But Godel was speaking of a small finite collection of axioms — jgill
But Godel was speaking of a small finite collection of axioms, not an axiomatic system that continues to increase without end. At what point does one initiate the drawing of conclusions? Tacking on the axiom of choice took math into new dimensions, as did infinity axioms. BOAK seems bewildering rather than enlightening, imo. — jgill
The incompleteness theorem applies to formal theories, — TonesInDeepFreeze
The first incompleteness theorem states that in any consistent formal system F
within which a certain amount of arithmetic can be carried out, there are statements
of the language of F which can neither be proved nor disproved in
https://plato.stanford.edu/entries/goedel-incompleteness/
For any model M for a language L, every sentence in L is either true or false, and not both, in M. — TonesInDeepFreeze
A theory T is complete if and only if, for every sentence S in the language L for T, either S is a theorem of T or the negation of S is a theorem of T. — TonesInDeepFreeze
For any model M, there is the theory T whose theorems are all and only the sentences true in M. It was Tarski who proved "the undefinability of truth" theorem, which says that the set of sentences true in the standard model for the language of arithmetic is not definable in the language of arithmetic. — TonesInDeepFreeze
Not a finite set of axioms, rather a countably infinite set of axioms. — TonesInDeepFreeze
Tarski's proof makes no false assumptions, no matter whatever incoherent ersatz pseudo formulations a crank on the Internet wishes to cook up. — TonesInDeepFreeze
(3) x ∉ Provable if and only if x ∈ True. — PL Olcott
grammar does not in and of itself make a general set of statements, even ones regarded as formalized by Montague grammar, a recursively axiomatizable theory. — TonesInDeepFreeze
Tarski says no such thing as claimed two posts above. — TonesInDeepFreeze
And, for the third time: The incompleteness theorem pertains only to recursively axiomatizable theories. — TonesInDeepFreeze
The body of all analytical knowledge — PL Olcott
Unless you can describe this vague notion as it might appear in a computer program - that is to say a list with #1, #2, . . . - I can't get beyond it to the conclusions you draw. TonesInDeepFreeze is recognized as a go-to source on these kinds of subjects. — jgill
It is easiest to simply imagine that all the [general] things known to humans that can be written down in language have already been written down. Now we have the {body of analytic knowledge}. — PL Olcott
So if listed, the listing might have to be refined as new knowledge is accrued. Still way to vague for me, but others may feel differently. I admire your tenacity on the subject. — jgill
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