Gödel's Argument for God

Hello, I want to talk to you today about Curt Girdle's

argument

for the existence of God. It was found that among the unpublished articles he gave a couple of pages to Dana Scott in which he demonstrated sometime in the 1940s that God exists on the basis of a few. Quite simple axioms, today we are going to talk not about his original axioms but about the slight revision of those axioms proposed by Anthony Anderson that solves both an intuitive and technical problem in the original presentation. The girdle himself was the best logician. of the 20th century demonstrated the integrity of first-order logic, the incompleteness of the language of arithmetic and then demonstrated in a second incompleteness theorem that no theory can prove its own consistency, in addition to all that he managed to show consistency and independence. of the axiom of choice in set theory and later showed that there are models that allow time travel in Einstein's theory of general relativity, so he sticks to an original member of the Vienna circle who spent most of his career once the Nazis took power at the Institute for Advanced Study in Princeton.

He studied and was a close friend of Einstein. There he managed to reach fundamental results. Indeed, in many ways the history of logic in the 20th century can be taught as really a history of Godot's results. Well, what was the

argument

that Faja proposed for the existence of God? was inspired by anselm's ontological argument with some embellishments due to descartes leibniz and of course squirtle himself, how does the argument go well? There are some simple axioms, some definitions and then some theorems, so let's first talk about a concept that is used throughout. The axioms are really fundamental is the idea that a property is positive now, what is a positive property?

Well, it's his version of something that Descartes would have called perfection or in Anselm, in the monologue in section 15, he talks about certain properties as positive in the sense that it is better to have them than not to have them it is better to be powerful or weak it is better to be strong or weak it is better to be alive or dead it is better to exist or not exist and so on what is better to have are the positive properties now in Girdle's original proof he assumes that each property or its negation is positive and that forces us to say that red or not red is positive, whether square or not square is positive and so on. "I think the revision has appeal partly because it avoids that assumption: we can say that certain properties are better to have than not to have or to have their opposites, but other properties are simply neutral, it doesn't matter whether something is colored red or not, no matters." It doesn't matter if something is square or circular, etc., so we can say that certain things like power, knowledge, wisdom, existence, are positive.

It is better to have them than not to have them and not commit to other properties. Well, that idea of ​​a positive property. It is one that embellishes itself in axioms, so before we continue, let's take a look at the axioms that Girdle proposes. The first axiom that Girdle proposes says this: if a property is positive, then its negation is not positive, so we never have a property. and since its positive negation is at most one of them can be and on Anderson's review none could be, it could turn out, for example, that neither red nor not red is a positive property but existence, if existence is a positive property, it is better to have it than not to have it then it turns out that nonexistence cannot be a positive property the same if being powerful is a positive property then being weak cannot be a positive property the second axiom says that any property implied by a positive property is in others In other words, any property that is a necessary condition for a positive property is itself positive, so if you have to have this property to have some other positive property, it is also a positive property, for example, let's say it cannot be powerful without existing, then if being powerful is a positive property, so is existence, if having knowledge is a necessary condition for having wisdom and having wisdom as a positive property, then having knowledge is also a positive property, this will have the consequence that all Logical properties: all properties, such as being identical to themselves or having p or not b, for example, will turn out to be positive properties.

Everything has it, so of course they are necessary conditions for a thing to be a thing and have properties. have positive properties the third axiom says that being divine is a positive property well, that makes sense better to be god than not to be god it is good to be god but let's think about exactly what it is to be divine faja da For us, a definition of the term similar to a god, something is god-like if and only if it has as essential properties all and only the positive properties, think of Descartes, who says that God is a perfect being, God has all the perfections, that is something like the idea behind Faja. definition something is like god if it has all and only positive properties it has all the positive properties it has all the perfections and has nothing else it does not have any property that is not positive there is no imperfection in other words in god then it has all the perfections none of the imperfections that is the central idea here god has all positive properties and has no properties that are not positive properties the fourth axiom if a property is positive then it is a necessary truth that is positive in other words being positive is not something that is merely a contingent property of a property, it is something that is necessary if it exists, so if, for example, being powerful is a positive property, then it is necessary that being powerful is a positive property, actually, what does this amount to? ?

