Counterintuitions and the well-ordering theorem
in [Logic]
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From: George Greene on 30 Nov 2009 08:07 On Nov 22, 12:51 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Well, there is one sense in which the power set of omega is > no more problematic. We just introduce the term, P(omega), together > with the rule: If X is a set, and every element of X is a natural, > then X is an element of P(omega). In the FIRST-order treatment, you don't get to say "If X is a set". In first-order ZFC, EVERYthing is a set. To EXIST AT ALL is TO BE a set. > This is basically just introducing "is an element of P(omega)" as > a synonym for "is a subset of omega". There is no ontological > commitment yet. OF COURSE there is!! BEFORE you introduced P(.) as a functor, the class of all subsets of a given set WAS JUST THAT: A CLASS AS OPPOSED to a set! Insisting that the collection of all subsets of a given set MUST ITSELF ALSO BE A SET *IS* a NEW ontological commitment!! > It becomes more complex when we try to make sure that the other > axioms about sets (extensionality, separation, replacement, etc.) > hold for this new set that we introduced by fiat. It ALREADY went WITHOUT saying that ALL the axioms would hold FOR ALL sets. THAT'S THEIR PURPOSE. > It's not immediately obvious that we can apply these other axioms consistently. Well, DUH, for ANYthing as complex as ZFC (or even as PA, for that matter), you are not going to be able to prove that it is consistent without invoking machinery that is even more powerful&complex and therefore even LESS well- confirmed-as-consistent. > For example, if we allow definitions and newly introduced terms > to count as sets, Again, there is a prior paradigmatic issue here. Term-wise, in first-order PA, *EVERYthing* in the domain is AUTOMATICALLY "a natural number". Similarly, in first-order ZFC, if it EXISTS AT ALL then it IS AUTOMATICALLY a set. You DON'T GET to DECIDE to allow or disallow!! You can make this different if you use NBG (which, in addition to explicitly recognizing the set/class distinction, is finitely axiomatizable), but I wasn't aware that we had officially migrated to a different paradigm. > then there is the problem that two definitions > may extensionally denote the same set, but this fact is not > obvious. Throughout the realm, there is the property that two propositions may be logically equivalent, BUT THIS FACT IS NOT OBVIOUS, e.g. the multitude of equivalents of the axiom of choice. This is almost not even a problem. This is just inherent in the nature OF ANY theory that is recursively-enumerable-as-opposed-to-recursive. > How can we treat sets extensionally if we allow formulas for sets? This is very ambiguous. I seriously don't quite see what "allow formulas for sets" even Might Mean. It has always been true that if you replace some term in any formula with a HOLE and allow the hole to be filled by ANYthing/set, then you create thereby an "open" formula with the property that some pegs with which you fill the hole will make the formula true, while others will make it false. This makes the unary "open sentence" equivalent to A CLASS of-things- from-the- domain. So it is safe to say that even in theories that don't observe the set- class distinction, this fact about formulas means that we ALREADY DO associate CLASSES with formulas. But the whole question THEN becomes ABOUT which of these classes do or don't ALSO get to be sets. The adoption of the power- set axiom is a decision of a question of that type and it absolutely DOES have ontological commitment DESPITE the fact that it is "just" an abbreviation.
From: Marshall on 30 Nov 2009 09:04 On Nov 29, 9:05 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > On Nov 29, 7:47 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> And that's the position one should consistently stay with (and it's not > >> that difficult as you might have feared.) > > > Such condescension! What arrogance! > > As it's often said when hungry eat thirsty drink, when one > needs to stop the other kind of arrogance one might have > to be arrogant! So you admit you are arrogant then. Interesting. Marshall
From: Nam Nguyen on 30 Nov 2009 09:32 Marshall wrote: > On Nov 29, 9:05 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Marshall wrote: >>> On Nov 29, 7:47 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> And that's the position one should consistently stay with (and it's not >>>> that difficult as you might have feared.) >>> Such condescension! What arrogance! >> As it's often said when hungry eat thirsty drink, when one >> needs to stop the other kind of arrogance one might have >> to be arrogant! > > So you admit you are arrogant then. Interesting. Arrogance with a cause. Nothing extraordinary though.
From: Marshall on 30 Nov 2009 20:01 On Nov 30, 6:32 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Marshall wrote: > > On Nov 29, 9:05 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> Marshall wrote: > >>> Such condescension! What arrogance! > >> As it's often said when hungry eat thirsty drink, when one > >> needs to stop the other kind of arrogance one might have > >> to be arrogant! > > > So you admit you are arrogant then. Interesting. > > Arrogance with a cause. Nothing extraordinary though. You are too modest about your arrogance. Marshall
From: Nam Nguyen on 1 Dec 2009 00:10
Marshall wrote: > On Nov 30, 6:32 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Marshall wrote: >>> On Nov 29, 9:05 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> Marshall wrote: >>>>> Such condescension! What arrogance! >>>> As it's often said when hungry eat thirsty drink, when one >>>> needs to stop the other kind of arrogance one might have >>>> to be arrogant! >>> So you admit you are arrogant then. Interesting. >> Arrogance with a cause. Nothing extraordinary though. > > You are too modest about your arrogance. Sure. If you say so! I wish I could be more "arrogant" than that but I believe things would take time. And it would take you and those on the same side sometime to see it through that a) we do actually need the rigidity of the game of manipulation of meaningless strings of symbols to have a solid reasoning framework [and you might want to read Schoenfield's thought on this] and b) there's a _real_ possibility that there exist formulas that you can't know their theoremhood status in _all_ formal systems "as strong as arithmetic". Until you either admit a) and b) or muster _sound technical refute_ on these 2 points, all what you've said about anyone else's being arrogant is just a smoke screen of your own _real_ arrogance. |