Godel`s ontological proof

[Nirvana]

I recently came across Godel`s argument for the existence of God. To give an overview I`m gonna refer to wikipedia- ``
The proof uses modal logic, which distinguishes between necessary truths and contingent truths. In the most common semantics for modal logic, many "possible worlds" are considered. A truth is necessary if it is true in all possible worlds. By contrast, a truth is contingent if it just happens to be the case. For instance, "more than half of this planet is covered by water" is a contingent truth, that relies upon which planet "this planet" is. If a statement happens to be true in our world, but is false in another world, then it is a contingent truth. A statement that is true in some world (not necessarily our own) is called a possible truth.

Furthermore, the proof uses higher-order (modal) logic because the definition of God employs an explicit quantification over properties.[8]

First, Gödel axiomatizes the notion of a "positive property":[note 2] for each property φ, either φ or its negation ¬φ must be positive, but not both (axiom 2). If a positive property φ implies a property ψ in each possible world, then ψ is positive, too (axiom 1). Gödel then argues that each positive property is "possibly exemplified", i.e. applies at least to some object in some world (theorem 1). Defining an object to be Godlike if it has all positive properties (definition 1), and requiring that property to be positive itself (axiom 3),[note 3] Gödel shows that in some possible world a Godlike object exists (theorem 2), called "God" in the following.[note 4] Gödel proceeds to prove that a Godlike object exists in every possible world.

To this end, he defines essences: if x is an object in some world, then a property φ is said to be an essence of x if φ(x) is true in that world and if φ necessarily entails all other properties that x has in that world (definition 2). Requiring positive properties being positive in every possible world (axiom 4), Gödel can show that Godlikeness is an essence of every Godlike object (theorem 3). Now, x is said to exist necessarily if for every essence φ of x the following is true: in every possible world, there is an element y with property φ (definition 3). Axiom 5 requires necessary existence to be a positive property.

Hence, it must follow from Godlikeness. Moreover, Godlikeness is an essence of God, since it entails all positive properties, and any nonpositive property is the negation of some positive property, so God cannot have any nonpositive properties. Since any Godlike object is necessarily existent, it follows that any Godlike object in one world is a Godlike object in all worlds, by the definition of necessary existence. Given the existence of a Godlike object in one world, proven above, we may conclude that there is a Godlike object in every possible world, as required (theorem 4). Besides axiom 1-5 and definition 1-3, a few other axioms from modal logic were tacitly used in the proof.

From these hypotheses, it is also possible to prove that there is only one God in each world by Leibniz's law, the identity of indiscernibles: two or more objects are identical (the same) if they have all their properties in common, and so, there would only be one object in each world that possesses property G. Gödel did not attempt to do so however, as he purposely limited his proof to the issue of existence, rather than uniqueness.``
So, start discussing and have a good time

 

[Polymath257]

As Godel was perfectly well aware, the term 'property' is rather vague and attempting to define it is problematic.

So, the concept of 'positive' divides all properties φ into two classes: those where φ is positive and those where ¬φ is positive. It then requires that implication in all possible worlds takes positive properties to other positive properties.

It is far from clear that these two requirements are consistent. Certainly, Godel never proves their consistency and any such proof would require a precise definition of the term 'property' to execute the proof. In fact, such a division has similarities to the concept of an ultra-filter and the existence of such requires some high-power assumptions.

Furthermore, it is far from clear that such a division of properties is unique even if such a division is possible (existence). If uniqueness fails, that would imply the existence of many different 'Godlike' objects, one for each definition of 'positive property'.

Next, it is far from clear to me that even first order modal logic is valid. But we know well from standard logic that quantifying over properties even in the 'easy' case leads to a large variety of paradoxes (for example Russell's paradox) because we are looking at properties of properties. Because of this, the mere definition of the term 'Godlike' is problematic and the existence of Godlike objects is likely to be contradictory (which means the whole system is). To then go further and demand that this quantified property is positive is a HUGE assumption that is very far from being justified.

Next, Godel goes even farther and requires that positive properties be possible in all worlds. This is very problematic, because we started by assuming the properties were NOT based on the particular possible world. Properties are things, from the start, that are generalized over possible worlds (so, the property 'red' isn't reduced to 'red in world w'). This is part of what allows us to talk about the same property in different possible worlds. So, the very act of quantifying again over possible worlds becomes a HUGE issue for even the definitions of the concepts we started with (on top of the ones I described earlier).

