Gödel – Harmonia Philosophica

Gödel’s incompleteness theorem: The non-Cretan way out…

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Gödel’s incompleteness theorem is well known for proving that the dream of most mathematicians to formulate foundations for a complete and self-consistent theory of mathematics is a futile exercise.

Gödel proved that any set of axioms you could posit as a possible foundation for math will inevitably be incomplete; there will always be true facts about numbers that cannot be proved by those axioms. He also showed that no candidate set of axioms can ever prove its own consistency (1).

In essence, the incompleteness syllogism by Gödel starts from talking abour logical propositions (or mathematical propositions if you like) and ends up with a proposition that talks about the validity of… itself. This proposition which we might as well call reads something like “I cannot be proved”.

This leads to a dead-end.

If it can be proved, then it means that it cannot.

And vice versa.

So it is essentially a logically true proposition (since indeed it cannot be proved) but which cannot actually be proved within the axiomatic system at hand.

Hence, the incompleteness.

Essentially this is something the ancient Greeks have thought of thousands of years ago; something which they formulated in the famous Epimenides paradox. Epimenides was a man from Crete who said the following simple thing: “All Cretana are liars”.

Well, this ends up in the same dead-end as the proposition mentioned above. If Epimenides is truthful, then he is a liar since he is Cretan and all Cretes are liars. If he is a liar, then he is telling the truth! And, thus, he is a liar!

A self-reference paradox which essentially destroys the hope of mathematicians around the world for a consistent and full way to formulate mathematics. It is weird, but also important to mention here, that self-reference is the basis of our existence. Consciousness, our ability to speak about our self and our own existence and being, is the foundation of our essence as human beings. Without that, we would be nothing than complex machines.

But how can this dead-end be surpassed or perhaps by-passed?

Well, it cannot actually.

Unless…

You ctu right through it.

I was in a discussion the other day about the above topics and when the Epimenides paradox was mentioned, one immediate reaction that I got was the simple “So the solution is that he is not from Crete” (!)

What?! I answered. But I told you he was a Cretan.

Sure. He was.

But…

What is he wasn’t?

Then there wouldn’t be any paradox!

In the same sense…

What if the logical proposition…

“I am false”

is not a… proposition?

Then all problems would be solved!

But if it is not a logical or mathematical proposition then what is it? Well, as I said above, self-reference is not mathematics per se. It is more of a metaphysical reference to existence and being. A proposition talking about… itself is no more a proposition but an attempt to speak with the abyss. It is more God talking to humans than humans trying to talk with God. Such a thing could be many things, but ‘simply’ a logical (mathematical) proposition not.

But this is gibberish, one might counter-argue.

Sure, it can be.

(Gibberish like the Russel way out of his paradox?)

If you really think a Cretan would ever call himself a liar.

Sure, it can be.

If you accept that a proposion can ever referto itself.

But it cannot.

In a cosmos where only humans can talk for themselves.

Gibberish.

In a cosmos where mathematics cannot prove themselves.

Gibberish.

In a world where endless-loop paradoxes exist.

Paradoxes.

In a life which is full with nothing but them.

Paradoxes were the end of the hopes of mathematicians. They alone can be the ones which will instil hope in the once again.

Look around Cretan.

Tell me.

If you cannot prove that there is a sea…

Will you ever lie that you are swimming?

We will know…

Neil deGrasse Tyson says: ‘We will know whether there’s life on other planets’. (1)

OK. Let’s for the sake of argument believe him.

What stops us from saying “We will know whether God exists”?
Or “We will know whether we can count to infinity”.
Or… “We will know whether we can know”?

But wait!

This is already proven! (thanks Gödel!)

Prove science with… Faith?

How can one believe science has no limits?

By the interpretation of Gödel’s theorem, no section of human knowledge (e.g. arithmetic) can prove its consistency by… it self. But this implies that it may be proved by means of another section of human thought.

So maybe the consistency of Science can be proved in… an un-scientific way? 🙂

Time reversal, “Time” and “time”…

Researchers at the University of Maryland have discovered how to transmit power, sound or images to a “nonlinear object” without knowing the object’s exact location or affecting objects around it using a “time-reversal” technique. [1]

The time-reversal process is like playing a record backwards. When a signal travels through the air, its waveforms scatter before an antenna picks it up. Recording the received signal and transmitting it backwards reverses the scatter and sends it back as a focused beam in space and time thus reaching the target-object.

