else's attempt to do justice to what Godel might have had in mind. Other versions may be more or less convincing.
Well then this is a significant problem because it distorts what Godel may in fact have meant. Someone wants it to be used to imply an entity God and CC erroneously used it in that way because the argument used the value laden word God. If there have been any people bamboozled in this discussion it's been CC, yourself, and Gad. In symbolic form, the def'n remains a def'n with no assumptions, just as in math and the conclusion does not conclude something which maps onto an actual world, nor even a conceptual one until someone attempts to do so.
Finally, you are correct to insist that the formal notions in the argument must correspond to something real in some sense. Positivity must be real. This is much easier if one has a prior commitment to neoplatonism as Godel did.
If one is a strict empiricist regarding what is real, or if one gives no ontological weight to mathematical and logical entities, then one would have to find a notion of positive that applies to things you consider real.
Just ask yourself if you think that there is an objective notion of positive in the sense of being "good" or "desirable".
Tarski, I really think you are off the mark on this, I don't have to find a notion of positive that applies to things I consider real. I can conceptualize a platonic ultimate knowledge, it doesn't have to be empirically known. I can imagine that that concept must exist. If we had the capability of being "all knowing" we would know every minute detail about the universe to the smallest possible particle, to how everything relates to everything else, to how the universe began etc. I can conceive of that platonic knowledge. It doesn't have to be real, doesn't have to be an entity, it can exist as a necessary concept.
I don't think there is such an absolute notion. To prove the existence of such would be another task like proving the existence of God.
But, if there were it would have a platonic existence not a physical existence.
In the end Godel's god would likely turn out to have the same reality as the integers (not real to you but real to Godel, Penrose and a few other familiar names).
I think you make it more complicated than it is.
Finally, notice that Godel asserts that necessary existence is positive. Thus we are treating existence as a property and this is the weak spot in the more traditional ontological arguments.
It's not a weak spot Tarski, it a weak spot to you because you assume an entity for the word "God" All along you've accused me of assuming some sort of Christian God and all along I've been trying to get a handle on Godel's God and what people in this discussion perceived it to be. But I never assumed anything of that God. You apparently are making assumptions it exists an an entity. Necessary existence Tarski, is different than existence in a the actual world. Necessary existence can exist in a conceptual world while not in our actual world we are able to perceive. Conceptual ultimate knowledge necessary exists (in a conceptual world) but it doesn't exist as an entity.
In anycase, you may rest assured that Godel's argument does not present a real challenge to atheism as we normally think of it.
Even to Godel the case is not closed. He fully realizes the burdens regarding the axioms etc that I have mentioned above.
The bottom line is that Godel argument seems to need platonism to be true and showing that is akin to (but not the same as) showing the existence of God in the first place.
The last sentence is nonsense Tarski. I don't think you know what you are talking about. Showing the existence of God, an entity with characteristics or properties does require empiricism. A conceptual platonic ulimate knowldege does not have properties or characteristics and needs no empiricism.