An uncountable countable set
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From: Tony Orlow on 20 Oct 2006 14:12 MoeBlee wrote: > Tony Orlow wrote: >> Also, upon which axioms is the definition of cardinality based? > > The usual definition is: > > card(x) = the least ordinal equinumerous with x > > The definition ultimately reverts to the 1-place predicate symbol 'e' > (and the 1-place predicate symbol '=', if equality is taken as > primitive). For the definition to "work out" ('work out' is informal > here) in Z set theory, we usually suppose the axioims of Z set theory > plus the axiom of schema of replacement (thus we're in ZF) and the > axiom of choice (thus we're in ZFC). However, there is a way to avoid > the axiom of choice by using the axiom of regularity instead with a > somewhat different definition from just 'least ordinal equinumerous > with'. Also, we could adopt a "midpoint" between the axiom schema of > replacement and the axiom of choice by adopting the numeration theorem > (AxEy y is an ordinal equinumerous with x) instead, which would be a > method stronger than adopting the axiom of choice, but weaker than > adopting both the axiom of choice and the axiom schema of replacement. > As to the more basic axioms of Z, for the definition to "work out", I'm > pretty sure we need extensionality, schema of separation (or schema of > replacement if we go that way), union, and pairing (pairing is not > needed if we have the schema of replacement). I'm not 100% sure, but my > strong guess is that we don't need the power set axiom for this > purpose. And we don't need the axiom of infinity. > > Why don't you just a set theory textbook? > > MoeBlee > I'm reading Non-Standard Analysis instead. Robinson agrees there's no smallest infinity, and that there are an uncountable number of countable neighborhoods with what he calls the '~' relation. I'm very encouraged to see essentially the same ideas as mine, put in technical terms. Maybe when I'm through with that, although I'm also reading Boole's treatise on logic where he eventually talks about probabilistic logic. Tony
From: Tony Orlow on 20 Oct 2006 14:16 David Marcus wrote: > Tony Orlow wrote: >> Your examples of the circle and rectangle are good. Neither has a height >> outside of its x range. The height of the circle is 0 at x=-1 and x=1, >> because the circle actually exists there. To ask about its height at x=9 >> is like asking how the air quality was on the 85th floor of the World >> Trade Center yesterday. Similarly, it makes little sense to ask what >> happens at noon. There is no vase at noon. > > Do you really mean to say that there is no vase at noon or do you mean > to say that the vase is not empty at noon? > If noon exists at all, the vase is not empty. All finite naturals will have been removed, but an infinite number of infinitely-numbered balls will remain.
From: Randy Poe on 20 Oct 2006 14:16 Tony Orlow wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> You have agreed with everything so far. At every point before noon balls > >> remain. You claim nothing changes at noon. Is there something between > >> noon and "before noon", when those balls disappeared? If not, then they > >> must still be in there. > > > > I thought you just said that the vase doesn't exist at noon. If the vase > > doesn't exist, how can the balls be in it? > > > > Either the experiment goes until noon or it doesn't. If all moments are > naturally indexed, then it doesn't. A set of moments is indexed. Those do not constitute "all moments". The clock ticks between the first and second insertion, for example, and we do not assign any index to any moment in that interval. - Randy
From: Tony Orlow on 20 Oct 2006 14:17 David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> Virgil wrote: >>>>> In article <4533d315(a)news2.lightlink.com>, >>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>>> Then let us put all the balls in at once before the first is removed and >>>>>>> then remove them according to the original time schedule. >>>>>> Great! You changed the problem and got a different conclusion. How >>>>>> very....like you. >>>>> Does TO claim that putting balls in earlier but taking them out as in >>>>> the original will result in fewer balls at the end? >>>> If the two are separate events, sure. >>> Not sure what you mean by "separate events". Suppose we put all the >>> balls in at one minute before noon and take them out according to the >>> original schedule. How many balls are in the vase at noon? >>> >> empty. > > Suppose we put ball n in at 1/n before noon and remove it at 1/(n+1) > before noon. How many balls in the vase at noon? > At all times >=-1 there will be 1 ball in the vase.
From: Tony Orlow on 20 Oct 2006 14:18
David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> Virgil wrote: >>>>> In article <4533d315(a)news2.lightlink.com>, >>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>>>> Then let us put all the balls in at once before the first is removed and >>>>>>> then remove them according to the original time schedule. >>>>>> Great! You changed the problem and got a different conclusion. How >>>>>> very....like you. >>>>> Does TO claim that putting balls in earlier but taking them out as in >>>>> the original will result in fewer balls at the end? >>>> If the two are separate events, sure. >>> Not sure what you mean by "separate events". Suppose we put all the >>> balls in at one minute before noon and take them out according to the >>> original schedule. How many balls are in the vase at noon? >> empty. > > Why? > Because of the infinite rate of removal without insertions at noon. |