We are interested in what is considered positive in every possible world, not just as a contingent matter, in fact in our world it might be that in our world, for example, having a degree in engineering is a positive thing, it is better to have it. than not having it, but that's a contingent fact about our world, it's not true in all worlds, so we don't want to conclude from this that God has an engineering degree, instead we're going to say look, we're interested in perfections, those properties that are positive in All Worlds are not simply positive in our world due to contingent matters of fact, so that's a way of saying look, we're interested in the properties that God has, those will be properties that are nice to have and we mean really absolutely. necessarily good to have not simply be good to have in this particular world the fifth axiom says that necessary existence is a positive property which is a way of saying in other words that since they can't say it, existence is a perfection of fact necessary existence It is a perfection to be able to be destroyed to be able to not exist seems like an imperfection It is something that makes you vulnerable There is something negative about it Would you rather be invulnerable or vulnerable Would you rather be someone who will be around no matter what Or would you rather be the kind of person who is mortal immortal or mortal?

A necessary juicy immortal existence seems like a really good thing, and that counts as a positive property. The last axiom is one that is easy to get confused with. Axiom Four also refers to properties being positive and necessity, but it is slightly different: it says that if something is a positive property, then the necessity of that property is also a positive property, so, for example, if being powerful is a positive property, so so is it. be necessarily powerful if existing is a positive property then so is necessarily existing so in summary if a property p is a positive property then necessarily p is also a positive property well those are the axioms now notice that they don't sound very controversial in fact They are Quite simple, some of them refer to the fact that counting as a positive property is not contingent and having a property necessarily seems positive if having that property is itself positive, so, in short, those axioms try to say that being positive is something that is supposed to be independent of the particular world you find yourself in, it is not something we are treating as a contingent fact, it is a matter of the nature of the property itself, really the only two axioms that have much substantial content refer to more than the modal status and the logic of positive properties are the two that reflect premises in Descartes' version of the ontological argument that God is a perfect being and has all perfections or here to be divine is a positive property and the other that existence is a perfection or in Girdle's terms, that necessary existence is a positive property to clarify the content of those axioms and also to make some concepts more precise, we need to improve the theorems that Girdle continues to prove, we need two more definitions, the first is the definition of an essence of a thing is like this, let's say that a is an essence of x if and only if it entails all and only the necessary properties of x, that is, for each property b, x has b necessarily if and only if a entails b, let's put it in In more intuitive terms, the idea is that an essence is something that entails all and only the necessary properties of something.

People often think that the essence of a thing is simply the collection of its necessary properties. Well, here it essentially counts as something that involves all and only the necessary ones. properties, so we could take it as a property that somehow combines all of them, but actually the central idea here is not that it is simply a collection of properties, but that it is a property that actually implies everything in that collection and nothing that is outside that collection. so it encapsulates, if you want to think about it this way, all the necessary properties of the thing, the second definition is of necessary existence and here faja means something a little stronger than what we normally understand by necessary existence, it says that x necessarily exists if y only if each essence of x is necessarily instantiated, so to be necessarily existent in these terms is for each essence to be necessarily instantiated.

If an essence simply encapsulates all necessary properties, that is a way of saying that all of its necessary properties are necessarily instantiated and I don't mean each one, I mean all together, the set of necessary properties is necessarily instantiated in every possible world. . There is something that has that set of necessary properties, so it's a somewhat unusual conception of necessary existence, but whatever. implies the usual and therefore we don't have to worry about that, however, it is important that when we talk about this being a positive property, that is the concept, it refers to a necessary existence in this sense, now you might object the proof of fact. because you don't like that axiom, you realize that wait, this is something stronger than what we would normally understand by existence or even necessary existence.

Am I so sure that it is a positive property? So Rob Coons, for example, has attacked Girdle's argument on the basis of attacking that axiom axiom 5 which says that necessary existence is a positive property, but for now I want to ignore that and think about the theorems that Girdle derives. from this. The first theorem reflects Girdle's main motivation for devising this proof in the first place. He thought that if there was an obvious flaw in Anselm and Descartes' proofs, they never actually show that the concept of god itself is consistent with there could be a being like God.

