There is a tag being placed on set theory questions that do not, a priori, have any connection to forcing arguments, such as:

is statement A consistent with ZFC?

is statement B independent of ZF? ( example: Mathematical statement with simple independence proof from $\mathsf{ZF}$ )

can statement C be proved without the Axiom of Choice? ( example: Proving "every set can be totally ordered" without using Axiom of Choice )

If the tags are meant to express properties of the questions then the correct label would be (consistency) or (independence) or something like that. Some of the answers would involve forcing but this is not requested in the question and in some cases there are answers that don't use forcing.

Also, (consistency) and (independence) are concepts that apply to any formal theory, not only set theory. I think that forcing is not as prominent for proving statements independent of Peano Arithmetic, for example. So perhaps it is better to have tags for (consistency) and/or (independence) and utilize those where forcing is not directly indicated in the question.

[update: added URLs as requested in the comments. In example C the first solution, which is discussed in the answer, was given in 1939 before forcing existed. In example B there are, say, sheaf or Boolean-valued models as alternatives to forcing. And there are many independence or consistency questions not from set theory.]

  • $\begingroup$ adding link to actual examples (of misuse of the tag) would be useful, perhaps $\endgroup$ – Grigory M Aug 18 '11 at 21:02
  • $\begingroup$ @Grigory M : updated with examples. I am saying that (independence) or (consistency) is better in many cases, not that (forcing) is a mis-use. Forcing tag could be applied in addition to a consistency/independence tag, but conceptually the latter is a category of question and the former is one method for answering the questions. $\endgroup$ – zyx Aug 18 '11 at 21:39

As the person who puts [forcing] tags on questions most of the times, here is my answer to this.

While it is indeed true that the questions are ultimately about independence (compare, for example, my question here: Forcing cardinality of a set which is strictly about forcing), the consensus is that given a detailed answer which fits into an additional tag, one may add it to the question (cf. http://meta.math.stackexchange.com/questions/2612/retagging-after-an-answer-is-given).

All these questions were answered (by me) in forms relating to forcing. Seeing how this tag is pretty small (and to be fair I don't expect it to reach even 100 questions over the next two years) there is no actual problem here.

With regards to the latest question regarding the independence of the ordering principle from the axiom of choice, while the original Mostowski proof is indeed from 1939 it is not a proof of the independence in ZF. The original question requested a proof over ZF, since the forcing argument is similar but a lot more technical I chose to bring the ZFA proof instead.

Lastly, to the point about topos based independence results - I don't see how they are simpler. Mostly because I know nothing about topoi.

  • $\begingroup$ In the absence of tags like [independence] (on Mathoverflow there is one called [independence-results] ) your answer is reasonable. But given that [independence] would describe the question and is very close in content to [forcing] (which describes the answers) does it not make more sense to add an independence/consistency tag(s) and use that, with a separate judgement about [forcing] if/when answers are posted? (Thanks for clarifying, by the way.) $\endgroup$ – zyx Aug 19 '11 at 7:53
  • $\begingroup$ Also, this is slightly different from the situation considered in the "Retagging after answer" discussion. The connection of the question to [forcing] is not due to details of the answers, but because it is the method of proof for most independence results, which is something known before any answer is posted. If forcing is implicit in almost all set-theory independence questions, could that not be indicated as well with an [independence] tag which is then available for questions outside set theory (such as logic, Peano Arithmetic, model theory, etc) that refer to independence per se? $\endgroup$ – zyx Aug 19 '11 at 8:08
  • $\begingroup$ Firstly, when I tagged the questions under forcing I already knew that I am going to write a forcing related proof. Secondly, you are somewhat correct in the sense that forcing is the "usual" way of proving independence results within set theory (at least in "logical" set theory, in contrast to algebraic set theory). For other questions regarding independence, if you feel that it is indeed suitable you can add a tag for independence and/or undecidable statements. $\endgroup$ – Asaf Karagila Aug 19 '11 at 14:15

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