An uncountable countable set
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From: Virgil on 20 Sep 2006 18:11 In article <45118b52(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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? The definition of 'successor' does not say that. The only standard definition of successor here has been y is the successor of set x, if and only if y = Union(x, {x}) 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. Ambiguous. Omega is a proper superset of each natural but not a proper superset of the set of all naturals. TO being sloppy about meanings again! > It is successor to the set with no > end, in that sense. Wrong! It IS the set with no end. The successor to that set has an end.
From: Virgil on 20 Sep 2006 18:36 In article <45118bde$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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. TO has no idea of what is or is not true. > We've > already established that the average value in the infinite set of reals > in [0,1] is 1/2. Provided that one defines average value as expected value under the uniform distribution on [0,1] Similiarly, we can apply such methods to the naturals Not so, as there cannot be any uniform distribution on N, nor any expected value of a member of N without assuming some possible distributions of probabilities on N. > 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. Not quite. It is merely not compatible with TO's notoriously erratic intuition.
From: Virgil on 20 Sep 2006 18:38 In article <45118c18$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > 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. Anyone approved by TO's intuition who hawks ideas contrary to common usage.
From: Virgil on 20 Sep 2006 18:44 In article <45119205(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> Given that all evens are natural, but only every other natural is even, > >> is is logically deducible that the naturals are twice as numerous in any > >> range as the evens. > > > > Define 'twice as numerous' for infinite sets. > > IFR does that and more, in terms of formulaic relations between sets > over a given range. Not quite. What TO's IFR does is produce formulas from order relations on sets, which are totally independent of any properties of the underlying sets themselves, except their cardinalities.
From: Virgil on 20 Sep 2006 18:47
In article <45119523(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> If omega is the successor to the set of all finite naturals, > > > > It isn't. It *is* the set of all finite naturals. By the way, omega is > > an ordinal number, not a cardinal number. > > Yes, and the transfinite cardinals are based on the limit ordinals, eh? Nope. One can develop cardinality and ordinality quite independently, despite the obvious connections between them. |