Time is more and more used in the “reverse” by scientists. From time travels (proved to be possible by… who else? Gödel) to “time reversal” techniques we are gradually getting acquainted with the notion that Time can be handled more as a… variable and less as a “physical entity with inherent properties”.

If time can be reversed, it we can go back to the past, then maybe time does not “exist” as we fantasize it does.

So “time” would be more correct than “Time”. Not so “inescapable” after all…

Gödel’s proof for God v2.0

Ontological Arguments

Many thinkers have attempted to prove the existence of an all-powerful being (like the one religions use to call “God”). These attempts are interesting not because they prove something beyond the shadow of a doubt (there are indeed logicians who think they are correct, but there are also others who think otherwise), but because the show that logic can be a tool that leads to God.

Gödel’s ontological argument

One of the greatest logicians of all times, Gödel, has made such an ontological argument which you can find at the book “Types, Tableaus, and Gödel’s God” (1) (3).

The argument can be summarized as follows.

We first assume the following axiom:

Axiom 1: It is possible to single out positive properties from among all properties. Gödel 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)” (Gödel 1995)

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

Axiom 2: If P is positive and P entails Q, then Q is positive.
Axiom 3: If P1, P2, P3, …, Pn are positive properties, then the property (P1 AND P2 AND P3 … AND Pn) 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 the respective Anselm’s ontological 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.

With the above reasoning, Gödel argued that in some possible world there exists God. Then he went on proving that since a Godlike object exists in ONE possible world, then it necessarily exists in ALL OTHER possible world (since “necessary existence” is one of its positive properties).

Thus, God exists.

The symbolic summarization of the above logical syllogism can be seen in the picture below (2), where Ax refers to Axioms, Th to theorems and Df to definitions used in the syllogism.

Gödel’s ontological proof symbolic notation

There are numerous objections with the above argument, the main of which are summarized in the next section.

Criticism to Gödel’s proof

The logic of the argument is not easily refuted. Of course as in any other argument there are counter-arguments and then arguments which counter those counter-arguments (4).

However the Achilee’s Heel of the argument (as that of any argument per se) is its foundations. The axioms that are innevitably stated when formulating an argument are considered as true based on the opinion of the author of the argument and, thus, can be refuted by others as simply invalid.

Gödel’s proof v2.0

In an attempt to clear things out regarding the proof, I have made a small addition to the debate on the validity of Gödel’s axioms so as to solve the issue once and for all: If some people argue that Gödel had defined “positive” too vaguely or that Gödel’s definition of “positive” is wrong altogether, then why not just accept their objections?!

And by doing that let’s say for a second that “existence” is indeed a “negative” property (and not a positive one as Godel claims in his axioms). Having that as granted, then the problem of God might not be solved but another similarly important is: All people should stop worrying about dying, since “not existing” is something good (i.e. a positive property)!

In that way all great philosophical problems of humans will be solved in a strange way. Philosophy does work in mysterious ways…

The problem of the existence of God is then solved indirectly: Since non-existence is a good thing, the phrase “God does not exist” takes a weirdly positive effect that could puzzle the greatest of atheists…

Instead of a Conclusion…

All in all, one might disagree with that argument. But the critical point here is that some other logicians agree! So even though this argument has not solved the great mystery of them all, it has given us a great lesson: Logic is not a tool for atheism only, it is a tool for theism as well…

References

Types, Tableaus, and Gödel’s God, Springer, Series: Trends in Logic , Vol. 12, Fitting, M., 2002, 196 p., Hardcover, ISBN: 978-1-4020-0604-3.
Gödel’s ontological proof, Wikipedia article, retrieved from here on 2021-02-04.
Graham Oppy, 1996, Godelian ontological arguments, Analysis 56(4), DOI: 10.1093/analys/56.4.226.
Curtis Anthony Anderson, 1990, “Some Emendations of Gödel’s Ontological Proof”, Faith and Philosophy. 7 (3): 291–303. doi:10.5840/faithphil19907325.