A new definition of Cardinality.
in [Logic]
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From: Jesse F. Hughes on 30 Nov 2009 13:56 George Greene <greeneg(a)email.unc.edu> writes: > On Nov 30, 8:59 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Every set is equipollent to some cardinal kappa. This is a simple >> theorem of ZFC. > > Therefore, NObody is disputing THAT. > So why are you even belaboring it? > This is called STUPIDITY. > ON YOUR part. Tell me, George, do you find it difficult to respond with such passion and vitriol when you know that you're wrong? Or does being right matter much less than being loud? Honestly, something ain't right with you. -- Jesse F. Hughes "Have we learned nothing, nothing, from the downfall of Vanilla Ice?" -- Time Magazine columnist Lev Grossman on James Frey's /A Million Pieces/.
From: zuhair on 30 Nov 2009 15:52 4 important questions? In my search for making a better definition of Cardinality and during my discussions with Prof. Dana S. Scott, some issues raised that makes me ask the following four questions: Question 1: is the following a theorem of ZF? For all x Exist y ( y is transitive & y equinumerous to x ) were y equinumerous to x <-> Exist f (f:x-->y, f is bijective). Question 2: If the above is not a theorem of ZF, then is its negation a theorem of ZF? Question 3: is the following a theorem of ZF? For all m Exist x ( x is transitive & for all y ((y is transitive & y supernumerous to m) -> x subnumerous to y)) were y supernumerous to m <-> Exist f (f:m-->y, f is injective) x subnumerous to y <-> Exist f (f:x-->y, f is injective) Question 4: if the above is not a theorem of ZF, then is its negation a theorem of ZF? Zuhair
From: K_h on 30 Nov 2009 20:06 "zuhair" <zaljohar(a)gmail.com> wrote in message news:f5c57b1d-d527-487b-a5b1-ebf0a2e3dc2b(a)e27g2000yqd.googlegroups.com... >4 important questions? > > In my search for making a better definition of Cardinality > and during Here is a fast and easy way to define cardinals -- I think. Define the surreal numbers and their order using Conway's simple construction convention and definition (these only require a few lines). Call surreals of the form {x|} "left surreals". Then define the cardinality of a set to be the least left surreal equinumerous to it. You're done! One reason this definition should be a good one is because infinite left surreals are equinumerous to all infinite left surreals less than themselves. Additionally, one can distinguish between finite and infinite left surreals by whether or not they can be equinumerous to proper subclasses of themselves. I don't see any problem with this approach but perhaps I missed something. Best, REF: http://planetmath.org/encyclopedia/OmnificIntegers.html
From: Jesse F. Hughes on 30 Nov 2009 20:57
George Greene <greeneg(a)email.unc.edu> writes: > On Nov 30, 1:40 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Oh. I guess I can't read. > > Not much guessing required. > > >> Here's the bit that gave me this odd notion that you doubted this >> simple theorem. >> >> ,---- >> | > If I assume the axiom of choice, then every set can be well-ordered, >> | >> | Right. >> | >> | > and is equipollent to some cardinal kappa, >> | >> | WRONG. >> `---- >> >> > So why are you even belaboring it? >> > This is called STUPIDITY. >> > ON YOUR part. > > This is just plain dishonest. These two things ARE NOT ADJACENT > in the original context. You have snipped without even acknowleding > RELEVANT context that you have removed!! What context have I removed? Those quotations *are* adjacent in the post on my server. And I included the paragraph following "WRONG." in the post you're replying to, so you surely aren't complaining about that snippage. >> Yes, I must be stupid for misunderstanding that very clear exchange. > > No, you must be stupid FOR REMOVING THE CONTEXT AND LYING > about what was being said. WHAT ACTUALLY OCCURRED after I said > "WRONG" was: >>> Ex[whatever] >>> does NOT mean that there exists a UNIQUE x such that whatever! >>> There are MANY DIFFERENT well-orderings of p(w) under ZFC and >>> they correspond TO DIFFERENT ordinals! > > This IS relevant, it DOES matter, and it DOES PROVE that Rupert was > wrong. No, it doesn't. There are many different well-orderings of P(w), true. And yet ZFC proves that P(w) is equinumerous to exactly one cardinal. Imagine! Just as Rupert said! > I was not disputing ANY of the THEOREMS! > YES, IT IS a theorem that any set is equipollent to some initial > ordinal. > YES, IT IS a theorem that any set (including p(w)) therefore has a > cardinality. > These truths DO *NOT MATTER*!! They are all that Rupert asserted. Right there, where you responded "WRONG." Can't you read? > They are NOT SUFFICIENT FOR ANYone to allege that ZFC *assigns* > or *determines* a cardinality FOR ANY uncountable set! Rupert's post did not use the terms "assigns" or "determines". What the heck are you replying to? Rupert's post has message id <9be040de-1f99-4f15-99b5-9e4a2ff35d0c(a)z35g2000prh.googlegroups.com>. Why don't you read it again and tell me where you think Rupert claimed something inconsistent with the independence of GCH. Or don't. Whatever. > DESPITE these theorems, the ACTUAL PARTICULAR cardinality of > ANY uncountable set IS UNdetermined by ZFC! > > This is A VERY BAD state of affairs! > I mean, if ZFC would just ADMIT that it couldn't determine these > cardinalities, > that would be BETTER. But INSTEAD it CLAIMS to fix them and then > FAILS to! > > In any case, my point is, I DID state this, and YOU DID *CUT* it while > TRYING > to claim that you were saying something rational about the argument! What I cut was irrelevant. All you said is that different well-orderings of P(w) correspond to different bijections from P(w) to some ordinal. Yes, but so what? Different well-orderings of w also correspond to different bijections from w to some ordinal. This trivial fact that there are different ways to well-order P(w) really doesn't have anything much to do with CH. > > MORE to the point, I PREFIGURED all of this: I STARTED OUT WITH > calm reasonable claims that you are NOT BOTHERING to acquaint > YOURself with before intervening! E.g., >>> Even WITH choice, the ZFC definition of cardinality STILL has a >>> serious drawback. You STILL CANNOT determine the cardinality of >>> p(w) under ZFC. The collection of all countable ordinals (aleph-one) >>> cannot possibly be bigger than p(w), but it could be equally large-- >>> the cardinality of p(w) could be aleph-one, aleph-2, or any finite >>> aleph. > > THIS IS THE ONLY sense in which I EVER said that Rupert was "WRONG", > and it WAS PRE-established by the context. You're right, I didn't read any of your contributions prior to your asinine reply to Rupert. But none of that matters. Rupert said something true, and you replied with "WRONG." Followed by an irrelevant observation about different well-orderings of P(w). Rest snipped, because I'm afraid that my laptop will run out of capital letters if I continue this tedious conversation. You were wrong, George, and all your shouting doesn't change that. Complain all you want about my reading comprehension. Rupert made a very clear and correct claim and you read something else and *even then* couldn't argue your point coherently. -- Jesse F. Hughes "Ultimately, I can bring the entire mathematical establishment to its knees... Live in a fantasy world if you wish, but to me that's just an expression of your intellectual inferiority." --James Harris |