Odifreddi: Godel's proof of the existence of God

Patrik Eklund peklund at cs.umu.se
Thu Nov 10 00:36:31 EST 2022

At Beziau's Square of Opposition 2014 I tried something out on the 
weakness of logical symbols.


The slides are from the conference, and a very brief abstract was 
published in the booklet, but the paper (the second link) that was 
submitted afterwards was never considered.

The message is two-fold. On the one hand I wanted to show how logic 
needs non-commutative operators, i.e. binary Boolean is too poor. On the 
other hand I wanted to underline how the problem with logic really is 
the existential symbol, like I have pointed out, in the Fuzzy Terms 
paper, also published in 2014, how the lambda symbol is problematic in 
lambda calculus. Church pointed out that problem in 1940, noting that 
lambda is an informal symbol, yet he treats in like a formal one. The 
existential symbol is similar. It's an informal one, yet we allow to 
generate paradoxes of all kind. Even worse, some of these paradoxes 
(like Liar and Richard) are used to create mathematical proof (like in 
Gödel's "Inconsistency").


In those Square of Opposition things from 2014 I used the case of 
Augustine's fight against the Arians, which is about the Holy Spirit's 
role to the other two in the Holy Trinity in Christianity. St Augustine 
added Filioque ("also from the Son") to the Creed, since Areios had said 
that the Holy Spirit comes "only from the Son" (solo Filio), and this 
eventually, some 600 years later lead to the splitting of the Christian 
Church into the Latin and Greek side, since the Orthodox side of 
Christianity never accepted that addition into the Creed. Some 500 years 
later, the Greek side accepts Palamas' hesychasm, which perhaps even 
more makes the Greek and Latin sides different. I don't know as I am not 
a specialist theologician, but I have been curious to try to understand 
such things from mathematical notation and language point of view.

In that analysis in Vatican 2014 I tried to argue that "and also" is a 
tricky operator, and a con-commutative one, so that e.g. "Father AND 
Son" is actually "Father AND ALSO Son". Boolean logic is unable to deal 
with this. Needless to say, "Father AND Son" is not "Son AND Father".

I also recalled the mathematically heretical "proofs" by Areios (the Son 
is not God) and Sabellius (the Father is the Son), which obviously are 
very strange since they deal strangely with existence ("there is").

In the unpublished Vatican 204 paper I mentioned e.g. how Kleene felt 
suspicion about the use of the existence operator, and my has always 
been that Gödel uses "mixed bags" in his proofs, i.e., terms and 
sentences are not properly typed. THe same is pointed out in Fuzzy 
Terms. It boils down to many things, e.g. to not treating the powerset 
type correctly in the original papers on logic and foundations. 
Logicians have never dealt with this strange situation appearing, 
basically since set theory is totally untyped, and logic allows symbols 
to be placed "one by one in a sequence", like Peano said, and then to 
provide ever so subtle "semantics" to those "placement principles".


On non-commutativity we come to many-valued logic and the use of 
quantales for many-valued truth.

On quantales you can see our treatment in our book.



We use these understandings also in real applications (industrial and 
health), and indeed not theological ones, as I tried out in Vatican 

If anyone wants to talk to me, I'm available for a confcall, anytime. 
Looking forward.



PS Does God exist? I don't know. How can I? I may believe or not 
believe, but that's another story, and what's the logic or math of 
belief anyway?

PPS And needless to say, the tricky thing here is also religions. Some 
believers believe only in God, and the Son and the Holy Spirit does not 
exist. Some say God (or Allah) exists, and do acknowledge a person (Isa) 
exists that others call Son, even if acknowledged in a "minor" role. 
Those who acknowledge so, write about the Christian Holy Spirit as St 
Mary, so Trinity takes a totally other form. And indeed, some say all 
three in the Trinity exist even if they are not in agree how they are 
related. And there are more types of believers, and within the larger 
groups there is disagreements, sometimes even leading to war and misery. 
In all these subtleties, Areios', Sabellius', Gödel's and other "proofs" 
tend to look very naïve.

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