An uncountable countable set
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From: Virgil on 26 Aug 2006 15:56 In article <1156585139.529628.290740(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David R Tribble schrieb: > > > Logic > > dictates that every set must have a cardinality, either zero, a > > finite cardinality, or an infinite cardinality. > > How could the logicians live happily during the 2500 years before > Cantor? > > > > I have never said I think there is a largest natural. I have said that > > > some of your assumptions lead to that conclusion. You have then said things which are false in several axiom systems, and which you have not not shown to be true in any axiom system. > > No number of the form 0.1, 0.11, 0.111, ... can index the number > 0.111... without simultaneously covering its smaller digit positions. One number needs only one index value, so 0 indexes 0.111..., leaving enough of the finite initial subsequences of 1's to index all the 1's in that endless string, separately.
From: Virgil on 26 Aug 2006 16:06 In article <1156585516.788671.21150(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > > > Wrong. In mathematics the concept of "infinite sum" is not defined, > > But if there are infinitely many naturals then the infinite sum is > necessary. What axiom makes it necessary? If it is no more that a claim unsupported by any axiom(s) then it is only being assumed and there is no requirement that it be the case. "Mueckenh" has asserted that the sum of all naturals must exist, and, presumably, itself must be a natural, since M rejects the possibility of its being anything else. And if the set of all naturals is finite, as "Mueckenh" also claims but its sum must still exist, that sum will be a natural number larger than every natural number, including itself, so that "Mueckenh"'s alleged finite number system has problems of its own.
From: Virgil on 26 Aug 2006 16:22 In article <1156609912.036360.51250(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > Again, definition: K is the number with K[p] = 1 for all p in N and no > > other digits. > > Do you agree that is a valid definition? If not, why not? > > Because there is no number which can be completely indexed while it > cannot be covered. Then get a larger tarpaulin. > > > K can be completely indexed because each digit position is a natural number. > > Do you agree with this? If not, why not? > > Because there is no number which can be completely indexed while it > cannot be covered. Then get a larger tarpaulin. > > > K can not be covered because there is no natural number n such that all > > digit positions are less than or equal to n. > > Do you agree with this? If not, why not? > > Because there is no number which can be completely indexed while it > cannot be covered. Then get a larger tarpaulin. There is none! This is simply proved by the numbers > of my list > > 0.1 > 0.11 > 0.111 > .... Except that I, among several others, have shown how it CAN be done for that list. > > > > With the axiom of infinity it is dead easy, see above. And I thought we > > were arguing with the axiom of infinity in mind. > > Yes, we do, but also with some fundamental understanding of the fact > that unary numbers are finite sequences of 1's which cannot index a > digit without covering its precursors. There is no axiom to that effect in any axiom system extant. One can permute the indexing in uncountably many ways in violation of that false edict, and still in the end "cover" every digit with one index. You cannot sacrifice the most > obvious and simplest truth only to safe the consistency of the axioms. You cannot make up your own inconsistent axiom sets only to safeguard what you want to believe. Until "Mueckenh" can show that there is some INTERNAL inconsistency in ZF or ZFC or NBG, which does not involve any of his additional assumptions, he must lie to call our systems inconsistent. > > You assert that all digits of 0.111... can be indexed by unary numbers > of my list without being covered by at least one of them. What do you mean by "being covered"? Absent any expanation of that obscure phrase, "Mueckenh" is mouthing mere moonshine. > > I have done so many times, and am doing it here again. > > You gave an example how a number of the form 0,111...1 with n digits > indexes the n-th digit but does not cover all digits with m =<n ??? "Cover"? Moonshine.
From: Virgil on 26 Aug 2006 16:33 In article <1156610108.784198.90920(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > If there are infinitely many numbers and, hence, infinitely many > differences, then there is a definition of an infinite sum. Addition, > as a mathematical procedure, is a precursor of subtraction. Thus if, as "Mueckenh" asserts, there re only finitely many naturals, the sum of the largest two cannot exist if the largest is greater than 1. Nor can the product exist if the largest is greater than 2. Or does "Mueckenh" choose to make his arithmetic "modulus the first non-existent natural" to allow all additions and multiplications? But this screws up successor defined order since one could then have x + y < min{x,y}. For an infinite set of naturals the binary operations of sum and product always work, and are always consistent with the natural ordering of the naturals.
From: Virgil on 26 Aug 2006 16:38
In article <1156610236.164255.101950(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > So, you cannot say that an infinite count like aleph_0 is greater than > > any finite count? That's fairly lame. > > Sorry, I did not follow the complete discussion. But that is also wrong > according to standard ZFC set theory. The transfinite ordinals and > cardinals stand in trichotomy with the finite numbers. If this is > denied, then my first goal is achieved. If one orders the ordinals either by membership or by subset, trichotomy holds. If one orders the cardinals by injectability of sets having those cardinalities, trichotomy holds. But if "Mueckenh" assumes that what either TO or RE claim represents standard mathematics, he will make even more mistakes than he makes now. |