An uncountable countable set
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From: Mike Kelly on 20 Sep 2006 14:38 MoeBlee wrote: > Mike Kelly wrote: > > It isn't. It *is* the set of all finite naturals. By the way, omega is > > an ordinal number, not a cardinal number. > > With the usual definition of 'cardinal', it is a theorem that w is a > cardinal as well as an ordinal, even if it is more common to emphasize > that w is a cardinal by using the notation 'aleph_0' rather than 'w' > when speaking of w in context of its being a cardinal. Thanks for the clarification. -- mike.
From: Tony Orlow on 20 Sep 2006 14:39 Randy Poe wrote: > Tony Orlow wrote: >> Given that all evens are natural, but only every other natural is even, > > Yes, the evens are a proper subset of the naturals. > >> is is logically deducible > > No, that's not a deduction. There's no theorem or axiom you > use to make this jump. IFR does that and more. Or, I can be a simpleton and refer to relative set densities. > >> that the naturals are twice as numerous in any >> range as the evens. > > Because you have an intuitive notion of range and > numerous, neither of which applies a priori to either the > set of naturals or of evens. That sentence no predicate. That clause. But, anyway, that I take certain principles as "a priori" in the sense of being more fundamental than the axioms of ZFC, indeed those priciples apply in that sense. > > The problems start when one tries to pin down a > meaning of "numerous" for such sets. Then the > way things are no longer follows "the way Tony > wants them to be". Indeed they do, if we make them. Hold onto the concept that less is not equal, and you'll see why proper subsets are never as large as their proper supersets. > > But I never worried about what you were thinking > when I learned mathematics, so it's not overly > bothersome to me that the deductions lead to a > different conclusion than the desires of some stranger > named Tony on the internet. When you learned transfinitology, had you seriously considered the mathematical notion of infinity previously? I had. > >> Intuition is a wholistic application of logic and >> association, > > No, intuition is guessing in the absence of logic. Um, you might want to do more research into the structure and behavior and laws of mind, before you claim to know what intuition is. That very statement could use some-nonverbal reflection on your part. > >> not to be rejected unless proven contradictory. > > Which it has been, over and over and over... Um, no, it hasn't, except in the very limited context of transfinitology. > > - Randy > - Tony
From: Tony Orlow on 20 Sep 2006 14:41 MoeBlee wrote: > Tony Orlow wrote: >> If omega is the successor to the set of all finite naturals, > > But it's not the successor of the set of all finite naturals. > > MoeBlee > It's not the least set that includes them all? For all finite naturals, they represent the set of all naturals less than them. The notion of limit ordinals is an extension thereof. Omega is the least ordinal which serves as a superset of the naturals. It is successor to the set with no end, in that sense. It is certainly more than any neN, as any n is more than each member of the von Neumann set with which it is equated. ToeKnee
From: Tony Orlow on 20 Sep 2006 14:43 MoeBlee wrote: > Tony Orlow wrote: >> Yes, wholeheartedly. In finite arithmetic, when you add a nonzero >> quantity, you increase the value - not so in transfinitology. You can >> remove elements, divide the set in half, > > No, you need to DEFINE a division operation on infinite sets or > infinite cardinals if you are to speak of "divding" as a kind of > arithmetic. It is the fact that there is NOT a unique result from such > a proposed "divsion operation" that blocks arguments that set theory is > inconsistent due to erratic divsion results. > > MoeBlee > That is simply not true for any well-defined infinite value. We've already established that the average value in the infinite set of reals in [0,1] is 1/2. Similiarly, we can apply such methods to the naturals, giving that the evens are half the set. The fact that transfinite "arithmetic" does not get a compatible result is evidence that it's not compatible with mathematics. ToeKnee
From: Tony Orlow on 20 Sep 2006 14:44
MoeBlee wrote: > Tony Orlow wrote: >>> Do you understand that Cardinality doesn't claim to be the only or best >>> analogy to "size" for finite sets? >> Mathematicians claim that. If not, then why so much resistance to >> attempts to improve upon cardinality? > > The resistance is not to different systems, but rather to nonsense that > is not even a system and resistance also to critiques of set theory > that are ill-premised and simply incorrect. You can read ALL KINDS of > meaningful debates among mathematicians and philosophers about widely > different systems. But such meaningful debate is not found in the > gobbeldygook of cranks. > > MoeBlee > Define "cranks" from primitive terms, please. ToeKnee Are they people with dumb names like these? ;) |