They argue that if it could exist, then the existence of God. It's necessary, but no matter how successful that argument is, they never really convince you that God's existence is possible. Girdle had been studying Leibniz, who he thought was a fundamental flaw in Descartes's proof and also in Anselm. Now Anselmo has an argument of sorts. Even the fool says, the person who says that there is no god shows that he understands what the statement is that there is or is not a god and the only way it can really be understood is if the person has the idea of ​​god right in his head. . that seems to show that apart from questions of existence or non-existence, the existence of god is conceivable, the atheist conceives it but then rejects it, however, that could be leibniz is worried about saying yes, but that doesn't really convince me that it is a completely consistent conception that such a thing could exist, for example, if someone says that there is no such thing as a round square, I understand what they are saying, but that does not mean that I can conceive of a round square or that a round square is actually possible, then Leibniz says that I want something more to convince me that this is something that is really possible.

In other words, the concept of God is consistent. He doesn't see Descartes really offering us anything in this regard. What if perfections are in some way? mutually inconsistent now Leibniz proceeds to argue that we will think about what are the various perfections that enter into our concept of god. He says that in the end they are indefinable primitive things and precisely because they are indefinable because they are primitive they can be combined in the only way. The way thatsays that we might discover some kind of inconsistency is to look at their definitions and discover that when we put them together there is some kind of inconsistency or contradiction, but he says let's look at whether they are all primitive terms that cannot be gidle thinks that it is not so easy to prove that all of these things are actually primitive terms, in fact, he thinks that's the wrong way to approach it.

Instead, his first theorem is intended to address Leibniz's problem, but to solve it in a different way, here is what he says that if a property is positive then it is consistent if a property is positive then it is consistent meaning that it is possible to instantiate it. there is no contradiction in it now at first glance you could say that it is quite surprising how we get that outside of our definitions and axioms, but in reality the proof is very simple and, as we will see, the goal here is to argue that the concept of god or , in their terms, being godlike is consistent because one of the axioms says that it is a positive property.

But how do we get there first? Suppose something is a positive property. How do we conclude that it is consistent? Which is possibly exemplified. Here is the proof. Let's say the property p is positive, then p is not positive according to the axiom. One p implies self. -identity, anything that has this property p must be identical to itself, so it turns out that also being self-identical is a positive property that follows from axiom two to axiom one, and it also follows that self-diversity or self-difference , in other words, not being what you are x not being equal to denial why?

Is it so important because one inconsistent property would imply that all properties are fine? Anything that is an inconsistency in classical logic implies everything, so in particular it would imply not only self-identity but self-diversity, it would imply not only x is equal to x but x is not equal to x, however a positive property can never imply a property that is not positive, so we conclude that any positive property is consistent, if we are inconsistent, it would imply all properties, including those that are not positive, well, one of our axioms says that the property of being divine is a positive property, so the property of being divine is consistent, let's move on to theorem two, if something is divine, the property of being divine is an essence of that thing, now this is how Faja has been fixed. actually it is the complicated proof this is the one that requires a little sophistication still it is not that complicated a proof it is not very difficult to understand this is how it works suppose that g, for example, is divine, then, according to our first definition, g has as essential properties all and only positive properties according to axiom 3 the property of being god-like is positive according to axiom 6 so is the property of being necessarily divine good the property of being god-like is essential for this g if and only if for each property b g has b necessarily if and only if being divine implies b In other words, an essence is something that implies all and only those properties that the thing necessarily has, so being divine will have to imply all and only the properties that that divine thing necessarily has.

