Godel's Ontological Theorem.

[LiamOS]

This thread is both for discussion of Godel's Ontological Theorem and a continuation of a debate which was in another thread.

Godel's Ontological Argument is expressed symbolically as:

For those unfamiliar with modal-logic, there is an article on the general Ontological Argument here.


With respect to the theorem's axioms, WikiPedia tells us the following:

We first assume the following axiom:

Axiom 1: It is possible to single out positive properties from among all properties. Gdel defines a positive property rather vaguely: "Positive means positive in the moral aesthetic sense (independently of the accidental structure of the world)... It may also mean pure attribution as opposed to privation (or containing privation)." (Gdel 1995)

We then assume that the following three conditions hold for all positive properties (which can be summarized by saying "the positive properties form a principal ultrafilter"):

Axiom 2: If P is positive and P entails Q, then Q is positive.
Axiom 3: If P₁, P₂, P₃, ..., P_n are positive properties, then the property (P₁ AND P₂ AND P₃ ... AND P_n) is positive as well.
Axiom 4: If P is a property, then either P or its negation is positive, but not both.

Finally, we assume:

Axiom 5: Necessary existence is a positive property (Pos(NE)). This mirrors the key assumption in Anselm's argument.

Now we define a new property G: if x is an object in some possible world, then G(x) is true if and only if P(x) is true in that same world for all positive properties P. G is called the "God-like" property. An object x that has the God-like property is called God.

For debate:
-Is the Ontological Theorem logically valid?
-Are all the axioms of the theorem valid?
-Can the argument hold without the axioms being valid, if they are not necessarily so?

[JoeyKnothead]

I 'pologize if I'm spamming or wrongly promoting my own, but...

The proof can summarized as:

IF it is possible for a rational omniscient being to exist THEN necessarily a rational omniscient being exists.

Is Wikipedia's summary accurate? If no, please help me out.

If the summary is accurate, then the question remains...

Is the possibility that something may exist sufficient to declare that that something does exist?

I say no.

[EduChris]

...The reason people seem to be making wild accusations is not because they hate Theists, rather it's either due to such an accusation being fully or party warranted or simply a misunderstanding.

I wish you were right. In my experience, I've had to repeat myself numerous times, saying essentially the same thing each time, and still it seems that the accusers aren't reading my posts at all--everything I say is being filtered through the lense of their preconceived notions of what they think theists always say, rather than what I'm actually saying.

I guess this just supports my assertion that one's presuppositions and prejudices will always affect one's understanding and one's conclusions--sometimes even more than the evidence and argument itself.

[Goat]

I 'pologize if I'm spamming or wrongly promoting my own, but...

The proof can summarized as:

IF it is possible for a rational omniscient being to exist THEN necessarily a rational omniscient being exists.

Is Wikipedia's summary accurate? If no, please help me out.

If the summary is accurate, then the question remains...

Is the possibility that something may exist sufficient to declare that that something does exist?

I say no.

You know what.. David Hume makes the same point

from that wiki article


In David Hume's Dialogues Concerning Natural Religion, the character Cleanthes argues that no being could ever be proven to exist through an a priori demonstration:

[T]here is an evident absurdity in pretending to demonstrate a matter of fact, or to prove it by any arguments a priori. Nothing is demonstrable, unless the contrary implies a contradiction. Nothing, that is distinctly conceivable, implies a contradiction. Whatever we conceive as existent, we can also conceive as non-existent. There is no being, therefore, whose non-existence implies a contradiction. Consequently there is no being, whose existence is demonstrable.

[cnorman18]

...The reason people seem to be making wild accusations is not because they hate Theists, rather it's either due to such an accusation being fully or party warranted or simply a misunderstanding.

I wish you were right. In my experience, I've had to repeat myself numerous times, saying essentially the same thing each time, and still it seems that the accusers aren't reading my posts at all--everything I say is being filtered through the lense of their preconceived notions of what they think theists always say, rather than what I'm actually saying.

I guess this just supports my assertion that one's presuppositions and prejudices will always affect one's understanding and one's conclusions--sometimes even more than the evidence and argument itself.

That's true in both directions. As one who stands between conservative Christianity and atheism, I spend as much time clarifying what I'm NOT saying as what I AM, and to both sides. Nature of the beast, and it's not going to change. That's why my own posts tend to be wordy and long; I want to be clear from the outset. Assumptions -- that the Bible is authoritative and true, and that objective evidence and rigid logic are the only rational standards for human thought, to name only two -- sometimes can't be set aside, even for the sake of argument. Frustrating, but that's human nature (and not actual, rational thought in either case).

[Board]

Desire in and of itself is good, in my opinion, though like eveything else it can become corrupted or debased--it can become "unreasonable" and uncontrolled and a hinderance to the flourishing of the individual and his/her/their society.

And this is where the west and the east collide. Eastern philosophy finds desire to be the root of all suffering. The west embraces desire whether good or bad. I too tend to lean toward the Western thinking on this.

I agree with you. I have often stated on this forum that many people (most people?) pretty much believe whatever they want to believe. And of course, every time I have suggested this, I have been subjected to a barrage of ciriticism from the usual folks here. I had not previously been aware of the terms "pleasure principle" and "reality principle." If I had used these terms I probably could have calmed my detractors down a notch.

I only found the terms reality and pleasure principle when I was reading up on the nature of desire... Freud, while crazy, had some pretty profound and simple statements.

Yes, I think that we all have a tendency to fear that our existence is ultimately meaningless. The hope that life may be meaningful is acquired partly on faith--not blind faith, as so many people like to say, but rather on those few moments in our personal experience where the act of "trusting" has brought us greater exhilaration and joy than the act of "fearing" ever could.

