An uncountable countable set
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From: MoeBlee on 20 Sep 2006 12:53 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
From: MoeBlee on 20 Sep 2006 13:04 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
From: MoeBlee on 20 Sep 2006 13:21 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
From: Mike Kelly on 20 Sep 2006 13:27 Han.deBruijn(a)DTO.TUDelft.NL wrote: > Mike Kelly wrote: > > > The limit of a sequence need not have a property that each of its > > elements has. > > The limit of the sequence 1,1,1,1,1,1,1,1, ... ,1, ... is 1. > > Han de Bruijn True. How does that refute anything I said? We have a sequence of sets.. {0}, {0,1}, {0,1,2}, {0,1,2,3} .... and for each we can define a uniform probability distribution to choose one element of the set. The limit of this sequence of sets is N. We construct a sequence of corresponding "Kolmogorov sums", the summations of the elementary probabilities. 1/1, 2/2, 3/3, 4/4, ... = 1, 1, 1, 1... The limit of this sequence of real numbers is 1. Now you want to conclude that there is a uniform distribution on N. However, you offer no justification for this. -- mike.
From: MoeBlee on 20 Sep 2006 13:30
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. It's a theorem of set theory is that there is bijection between the set of natural numbers and the set of even numbers. It's also a theorem of set theory that every other natural is an even natural in the standard ordering of naturals. This is not a contradiction. MoeBlee |