Cantor Confusion
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From: David Marcus on 6 Nov 2006 16:08 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > Modern mathematics does not use the term "finished infinity". If you > > wish to use it, please define it. > > Modern mathematics assumes that there are sets which are larger than > infinite sets. That is the same as to say after one finished infinity > we consider the next infinity. I asked you for a definition of "finished infinity". Is what you wrote supposed to be the definition? Normally, a definition should start out something like "Finished infinity is ... ". -- David Marcus
From: William Hughes on 6 Nov 2006 16:16 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > No. I believe that the length of the diagonal is the supremum of > > > > the length of the lines. This supremum does not have to be the > > > > length of any line. Call it omega. Then omega is the *supremum* > > > > of the set of lines (equally the supremum of the set of columns). > > > > > > The set of lines has infinitely many elements, no line has an infinite > > > number of columns. That is a difference. > > > > Correct (in ZFC). > > Correct in any theory which claims "an infinite set of finite natural > numbers". > > > > > If you consider omega to be not the maximum but only the supremum of > > > the set of lines, then we agree that actual infinty does not exist. > > > > Omega is the sup of the set of *lengths* of lines. > > > > > But we both then disagree with ZF which states *there exits* the set of > > > lines. They all are there. > > > > Why would William disagree that there is a set of lines? I'm sure he > > doesn't. > > Please read carefully: > If he considers omega to be not the maximum but only the supremum of > the set of lines, then we agree that actual infinty does not exist. No. You seem to be saying that: if the set of lines does not contain a line of maximum length then the set of lines does not exist. However, my claim is that: the set of lines does not contain a line of maximum length and the set of lines exists. It is not true that a set cannot exist unless it has a maximum element. - William Hughes > > > > > And there does not exist a set of infinitely > > > many columns (= an infinite natural number). > > > > What do you mean a "set of infinitely many columns"? We can consider the > > set of column numbers. Is that what you mean? > > The number of columns is finite for every line considered. The numbers > of lines is infinite for every column considered. > > Regards, WM
From: David Marcus on 6 Nov 2006 16:15 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > 1 > > > 11 > > > 111 > > > ... > > > > > > The length of each column is omega. > > > > I assume that we are using ZFC as our logical system. > > > > Assuming by the "length of each column" that you mean the number of 1's > > in each column, then the number of 1's in each column has cardinality > > aleph_0. So, what you wrote is essentially correct. > > > > > The length of each line is less than omega. > > > > Each line has a finite number of 1's. Finite cardinality is less than > > aleph_0. So, what you wrote is essentially correct. > > > > > The length o the diagonal is less than omega. > > > > The number of 1's in the diagonal is aleph_0. So, what you wrote is > > false. In fact, I have no clue what you could possibly be thinking that > > would lead you to make such an obviously incorrect statement. The length > > of the diagonal is clearly the same as the length of the first column, > > and you just wrote above that the length of each column is omega. > > The length of the diagonal is clearly not more than the length of any > line. Do you disagree that the length of the diagonal is the same as the length of the first column? As for your statement that "the length of the diagonal is clearly not more than the length of any line", the length of the first line is 1, and this is less than the length of the diagonal, so your statement as written is false. So, what do you really mean? -- David Marcus
From: Virgil on 6 Nov 2006 17:00 In article <1162820871.886446.129490(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > But let's take the actually infinite set of natural numbers. > > Suppose it contains your non-natural number. Now remove that non-natural > > number from that set. Isn't the set still actually infinite? > > That shows only the self-contradiction of the notion actual infinity. It only marks the misrepresentataions made by mueckenh.
From: Virgil on 6 Nov 2006 17:04
In article <1162821418.935074.235580(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1162557998.763485.221280(a)i42g2000cwa.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > > > > > > No, something quite different was argued there. Namely that the > > > > > > limit > > > > > > *also* is the number of edges in the infinite tree (or somesuch) > > > > > > requires transfinite induction. > > > > > > > > > > But is does not. > > > > > > > > Indeed, it does not. But I thought you were maintaining that it would > > > > be? > > > > > > Good heavens! I should require transfinite induction? > > > > Sorry, I misread. You insist that the limit *also* is the number of edges > > in the infinite tree. To prove that you need transfinite induction. > > Then you need transfinite induction to prove that the number of > rationals in the case of an infinite set Q is countable. When one can construct explicit injections from Q into N, no sort of induction is needed at all. |