And if I might interject a bit of "Christianizing" here, I link this "fear of meaninglessness" to the Christian concept of "Satan," and I link the "hope/desire/trust for meaning" to the Christian concept of "God's Spirit."

I can see where you are coming from on the Christian concepts of Satan and the Spirit. That is probably very close to what I used to believe.

It is the part of your statement I marked in bold that I find intriguing. So you are liking fear to pain and trust to pleasure? I would say that is a safe comparison.

And coming back to the concept of meaningfulness being a reasonable concept. I would say that it is reasonable in light of the human condition as it pertains to the pleasure principle. Meaningfulness is seen as pleasurable so in our mind it is reasonable and sought. I can see where this logic holds true under this condition.

However, I am not sold on the existence of objective meaningfulness. If we recognize the pleasure principle may be what is telling us to seek meaningfulness then we would have to step outside of this human condition to seek the existence of objective meaning.

In a nutshell I think we need more proof outside of human feelings like trust and fear that could be passed off as a larger psychological condition of our species.

I tend to confuse myself at times when I try to regurgitate what is in my head... did any of that babble make sense?

[JoeyKnothead]

At least someone'll talk to me

I 'pologize if I'm spamming or wrongly promoting my own, but...

The proof can summarized as:

IF it is possible for a rational omniscient being to exist THEN necessarily a rational omniscient being exists.

Is Wikipedia's summary accurate? If no, please help me out.

If the summary is accurate, then the question remains...

Is the possibility that something may exist sufficient to declare that that something does exist?

I say no.

You know what.. David Hume makes the same point

from that wiki article


In David Hume's Dialogues Concerning Natural Religion, the character Cleanthes argues that no being could ever be proven to exist through an a priori demonstration:

[T]here is an evident absurdity in pretending to demonstrate a matter of fact, or to prove it by any arguments a priori. Nothing is demonstrable, unless the contrary implies a contradiction. Nothing, that is distinctly conceivable, implies a contradiction. Whatever we conceive as existent, we can also conceive as non-existent. There is no being, therefore, whose non-existence implies a contradiction. Consequently there is no being, whose existence is demonstrable.

If it is possible for a rhetorical you're god to NOT exist, then does that mean he doesn't?

What's the fancy term for an argument from possibility?

[EduChris]

...The proof can summarized as: IF it is possible for a rational omniscient being to exist THEN necessarily a rational omniscient being exists...

I don't believe that summary really gets at the heart of Godel's theorem. A physicist summarized the theorem for me as, "If anything at all exists, something must necessarily--i.e., non-contingently--exist."

Here is another link and a highlighted section from the paper. To the extent that the author is an "expert witness" himself, one should not be too quick to claim "fallacy" when presented with Godel's argument:

The ontological argument attempts to ground all existing things in a necessary being, namely God....It takes a certain amount of courage to state publicly that you believe Gods existence can be proved. Many scholars who admire Kurt Gdel as the greatest logician of the twentieth century have found themselves a bit embarrassed by his ontological argument. His Incompleteness Theorems are undoubtedly masterpieces of mathematics. His rotating universe is respectable in physics because its properties can be explained without reference to his belief that Becoming is in some sense illusory. By showing that the rotating universe is conceptually consistent with general relativity, he helped make academic discussion of time travel acceptable for later researchers such as Kip Thorne. As a proof theorist, he helped make modal logic respectable by demonstrating the utility of modal operators in the foundations of mathematics. By establishing the consistency of the continuum hypothesis with Zermelo-Fraenkel set theory, he helped make Cantors transfinite arithmetic respectable. Yet the ontological argument remained on the margin of intellectual thought. It is now emerging from this intellectual wilderness to a more prominent place where it belongs. Gdels research forms a unified body of thought in the Platonic tradition. Within that body of thought, his ontological argument is no aberration, but an essential part of one of the most remarkable thinkers of modern times (pp. 27-28).

[EduChris]

...I am not sold on the existence of objective meaningfulness. If we recognize the pleasure principle may be what is telling us to seek meaningfulness then we would have to step outside of this human condition to seek the existence of objective meaning...we need more proof outside of human feelings like trust and fear that could be passed off as a larger psychological condition of our species...

Which is precisely what we cannot do. The choice available to us is, do we choose "the fear of meaninglessness," or do we choose "the hope of meaningfulness."

Given that neither assumption can be proven in any objective fashion, and neither is inherently implausible, I cannot imagine why anyone would choose fear over hope.

[cnorman18]

...I am not sold on the existence of objective meaningfulness. If we recognize the pleasure principle may be what is telling us to seek meaningfulness then we would have to step outside of this human condition to seek the existence of objective meaning...we need more proof outside of human feelings like trust and fear that could be passed off as a larger psychological condition of our species...

Which is precisely what we cannot do. The choice available to us is, do we choose "the fear of meaninglessness," or do we choose "the hope of meaningfulness."

Given that neither assumption can be proven in any objective fashion, and neither is inherently implausible, I cannot imagine why anyone would choose fear over hope.

Not everyone casts the choice(s) in those terms. One may also choose one's own meaning, without claiming that it applies to everyone or is an objective truth.

[ScotS]

I 'pologize if I'm spamming or wrongly promoting my own, but...

The proof can summarized as:

IF it is possible for a rational omniscient being to exist THEN necessarily a rational omniscient being exists.

Is Wikipedia's summary accurate? If no, please help me out.

If the summary is accurate, then the question remains...

Is the possibility that something may exist sufficient to declare that that something does exist?

I say no.

That's not what the argument says. It's not talking about just any old "something".

The way I read it is: if it is logically possible for a being to possess all positive properties, then that being exists necessarily since necessary existence is also a positive property. Meaning that it is not possible that the being does not exist.