`Hal Ruhl wrote:`

Hi Jesse:

I think you miss my point. The All contains ALL including Turing machines that model complete FAS and other inconsistent systems. The All is inconsistent - that is all that is required.

You mean because "the All" contains Turing machines which model axiomatic systems that are provably inconsistent (like a system that contains the axiom "all A have property B" as well as the axiom "there exists an A that does not have property B"), that proves the All itself is inconsistent? If that's your argument, I don't think it makes sense--the Turing machine itself won't behave in a contradictory way as it prints out symbols, there will always be a single definite truth about which single it prints at a given time, it's only when we interpret the *meaning* of those symbols that we may see the machine has printed out two symbol-strings with opposite meaning. But we are free to simply believe that the machine has printed out a false statement, there is no need to believe that every axiomatic system describes an actual "world" within the All, even a logically impossible world where two contradictory statements are simultaneously true.

You mean because "the All" contains Turing machines which model axiomatic systems that are provably inconsistent (like a system that contains the axiom "all A have property B" as well as the axiom "there exists an A that does not have property B"), that proves the All itself is inconsistent? If that's your argument, I don't think it makes sense--the Turing machine itself won't behave in a contradictory way as it prints out symbols, there will always be a single definite truth about which single it prints at a given time, it's only when we interpret the *meaning* of those symbols that we may see the machine has printed out two symbol-strings with opposite meaning. But we are free to simply believe that the machine has printed out a false statement, there is no need to believe that every axiomatic system describes an actual "world" within the All, even a logically impossible world where two contradictory statements are simultaneously true.

Godel's theorem is a corollary of Turing's.

As you say a key element of Godel's approach to incompleteness is to assume consistency of the system in question.

But do you agree it is possible for us to *prove* the consistency of a system like the Peano arithmetic or the axiomatic system describing the edges and points of a triangle, by finding a "model" for the axioms?

But do you agree it is possible for us to *prove* the consistency of a system like the Peano arithmetic or the axiomatic system describing the edges and points of a triangle, by finding a "model" for the axioms?

The only way I see to falsify my theory at this location is to show that all contents of the All are consistent.

Hal

I think you need to give a more clear definition of what is encompassed by "the All" before we can decide if it is consistent or inconsistent. For example, does "the All" represent the set of all logically possible worlds, or do you demand that it contains logically impossible worlds too? Does "the All" contain sets of truths that cannot be printed out by a single Turing machine, but which could be printed out by a program written for some type of "hypercomputer", like the set of all true statements about arithmetic (a set which is both complete and consistent)?

I think you need to give a more clear definition of what is encompassed by "the All" before we can decide if it is consistent or inconsistent. For example, does "the All" represent the set of all logically possible worlds, or do you demand that it contains logically impossible worlds too? Does "the All" contain sets of truths that cannot be printed out by a single Turing machine, but which could be printed out by a program written for some type of "hypercomputer", like the set of all true statements about arithmetic (a set which is both complete and consistent)?

Jesse