Cantor Confusion
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From: Virgil on 8 Nov 2006 23:31 In article <1162983690.750572.35670(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > It is tedious to explain things to you. Please consult a book. Hint: > "Some mathematicians object to the Axiom of Infinity on the grounds > that a collection of objects produced by an infinite process (such as > N) should not be treated as a finished entity." (Karel Hrbacek and > Thomas Jech: "Introduction to set theory" Marcel Dekker Inc., New York, > 1984, 2nd edition.) > > Now substitute "entity" by the most significant property of the set N. > > Regards, WM That might lead to someone objecting to claims of a set having the property of being infinite, but does not explain what you mean by "finished infinity".
From: Virgil on 8 Nov 2006 23:38 In article <1162984299.823829.205490(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > 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. > > That means: All its elements exist? > > > > It is not true that a set cannot exist unless it has a maximum element. > > It is true, at least if all of its elements do exist. Not in ZF or NBG or NF, so what is the system WM is proposing is which every ordered set must have a maximal member? > But in order to > force you to understand that simple truth, consider the bijection > between lines and columns. All lines do exist, all columns do not > exist. Unless there is some finite upper bound on line length, so that one can find some n in N for which there is no column beyond column n, WM is wrong.
From: Virgil on 8 Nov 2006 23:43 In article <1162984577.878417.221660(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > 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? > > No it is the same. I distinguish beween lines and columns. But the length of the first column equals the number of lines which WM says is greater than then length of any line (number of columns in any line). WM is clearly confusing himself. > > > > 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. > > Maybe that my use of "any" was wrong. Please replace it by "every": the > length of the diagonal is clearly not more than the length of every > line. If every line is finite and the diagonal is, as WM admits above, equal to the (non-finite) number of lines, then WM reveals once again his continuing foolishness. > > Regards, WM
From: Virgil on 8 Nov 2006 23:45 In article <1162984762.830387.209070(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > 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. > > > When one can construct explicit injections from the set of edges into > N, no sort of > induction is needed at all. That only proves the countability of the set of edges, it does not, and cannot, prove countability of the uncountable set of paths.
From: Virgil on 8 Nov 2006 23:56
In article <1162985144.480732.105800(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > The length of the matrix is omega, as a maximum taken. > > > > > > Does WM claim a line numbered omega? > > No. You claim omega lines where omega is actually a number, though not > a natural one, but a number which can be larger than any natural. What I claim is that there is a line for every MEMBER of N (or omega). and if each line numbered n has at least n entries, there is a column for each member of N ( or omega). I do not say that the "number" of either is "equal" to omega, as I do not know what that means unless it means no more than what I said above that there is a member for each member of omega. > I > prove that this is false by showing that in this case a number omega > would be required by the simple bijection: > > 1 <--> 1 > 1,2 <--> 2 > 1,2,3 <--> 3 > ... > > 1,2,3,... <--> omega > 1,2,3,...,omega <--> omega+1 > > > > If so then what is the line number of the immediately preceding line? > > If not then it is only a supremum on the line numbers and not an actual > > maximum. > > > > > The width of the matrix is omega, as a supremum not taken. > > > > > > This small problem contradicts ZFC. > > > > No, it contradicts WM. > > It contradicts the square matrix which is required for Cantor's > diagonal argument. I do not recall anything in the Cantor proof about square matrices. The only requirement in the Cantor "diagonal" proof is that the nth number listed have a determinable nth decimal digit. And that does not require any sort of "square matrix". |