I am trying to follow discussions regarding provability and paradoxes, particularly in the domains of logic and set theory. It is my belief that there is an assumption that any question expressable in english must have a binary answer - yes or no, true or false, but I believe that much of the confusion stems from questions in which the scope is not clearly defined, etc.
The obvious solution would be to express the problems in terms of a mathematics - a group.
So, I ask if anyone has already created such a group, before i start wandering down the garden trail.
Some of the concepts I am toying with include Questions Answers - even for a boolean question we can have many answers, such as
- True or
- False. However, the system needed more possible answers to work.
- Bad question
Now there are multiple subtypes of 3. Bad Question. 3a. The question is understood, but it is not answerable in a true or false sense. 3b. The question is not understandable. It is badly phrased. 3c. The question is not answerable because there is not sufficient information provided within the question. (e.g. if x = 5, is x > y) 3d. It depends on the scope of the question. (e.g. which mathematics theorems are you using). 3e. One or more theories assumed in the scope of the question is incorrect, or inadequate to the nature of the question (. E.g using Newtonian mechanics when Relativity is required)
And just within the scope of "Is statement x provable?" there are multiple meanings for provable: a. Provably true, b. Provably false, c. No known proof has been found yet d. No known disproof has been found yet e. Provably unprovable. i.e. it can be proven that it can never be proven.
From this we can clearly see that the Boolean space is not sufficient.
Group: a mathematical system, containing Elements Operators Axioms Capable of expressing statements Some of which are provably true Some of which are provably false Some of which are not provable
Valid Group – a group which is incapable of expressing paradoxes. One that is fully internally consistent. Broken Group – a group which has one or more incorrect axioms, and so is capable of expressing paradoxes. (Gödel's second incompleteness theorem) Incomplete Group – a group which has axioms yet to be discovered, and so is incapable of proving all valid statements
An example of an incomplete group would be ZF set theory without Axiom of Choice. Apparently there are numerous unprovable statements there.
Please be gentle with the jargon. I am almost a lapsed applied mathematician (applied physics), not theoretical, but my work in ICT is leading me back to categorisation theory in the context of metadata.