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first of all I must specify that I am a mathematician so I hope I am using the appropriate words to formulate my question.

Recently, while talking with some colleagues, a question arose; Based on the fact that all facts can be expressed by means of mathematics (it seems to me that this is known as "mathematicism"), would it then be possible to give the world a "formal system" structure?

In case this is possible, could God be a Gödelian sentence?

I would like to know if there are any theoretical studies about this and if so, provide me with a bibliography to be able to address the issue

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  • "could God be a sentence????"
    – Mauro ALLEGRANZA
    Jul 4 at 5:43
  • more than a sentence I mean something simply unprovable (assuming everything already described)
    – Luis Alexandher
    Jul 4 at 5:52
  • There are object and there are sentences, i.e. linguistic expressions. Sentences express facts, i.e. "talks about objects". We prove sentences (in the context of some theory): so, in order to prove a sentence expressing the existence of God we need a "theory of God". The unprovable sentence that Godel manufactured was an arithmetical sentence , i.e. a sentence about numbers.
    – Mauro ALLEGRANZA
    Jul 4 at 5:59
  • How would you express facts like "I'm afraid of ghosts" or "The crescendo at the end of that piece shook me." in mathematics?
    – David Gudeman
    Jul 4 at 19:52
  • What aspects does God have in common with Godelian sentences that would lead you to suggest this?
    – David Gudeman
    Jul 4 at 19:53

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As far as I understand, a Gödel sentence is one which can neither be proved nor disproved within a formal system.

Applying this to God sounds like you're trying to describe agnosticism in a fancy way:

Agnosticism is the view or belief that the existence of God, of the divine or the supernatural is unknown or unknowable. It can be categorized as an indifference or absence of firm beliefs in theistic religions and atheism on that basis. Another definition provided is the view that "human reason is incapable of providing sufficient rational grounds to justify either the belief that God exists or the belief that God does not exist."

Note: this is not my position.

That said, this is more about human knowledge than objective truth. God would either exist or not exist as an objective truth*, regardless of whether we can know this.

*: Unless you hold to metaphysical subjectivism, perhaps.

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For God to be represented by a Gödel sentence or a set of Gödel codes would require, initially, that a theory of God was required (or at least able) to prove arithmetic past a certain point. More broadly, theories of sets of such codes are theories of sharps, e.g. the most famous, and simplest, example is 0♯. But there are posits of other sharps, e.g. various Icarus sets.

Generally, some X♯ exists if and only if there exists a nontrivial elementary embedding j: L[X] → L[X]. So if God was a sharp, It would be generated modulo some n.e.e., which sounds peculiar and anyway out-of-line with the idea that God is self-existent and self-explanatory (if existent and explainable at all).

Even on the level of individual codes, I would be curious as to how, "God exists," would allow us to prove sufficiently complex arithmetical statements. Or, "God is metaphysically simple," "God is eternal," etc.: do these have an arithmetical use? Some Christian analysts do think such things (Cornelius van Til claimed that the concept of the Trinity provides a solution to the "one over many" problem, for example) but I don't know that those analysts have tried to develop such doctrines to the relevant extent (to the extent of justifying higher theories of arithmetic).

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