Talk:Independence (mathematical logic)
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When I wrote this article, I was unaware of the existence of the logical independence article (I think I had seen it before; I just forgot). That article is more complete, but this one has the better name, so I put the merge tags from logical independence to independence (mathematical logic). The usage notes in independence (mathematical logic) about the word "undecidable" and about the sense of "independent" meaning simply "not proved" (rather than "not proved nor refuted") should in any case be maintained. --Trovatore 05:15, 24 January 2006 (UTC)
- Instead of a merge, I had what I had written at independence (mathematical logic) deleted (after saving a copy locally), and logical independence was then moved here. Then I selectively reincorporated my text from the previous independence (mathematical logic) into the current article. --Trovatore 18:01, 16 February 2006 (UTC)
Here are my objections to DesolateReality's latest edits:
- "Independent of a maximally consistent body of propositions". The only thing I can understand by "maximally consistent" is that any proper extension is inconsistent. But then nothing can be independent of that! (In the sense of "independent of and consistent with"; see next objection.)
- The latest edits have as the primary meaning of "independent" the sense of "cannot be proved" rather than "can neither be proved nor refuted", and refers to the second sense as "informal". I kind of doubt that this really reflects general usage.
- The claim The existence of independent statements is of philosophical interest. It puts into question Hilbert's program, casting doubt as to whether a complete formalism of mathematics is possible. is not really accurate; the fact that a theory fails to prove or refute something may just mean that you haven't made the theory strong enough. The argument against Hilbert's program has to do with the necessary incompleteness of any theory satisfying certain hypotheses. Without quantifying over theories, the claim is severely misleading. (A related problem is that no sentence is "independent" full stop; it can only be independent of some specified theory.)
- The "standard technique" section is mostly accurate but does not strike me as being written in encyclopedic style. --Trovatore 21:45, 10 June 2007 (UTC)
- Thank you, Trovatore, for the revert. I agree with you generally about your objections. Here are my specific replies:
- I wanted some way to refer to the term "theory" for a general audience. I agree that "maximally consistent" doesn't make sense.
- I agree.
- My intention here is to bring out why logicians are interested in independence proofs. Until I find a better way to phrase this, I agree with the revert.
- I think later editions of this article should try to incorporate the observation that independence of σ from T is usually proven by exhibiting a model of T + ¬σ. Such a method of proof is usually the first thing taught to logic students immediately after the notion of independence is explained.--DesolateReality 04:43, 11 June 2007 (UTC)
"Undecidable" in the sense of decision problems
I undid edit, which had the following edit summary:
- This is in fact a specific application of the meaning of "decidable" as applied in a decision problem, as a sentence is independent if and only if its truth value can be decided by an algorithm that enumerates all proofs.
That's not the same thing at all. The IP is arguing that a statement independent of a formal theory is one that is not decided by a particular program, and he/she is right about that. But an undecidable problem in the sense of decision problems is one that cannot be decided by any program whatsoever. There is no such thing as a problem with only one (or finitely many) instances that is undecidable in this sense. You can always write a program that will just say "yes" or "no" unconditionally, and that "decides" the problem for that single instance. --Trovatore (talk) 19:46, 9 January 2015 (UTC)