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guslThe value of Σ is not known for n >= 5, because there are machines that we haven't been able to run long enough to halt or prove that they never halt (also, there is no hope of proving that they halt if we use a small axiomatic system: Google "ten pounds of axioms" for philosophical controversies).
Lacking evidence to the contrary, could it be justified to assume that a particular TM never halts? (note that many of these *deserve* to be axioms, since they are true and don't follow from our small axiom sets; and if we are wrong, we *will* eventually find out)
Such assumptions would allow us to solve open Pi_1 problems (like the Goldbach Conjecture) that can be expressed in n bits where Σ(n) is known, i.e. the assumption is that Σ(n) isn't bigger than the current champion. (the argument is that KC(1st counterexample) <= KC(statement). Thus 1st counterexample <= Σ(n) where n is the length of the statement)
This seems like a fine working assumption, analogous to axioms used in the natural sciences.
I can imagine proceeding this way, by choosing axioms that best explain mathematical regularities (including unproven ones), perhaps in a Bayesian way. Having said that, even if our assumption is falsified (if the TM is shown to halt afterall), might it be useful to continue pretending otherwise? (do axioms that take a long time to be refuted tend to be, for application purposes, less significant?)
Chaitin said that we need to be empirical about axioms, and that's what I'm suggesting here.
Lacking evidence to the contrary, could it be justified to assume that a particular TM never halts? (note that many of these *deserve* to be axioms, since they are true and don't follow from our small axiom sets; and if we are wrong, we *will* eventually find out)
Such assumptions would allow us to solve open Pi_1 problems (like the Goldbach Conjecture) that can be expressed in n bits where Σ(n) is known, i.e. the assumption is that Σ(n) isn't bigger than the current champion. (the argument is that KC(1st counterexample) <= KC(statement). Thus 1st counterexample <= Σ(n) where n is the length of the statement)
This seems like a fine working assumption, analogous to axioms used in the natural sciences.
I can imagine proceeding this way, by choosing axioms that best explain mathematical regularities (including unproven ones), perhaps in a Bayesian way. Having said that, even if our assumption is falsified (if the TM is shown to halt afterall), might it be useful to continue pretending otherwise? (do axioms that take a long time to be refuted tend to be, for application purposes, less significant?)
Chaitin said that we need to be empirical about axioms, and that's what I'm suggesting here.