On 24 Jan 2014, at 08:23, Brian Tenneson wrote:
There are undecidable statements (about arithmetic)... There are
true statements lacking proof. There are also false statements about
arithmetic the proof of whose falsehood is impossible; not just
impossible for you and me but for a computer of any capacity or
other forms of rational processing.
OK. But this is not really prove for computer having the ability to
transform themselves, like acquiring new axioms. But then that
acquisition cannot be programmed at the start (in which case Gödel can
be applied).
Without comp, there is no means to prove the existence of an
absolutely undecidable proposition. We can only prove that for a fixed
machine, or a fixed RE sequence of machine, there is a proposition
undecidable for that machine, or for any machine in that sequence.
We'll never have a computer, then, that will work as a
mathematically-omniscient device. By that I mean a computer such
that every question that has a mathematically-oriented theme having
an answer truthfully can be answered by such a device. Calculators
demonstrate the concept but are clearly not mathematically-
omniscient: you ask the calculator what is 2+2 and press a button
and "presto" you get an answer. What I'm talking about would be
questions like "is the set of rational numbers equal in size to the
set of real numbers", and get the correct answer. So we will never
have such a computer no matter what its capacities are, even if
computer encompasses the entire human brain. Unfortunately, that
means that even for humans, we will never know everything about
math. Unless something weird would happen and we suddenly had
infinite capacities; that might change the conclusions.
Yes, arithmetical omniscience needs a "god", like the Arithmetical
Truth. Even a theory as powerful as ZF or NF can only scratch the
arithmetical truth. OK.
Bruno
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