Lesson 11: Gödel’s Mathematical Proof Of God

The strange looking symbols above is Gödel’s Mathematical Proof Of God. This is the ontological proof we discussed in lesson 10 put in math form. You will remember the ontological proof said, if it is greater to exist in reality than in the mind, and God is the greatest thing we can think of, then God MUST exist in reality and not just in our mind. What Gödel did was present this argument using the mathematical argument above.

When you develop a proof you must begin with Axioms (Ax above). Axioms are considered to be an obvious truth accepted by all. The second part of a proof are the definitions (Df above). Definitions define the objects in a proof. Lower case letters such as x and y are objects and the Greek letters are properties. For example a lemon is an object. Yellow and sour are its properties. The final part of a proof is the Theorem (Th above) which are state what you are trying to prove using the Axioms and Definitions.

The other symbols are shorthand notation used by mathematicians. For example the upside down A means “for every”, the backward E means “there exists.”

Now lets look at the proof line by line. I will translate each line into English for you and then we will see what it means.

The first line is an Axiom which remember is a simple truth accepted by all.

The line reads: If phi is a positive property and it is necessary that for every x if x has the property of phi then x has the property of psi, the psi is a positive property.

Another way of looking at this is a property must be positive or negative. It cannot be both.

The second line is another Axiom which can by common sense.

The line reads: Not-phi is a positive property if and only if phi is not a positive property.

Lacking a positive property is the same thing as having a negative property. A property has to be good or bad. There is no in between. Lacking a cancer-free property is the same as having cancer.

The third line is the first statement of what Gödel is trying to prove.

The line reads: If property phi is a positive property then it is possible that there exists an x such that x has property phi.

Another way of putting this is there are good objects. God is a good object.

The fourth line is a definition Gödel uses to define God.

The line reads: Object x has the godlike property if and only if for every property phi, if phi is a positive property, then x has property phi.

This definition defines God as a good object having every positive property and from axiom two having every positive property means you can have no negative properties.

The fifth line gods to to a simple logical axiom.

The line reads: The property of being godlike is a positive property.

God can have only positive properties. God cannot have negative properties.

The sixth line is Gödel’s second theorem.

The line reads: It is possible that there exists an object x that has the godlike property.

Positive properties exist so it is possible and being Godlike is a positive property so there exists a Godlike object.

The seventh line goes back to the definition of an essential property.

The line reads: Phi is an essential property of x if and only if object x has the property phi and – for every psi, if object x has property psi then it is necessary that for every y if object y has property phi then object y has property psi.

There is an essential property of an item that let you know all its other properties. If you know a piece of bread is banana nut bread (the essential property) you know all its other properties such as it is food, it tastes good, it is light and fluffy. Because it is banana nut bread it causes all other properties.

Line eight states another common sense axiom.

The line reads: If the property phi is a positive property then it is necessary that the property phi is a positive property.

Again we go back to the fact positive can only come from positive.

Line nine is another theorem.

The line reads: If object x has the godlike property then the godlike property is the essential property of object x.

A godlike object has every positive thing you can think of. This is similar to the ontological argument. If it is greater to exist in reality than in the mind, and God is the greatest thing we can think of, then God MUST exist in reality and not just in our mind.

Line ten is a defining existence.

The line reads: Object x has necessary existence if and only if for every property of phi, if phi is an essential property then it is necessary that there exists an object y that has property phi.

An object has to exist if its essential property exists.

Line eleven is definitely a no-brainer axiom.

The line reads: Necessary existence is a positive property.

It is good to exist.

Line twelve is the concluding theorem in Gödel’s proof.

The line reads: It is necessary that there is an object x that has the godlike property.

This is the conclusion of the proof. It all comes down to the fact that God has all good properties and necessary existence is a good property therefore God must exist.

Again this is similar to the ontological argument. If it is greater to exist in reality than in the mind, and God is the greatest thing we can think of, then God MUST exist in reality and not just in our mind.

Don’t worry if you don’t quite grasp everything. Gödel may have been the greatest mathematician to ever live. Many of today’s mathematicians don’t always understand his work.

Just keep in mind the conclusion of the proof.

If God has ALL good properties and necessary existence is a good property then God must exist.

Keep in mind we are not proving the God in the Bible, but a higher, creative, infinite intelligence responsible for the universe we live in.  There is more than just this finite world we experience.