I could go on, but I think the problems are clear at this point. Godel was good enough at logic, I think he was playing a joke with this.

 

[Revoltingest]

As I recall (from looking at this years ago), Godel's premises don't apply to the natural world.
Mathematical models of phenemona in our universe are useful in describing it & predicting
more phenomena, but we shouldn't let the tail wag the dog. The universe reserves the
right to not behave as our models say it should.

 

[viole]

I recently came across Godel`s argument for the existence of God. To give an overview I`m gonna refer to wikipedia- ``
The proof uses modal logic, which distinguishes between necessary truths and contingent truths. In the most common semantics for modal logic, many "possible worlds" are considered. A truth is necessary if it is true in all possible worlds. By contrast, a truth is contingent if it just happens to be the case. For instance, "more than half of this planet is covered by water" is a contingent truth, that relies upon which planet "this planet" is. If a statement happens to be true in our world, but is false in another world, then it is a contingent truth. A statement that is true in some world (not necessarily our own) is called a possible truth.

Furthermore, the proof uses higher-order (modal) logic because the definition of God employs an explicit quantification over properties.[8]

First, Gödel axiomatizes the notion of a "positive property":[note 2] for each property φ, either φ or its negation ¬φ must be positive, but not both (axiom 2). If a positive property φ implies a property ψ in each possible world, then ψ is positive, too (axiom 1). Gödel then argues that each positive property is "possibly exemplified", i.e. applies at least to some object in some world (theorem 1). Defining an object to be Godlike if it has all positive properties (definition 1), and requiring that property to be positive itself (axiom 3),[note 3] Gödel shows that in some possible world a Godlike object exists (theorem 2), called "God" in the following.[note 4] Gödel proceeds to prove that a Godlike object exists in every possible world.

To this end, he defines essences: if x is an object in some world, then a property φ is said to be an essence of x if φ(x) is true in that world and if φ necessarily entails all other properties that x has in that world (definition 2). Requiring positive properties being positive in every possible world (axiom 4), Gödel can show that Godlikeness is an essence of every Godlike object (theorem 3). Now, x is said to exist necessarily if for every essence φ of x the following is true: in every possible world, there is an element y with property φ (definition 3). Axiom 5 requires necessary existence to be a positive property.

Hence, it must follow from Godlikeness. Moreover, Godlikeness is an essence of God, since it entails all positive properties, and any nonpositive property is the negation of some positive property, so God cannot have any nonpositive properties. Since any Godlike object is necessarily existent, it follows that any Godlike object in one world is a Godlike object in all worlds, by the definition of necessary existence. Given the existence of a Godlike object in one world, proven above, we may conclude that there is a Godlike object in every possible world, as required (theorem 4). Besides axiom 1-5 and definition 1-3, a few other axioms from modal logic were tacitly used in the proof.

From these hypotheses, it is also possible to prove that there is only one God in each world by Leibniz's law, the identity of indiscernibles: two or more objects are identical (the same) if they have all their properties in common, and so, there would only be one object in each world that possesses property G. Gödel did not attempt to do so however, as he purposely limited his proof to the issue of existence, rather than uniqueness.``
So, start discussing and have a good time

Mmh. i don't like the request that having all possible positive properties is a positive property. It smells like a sleight of hand to bypass the composition fallacy. And seems to lead to a loop.

For if having all positive properties is a positive property, and if I was Godlike, then I also have the property of having all possible properties.

Which entails that I also have the property that consists in having the property of having all properties.

Which means, I have the property of having the property that consists in the property of having the property that consists in having all positive properties.

......


Which means, I have the property of having the property (that consists in having the property)^n to have all positive properties.

With n being an arbitrary natural number.

Ciao

- viole

 

[Polymath257]

I got to thinking about this a bit more. While I think there are issues with axioms 1-3, the really potent one is axiom 4: that any property that is positive is positive in all possible worlds.

In other words, we *assume* that any positive quality is necessarily positive. That is a huge, huge assumption! It allows us to go from a single possible world to *all* possible worlds!