An uncountable countable set
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From: Virgil on 25 Aug 2006 14:16 In article <1156500895.295710.85000(a)75g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > The set of edges of an infinite binary tree are certainly countable, and > > it is not difficult to construct an explicit bijection between them and > > the set of binary representations of the naturals. > > > > The set of paths, however is not countable, as may is shown by > > constructing an explicit bijection between the set of all such paths and > > the set of all subsets on the set of naturals. > > > > Both the bijections referred to above have been presented here with no > > one able to show either to be anything other than as advertised. > > Ok. Let's accept that. But by rational relation we find the set of > paths being *not* larger > than the set of edges. How is that possible? Because the "rational relation" in question requires that both be finite.
From: Tony Orlow on 25 Aug 2006 14:20 MoeBlee wrote: > Tony Orlow wrote: >> If the naturals are not a subset of the reals in ZFC and NBG, then those >> theories are even more screwed up than they already seemed. > > With either of the usual constructions of the reals (Dedekind cuts or > equivalence classes of Cauchy sequences), both the system of naturals > and the system of rationals are isomorphically embedded in the system > of reals. Mathematicians usually speak of the system of natural numbers > and the system of rational numbers as subsystems of the the system of > reals. Strictly speaking, that is incorrect, but it is harmless given > the isomorphism. Harmless, even though incorrect? What makes it incorrect, if not inconsistency? Doesn't inconsistency cause a problem? > >>>>> I don't know what is meant by '(y=2)' in the larger formula. >>>> I mean that for x>2, 2*x < x^2 < 2^2. For x=2 they're equal. >>> Then (1) is a very peculiar way of trying to convey that claim. >> What's peculiar about it? > > The notation of appending the formula with the parenthetical is unclear > without your explanation of its meaning. This is a minor point only > regarding your notation. I used y in the equation (or statement of inequality) and then specified in parentheses at the end, with a space, which condition for y made that statement true. I didn't think that was tooooo confusing. > >>>> Is aleph_0>2? >>> AS TO's comparison operator is according to TO defined only for reals, >>> the question is nonsense. >>> >> Then aleph_0 is not a count of any sort, or it would exist on the >> hyperreal line. It's not a number of any real sort. > > 'number of any real sort' is not a defined predicate of set theory. w > is not in the standard ordering of the reals. That does not make w an > undefined object. My experience is that asking amthemticians for a definition of "number" results in.....nothing. > >> you don't see scientists accepting transfinitology either. > > Of course, you have a survey of scientists to support your claim. Uh, yes, right here. Why don't you survey thise that object, regarding what they do? > >>> As we have no definition of length valid for intervals whose endpoints >>> are not finite reals, it means nothing. >>> >> That's because you have no infinite values which behave sufficiently >> like numbers to support such a definition. > > You have no coherent system of definitions at all. It's not the job of > set theory to define objects that obey the whims of your informal > notions. It's the job of mathematicians to work with numbers. > >> The selection of any unit is done by simply choosing a point separate >> from the origin. When it comes to division using infinite values, one >> translates the geometric definition into symbolic form and applies >> induction formulaically. > > "Translates the geometric definition into symolic form and applies > induction formulaically" is no less doubletalk than "Coordinates the > numeric form into geometric postulates and applies the recursive > definition metrically." > Is that a question? > MoeBlee >
From: Virgil on 25 Aug 2006 14:23 In article <1156501289.435365.119480(a)75g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1156189552.184903.323170(a)m79g2000cwm.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Dik T. Winter schrieb: > > > > > But often some expression like 0.111... is ivolved where not all digit > > > positions can be indexed by natural numbers and n is understood to > > > approach something I do not know. > > > > Your ignorance is not an acceptable excuse to those of us who have no > > trouble indexing every digit of "0.111..." by a natural number. > > The problem is not indexing but "indexing without covering". > That is easy to prove impossible for any finite natural number in unary > representation. And, alas, there are only finite natural numbers. I have no idea what "indexing without covering" means, and until it has a clear definition, I will continue to state that indexing all of them in any way indexes all of them. > > > > Division was possible and was practised in fact long before rings and > > > fields were known. > > > > Not outside of what later became known as rings. > > > > > Division of rationals by rationals or reals by reals is quite different. > > So we have no reason to suppose that divisions involving other than > > naturals need behave like division of naturals, or even be possible > > without specific definitions of what it is and how it works.. > > > The old Greek already developed the method of geometric division, using > the so-called Gnomon. This method makes no difference between division > of naturals, rationals, reals or whatever deserves the name "number". > Every well educated mathematician knows it. (Geometry is a certain > language of mathematics.) But they HAD those specific definitions that are required. It is division in the absence of such definitions that I object to. > > > > > > One must have a very restricted mind to believe that without fields and > > > rings division was impossible. > > > > The set of positive naturals is not even a ring, but it allows at least > > two forms of division, and this has been recognized by most of us from > > well before "Mueckenh". > > Your frequence of self contradicitions increases. You just said "Not > outside of what later became known as rings." The naturals, or at least an ismorphic image of them , are contained in lots of rings, so that arithmetic is "within" a ring.
From: Virgil on 25 Aug 2006 14:25 In article <1156501377.098380.50700(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > But as we just investigate consistency, you cannot presuppose it. With > > > your attitude it is impossible to find any inconsistency even in an > > > inconsistent theory. Deplorably you are too simple to recognize that. > > > > If "Mueckenh" can deduce from any axiom system both a statement within > > the system and its negation, "Mueckenh" will have found his > > inconsistency. > > Which part of my proof concerning the binary tree is not in accordance > with the ZFC axioms in your opinion? The part that says a bijective image of the naturals bijects with a bijective image of the power set of the naturals.
From: Virgil on 25 Aug 2006 14:37
In article <1156501674.444594.242900(a)74g2000cwt.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > > Mathematics is not about reality... > > It is. You have not yet recognized it. Mathematics itself occurs only in the imagination, and is no more 'real' than anything else that occurs there. It is only in attempts to apply mathematics to reality that there is any point of contact between the two. > > The notion n --> oo is not clear It is to me. it means "as n increases through the naturals without limit". > , often, because it is intermingled: > 1/n = 0 in the limit and n remaining a natural which leads to 1/n > 0 > in the limit. According to one standard definition of "lim{n -> oo} f(n) = L, where L is some real number and f is a function from the natural numbers to the real numbers, this is logically equivalent to saying that for every positive real number epsilon, Card({n \in N: | f(n) - L | >= epsilon}) \in N. Any interpretation which conflicts with this one is improper. |