We have to prove that, so we have to show that for any b g has b necessarily if and only if being divine entails b well, there is a left-to-right direction and a right-to-left direction for this like any other if and only if for conditional type of proof, so let's first look at the direction from left to right, let's say g has b necessarily, then b is positive, in fact, according to axiom four, it is necessarily positive, anything divine has to be, so being divine implies b, how about the direction from right to left? Let's say that being divine implies having the property b then by axioms two and three b is positive b and g being divine as b necessarily now the theorem three is exemplified necessarily the property of being divine that is the way in which faja says that God necessarily exists like this that this is a very powerful way result here is the proof suppose that something is similar to a god then it has all the positive properties according to axiom 5 it also has the property of necessarily existing now we might think that we are done there however sash really means something strong my necessary existence so let's continue The essence of x is necessarily instantiated, so the property of being divine is an essence of x according to theorem two, so if something is divine, then necessarily the property of being divine is instantiated necessarily, something is divine having been shown that you use only necessary truths. that in itself has to be a necessary truth, well, according to our first corollary, it is possible that something has disappeared, so it is possible that something is necessarily god-like, so something is necessarily god-like, It can easily impress you that wait, something very strange is happening. in that last theorem and in fact there is something a little controversial now in a sense not so controversial but a little controversial so let me explain that there are actually only two modal principles that are being used in this proof, one simply says that if a implies b and a is possible, then b is possible and one is fairly uncontroversial, that is true in all normal modal logic systems, but the second is much more controversial, it is the characteristic principle of a system known as s5, now s5 is The most popular.

Modal logic system for analyzing metaphysical necessity is something that was simply assumed by Aquinas, Descartes, Kant, really by everyone before the 20th century, in a way in which the axiom says that if something is possible, then it is possible. necessarily possible or in the way it is used here, but it is an equivalent formulation is that if it is possible for something to be necessary then it is necessary now clearly these things do not apply to all senses of necessity after all the first that if something is possible is necessarily possible implies that being possible is not something that depends on contingent matters of fact, but yes, our intuitive conception of possibility says: well, of course yes, it could be possible for me to bench press 300 pounds if I were younger, if I trained a lot more than I do now, well, okay, maybe it's possible. for me in those circumstances, but it is not possible for me to do it now, so we would think that the possibility of something like that depends on several contingent questions of fact if we are thinking about metaphysical possibility, however, it is not obvious, in fact, It would seem that it is not true that that is the sort of thing that should depend on contingent matters of fact, and so for metaphysical possibility and necessity it seems quite plausible to say that if something is possible from a metaphysical point of view, it is a necessary truth, it is necessarily possible. and similarly, if it is possible for something to be necessary in this metaphysical sense that it is necessary, it is metaphysically possible for this to be necessary, so it is necessary, that is the modal principle assumed here, in terms of modal logic It's something like a symmetry assumption.

That is, if a world is relevant to evaluations of possibility and necessity from the point of view of another world, it also goes in the other direction, so let's say that something is possible in our world in this metaphysical sense, which means that intuitively true in some other metaphysically possible world, but then our world must also count as metaphysically possible from the point of view of that other world; that assumption is necessary for this argument to work; In the end, it is also necessary that we understand others such as modal. The form of Anselm's original ontological argument, the prozlogian form iii, so s5, as I mentioned, is somewhat controversial, it seems to me clearly incorrect for certain senses of possibility of necessity, but it is actually quite plausible for metaphysical possibility and necessity and , in fact, is the most popular logic. used to analyze those terms was assumed by all the major figures of medieval philosophy, as well as early modern philosophy, it was not until ci lewis began to develop a variety of modal logic systems in the 20th century that people realized Realizing that there are alternatives, let me finish with A brief mathematical note, the first three axioms could be summarized by saying that the positive properties form an ultrafilter in the order of entailment and definition 1, as well as axiom 4, established the divine property as the main element of that ultrafilter now for those who don't.

I don't know what an ultrafilter is, don't worry, but for those who do, I think it's actually a useful conception, because what it suggests is that those axioms and definitions together construct the concept of god as, in some way, a maximum. being and that is the central idea behind the ontological arguments in and some gods is that the greatest that cannot be conceived in Descartes god is being with all possible perfections and therefore god is maximum god is the main element of a ultrafilter and in logic In mathematical terms, that is a conception of maximality, so we did not directly say that God is maximal, but we actually incorporated it through a combination of definitions and axioms and that is really, I think, the strategy central to what Girdle was doing, in fact his original axioms make that strategy even more obvious, so there is a notion of maximality (the maximal nature of God) that is at the center of this argument, as well as of the classical ontological arguments of Anselm and Descartes, so we could say that the underlying strategy of All those arguments is to move from something mental to the actual existence of a thing, that is the same move that bothered Thomas Aquinas and a variety from others, but the idea is that if something is maximum, then it must exist.

Maximality implies existence, indeed necessary existence.