An uncountable countable set
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From: David Marcus on 18 Oct 2006 18:15 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? -- David Marcus
From: Ross A. Finlayson on 18 Oct 2006 19:38 MoeBlee wrote: > Ross A. Finlayson wrote: > > > I actually recommend that you read the book "Counterexamples in Real > > Analysis." That contains scores of what are called "counterexamples in > > standard real analysis." > > I have browsed that book. It does not give counterexamples that shown > an inconsistency in analysis. What it gives are counterexamples to > generalizations that are NOT theorems. The counterexamples themselves > are theorems, not contradictions of theorems. It's of the form (though > of course, this is not an actual example): A counterexample to the > statement that "all prime numbers are odd" is the prime number 2. > > > Did you know model theory posits the existence of a maximal ordinal of > > which there is none in ZF? > > Of course no one knows that, since it is not true. > > MoeBlee Hi MoeBlee, What do you make of where it shows there's a least positive real, then? If the differential is finite the sum over them does not necessarily represent the integral, and if the differential is zero it doesn't necessarily either. MoeBlee, if you read Cohen's definition of forcing, it is as I say. Ross
From: MoeBlee on 18 Oct 2006 19:56 Ross A. Finlayson wrote: > What do you make of where it shows there's a least positive real, then? What page number? I'll look at it. > MoeBlee, if you read Cohen's definition of forcing, it is as I say. Sure it is. Sure it is. MoeBlee
From: Ross A. Finlayson on 18 Oct 2006 20:48 MoeBlee wrote: > Ross A. Finlayson wrote: > > What do you make of where it shows there's a least positive real, then? > > What page number? I'll look at it. > > > MoeBlee, if you read Cohen's definition of forcing, it is as I say. > > Sure it is. Sure it is. > > MoeBlee Hi, I don't have a copy here with me, I believe it was about on page one, and the value was labelled s. M, the model, is maximal. There is no universe in ZF. So, quantify over sets. Ross
From: David Marcus on 18 Oct 2006 23:23
imaginatorium(a)despammed.com wrote: > Ross A. Finlayson wrote: > > I actually recommend that you read the book "Counterexamples in Real > > Analysis." That contains scores of what are called "counterexamples in > > standard real analysis." > > Sorry, Ross, perhaps I misread. Um, well, on looking closely, perhaps I > didn't. Here's what you refer to: > > "counterexamples to standard real analysis" > > I read that as meaning that there are counterexamples that show that > standard real analysis is inconsistent, wrong, or otherwise defective. > But the title of the book you mention is subtly different: > > "Counterexamples in Real Analysis" > > I went to Amazon.com, and found a book with that title, and looked in > the front. I saw a list of "counterexamples", and noticed that one of > them was (from memory) "There exists a set with an infimum greater than > its supremum". Sounds very odd - intuitively, it seems obvious that, > assuming the Latin words mean greatest lower bound and least upper > bound, the glb must be less than the lub, *until* you think about > proving it, and think about the definitions. Aha! Consider the reals in > [0, 1] with the usual ordering, then these have a subset whose glb is 1 > and whose lub is 0. (At least if I've guessed the standard definitions > correctly.) Can you see what it is? "inf" is glb and "sup" is lub, as you surmised. According to Amazon, the book says there exists a subset of R, not a subset of [0,1]. I wonder how the book is defining inf and sup, since with the definitions I usually see, inf A <= sup A, where A is any subset of R where the inf and sup exist. > Thus, it appears that this book is about "counterexamples", in the > sense of counterexamples to statements that might seem intuitively > obvious, but which are not true. Or, even counterexamples to statements that aren't intuitively obvious, but which you might conjecture as being true, e.g., take a theorem and remove one or more hypotheses. -- David Marcus |