On 12/15/2018 2:58 PM, Jason Resch wrote:
On Saturday, December 15, 2018, Brent Meeker <meeke...@verizon.net <mailto:meeke...@verizon.net>> wrote:On 12/15/2018 7:43 AM, Jason Resch wrote:On Sat, Dec 15, 2018 at 1:09 AM Brent Meeker <meeke...@verizon.net <mailto:meeke...@verizon.net>> wrote: On 12/14/2018 7:31 PM, Jason Resch wrote:On Fri, Dec 14, 2018 at 8:43 PM Brent Meeker <meeke...@verizon.net <mailto:meeke...@verizon.net>> wrote: Yes, you create a whole theology around not all truths are provable. But you ignore that what is false is also provable. Provable is only relative to axioms. 1. Do you agree a Turing machine will either halt or not? 2. Do you agree that no finite set of axioms has the power to prove whether or not any given Turing machine will halt or not? 3. What does this tell us about the relationship between truth, proofs, and axioms?What do you think it tells us. Does it tell us that a false axiom will not allow proof of a false proposition? It tells us mathematical truth is objective and doesn't come from axioms. Axioms are like physical theories, we can test them and refute them if they lead to predictions that are demonstrably false. E.g., if they predict a Turing machine will not halt, but it does, then we can reject that axiom as an incorrect theory of mathematical truth. Similarly, we might find axioms that allow us to prove more things than some weaker set of axioms, thereby building a better theory, but we have no mechanical way of doing this. In that way it is like doing science, and requires trial and error, comparing our theories with our observations, etc.Fine, except you've had to quailfy it as "mathematical truth", meaning that it is relative to the axioms defining the Turning machine. Remember a Turing machine isn't a real device. BrentAhh, but diophantine equations only need integers, addition, and multiplication, and can define any computable function. Therefore the question of whether or not some diophantine equation has a solution can be made equivalent to the question of whether some Turing machine halts. So you face this problem of getting at all the truth once you can define integers, addition and multiplication.
There's no surprise that you can't get at all true statements about a system that is defined to be infinite.
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