User talk:Dfeuer

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 * --Your friendly ProofWiki WelcomeBot 16:40, 29 November 2012 (UTC)

BPIT
I note that you're enthusiastically banging in the BPIT template all over the place in pages which in several cases already have AoC etc. templates in them. I have reservations about this for several reasons:

a) It has not completely been established exactly what BPIT actually is. Yes, there is a partial proof in place, but it might or might not work, but at the moment it is unfinished and, ultimately, incomprehensible because we don't even know what a prime ideal is yet.

b) Places where the BPIT already exists do not indicate exactly where it is used, and why it is relevant, and what the point is. If you look at existing pages using AoC (where the work has been done properly, of ocurse) you will see that there is a specific point in the proof where it is invoked, or if not, in the use of the template there is an indication of which result in that proof does use the AoC (e.g. "... by Zorn's Lemma" etc.). The BPIT work does not.

c) Even when we have sorted all that out, the philosophical significance of BPIT remains obscure. It's couched in technical language about a collection of objects that need considerable work to understand, and even when you do understand them all, it's like: so what? The AoC is a short, pithy fact that can be put into a single sentence's thought and therefore means something. The BPIT in consequence just looks like something clever-sounding that we read off the internet and include because it makes us look smart. Personally I don't think it *does* make us look smart.

Thoughts? --prime mover (talk) 06:24, 14 December 2012 (UTC)


 * Most set theory is pretty strange, but the set theorists keep making it anyway. The Boolean Prime Ideal Theorem is really no stranger than Zorn's Lemma. It apparently has several forms, some of them simpler than the one I started the page on, which all turn out to be equivalent (somehow), the simplest of which that I've seen is the Ultrafilter Lemma. It appears to have been studied extensively at least since the 1960s (and the Ultrafilter Lemma since the 50s), so there's plenty of source material out there—it's not some new experimental axiom that only a handful of specialists have heard of. Personally, I enjoy how it's helping me explore set theory some more to try to understand why different bits relate to each other in certain ways. I don't see how acknowledging on a page that a proof from weaker principles is possible and we'd like to have one makes us look stupid. As long as ProofWiki is a work in progress and not a finished product to be shown off in a library that seems fine to me. --Dfeuer (talk) 07:45, 14 December 2012 (UTC)


 * It would be a good idea to at least indicate that we are still looking for proofs from e.g. BPIT. This can be accomplished by amending the template. Upon establishing more relevant context for the BPIT it will be torn out of obscurity; and if not, there's always the statement that it is a consequence of AC so that any willingly ignorant visitor can content himself with accepting AC and not bothering. For the interested visitor, who may be pleased by accepting a varied assortment of axioms these sections contain valuable and interesting information. There should be more rather than less, IMHO. --Lord_Farin (talk) 12:14, 14 December 2012 (UTC)


 * What might be useful is a "overview" page like we have for, for example, Definition:Separation Axioms in which the various axioms and their relative strengths are presented and explained briefly. Extracting that brevity may be a challenge. In the meantime I recommend we complete the train of thought which goes towards the proof of BPIT (including the definition of the various objects - Prime Ideal comes to mind) and make an attempt to justify the claims that are made in its name. Finally, there are various pages which state "This theorem depends on BPIT" without any indication as to which particular point in the proof it is needed. IMO this is of paramount importance.


 * Demonstrations of relative strength and weakness of the axioms should ultimately be illustrated by examples of objects / constructs which are satisfied by one axiom but not another, proving that while A --> B there are objects that are B but not A. Again, this will be challenging, but should ultimately just be a literature search.


 * I understand the drive to raise the reader's awareness of these things - they fit into the strategy of where we want ProofWiki to go - but without making a solid attempt to justify the underlying foundations there's a danger of the information on this site becoming disconnected. In the extreme would fall into danger of becoming an alternative wikipedia - a conglomeration of cherry-picked interesting / important / noteworthy theorems with no solid underpinning. --prime mover (talk) 14:23, 14 December 2012 (UTC)