I read a very good book many years ago, published in 1979, I think. It won the Pulitzer prize, and it was concerned with Godel’s theorem, entitled “Godel, Escher, Bach”.
I had to read it several times before I grasped the basic ideas, But the author, Douglas Hofstadter, also wrote a book a bit simpler, called “I Am A Strange Loop”. A pretty good read.
In “Godel, Escher, Bach”, Hofstadter would introduce his ideas with a conversation, in Lewis Carrol fashion, between Achilles and the Tortoise. The conversations were based loosely around Zeno’s paradox.
At one point, Achilles was trying to use logic to convince the Tortoise that Achilles’ conclusions simply could not be denied if the tortoise accepted logic as the ultimate arbiter of truth.
The Tortoise, like most humans, said that he was very reasonable, and would never accept a conclusion that was illogical. Achilles just knew the argument as won.
Achilles began, “If I show that A is true, and B follows from A, then we must also conclude that C is also true as a conclusion following from the premises of A and B”.
“Of course” said the Tortoise, “It’s obvious that this should be the case”.
“Then you agree?”
“In principle”.
“What do you mean, ‘In principle’?”
“I mean that if A is true, and B is true, the we would conclude that C s true as a premise following from A and B”.
“That’s what I said”
“Yes” said the Tortoise, “But if A is true, and B is true, and C follows logically as a premise resulting from A and B, then we would conclude that this is summed up in premise D”.
“What?”
“You see, if A, B, and C are true, we would establish this as premise D, which simply states the fact of the truth of A,B,and C”.
“Okay” said Achilles, “I’ll accept that. But then surely you must accept D as the final statement of truth”.
“Not at all” said the Tortoise, “If D is the statement of truth summing up A,B.and C, then certainly we would establish this as a subset of premise E, which states that if A,B, C, and D are true, then E”.
Achilles smelled a rat. “But surely there would be a stopping point. Let’s say that Z is the statement that sums up the truth of A through Y. Surely you then have to accept that Z is the truth?”
“Of course” said Tortoise, “But then we would also have to say that A1 is the statement that demonstrates the truth of A through Z, and then A2, A3, A4, until we finally arrive at the ultimate truth”.
“Which is?”
“I don’t know. How many numbers are there?”
“Infinite numbers. So if I say that infinity sums up the truth of all previous statements, we can stop there?”
“I suppose, but where does infinity stop?”
What occurred was that both Achilles and Tortoise were discussing a kind of “schematic” of truth. not the truth in terms of each statement itself, but in terms of statements that represented truth or statements about truth, as each one saw it.
This is my point in talking “about” truth. We can develop processes of organization, mechanical representations of ‘truth’, but in fact, the arguments can proceed into infinity. The system of mathematics simply cannot define truth in any limited fashion. It can go on forever….
That is the essence of Godel’s theorem. In order to find out if truth could be represented in mathematics, Godel had to develop a system in which the system of math actually referred back to itself. To do this, he had to develop a “Godel number” system in which the axioms of math( plus, minus, division, multiplication, etc) were represented as numbers themselves, so that the system was “self referencing”.
What Godel demonstrated b this was that a system of complex mathematics would produce a statement which said of itself, “I exist, but I cannot be proven within this system”.
From “outside” the system, the person could see if it was true, but the system itself simply had nothing to say about it! it was undecidable, therefore making the system incomplete.
The result was “in any consistent axiomatic formulation of number theory, there exists undecidable propositions”.
This same process may also be admitted by looking at Jeremiah 17:9 and Romans 8:7. When the human mind looks inside itself for truth, “self references”, it will come up with an infinity of possibilities as to what is true, especially in regard to God!
If we seek to organize truth about God in the form of rules and laws, that organization will have to reflect the limitations of our own minds. It will reflect also the incompleteness of our mathematical systems and our systems of logic as well.
We can’t get “there” from “here”.
That’s why, if we seek to “convert” others to a certain truth, that truth will ultimately split into an infinity of different ideas and concepts! No human mind can represent God in a complete sense, and that’s what both Paul and Jesus tells us.
