An uncountable countable set
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From: Virgil on 27 Aug 2006 12:48 In article <1156688926.886747.19010(a)74g2000cwt.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <44eef4aa(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> > > writes: > > > Dik T. Winter wrote: > > ... > > > > The set of all naturals numbers consists of only natural numbers. There > > > is NO natural number where the count becomes infinite. So there is no > > > point in the set, even if you COULD get to the "last" one, where any > > > infinite set has been achieved. > > > > And there is no point in the set where you have the complete set. Yes. > > Indeed. So what? > > And there is no such point outside either. That is why the complete > number 0.111... does not exist. It would be the complete set, it would > be the end of the set. It would mean a number larger than any natural > number. None of which prevents it from existing. At least in ZF, and NBG. Where can one find the list of axioms for any system in which it does not exist?
From: Virgil on 27 Aug 2006 12:53 In article <1156689269.210304.49390(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Should you be able to quote a serious scientist who asserted that the > list of all algebraic numbers was computable? When mathematicians are allowed, and expected, to dictate science to scientists, ONLY THEN will scientists be allowed to dictate mathematics to mathematicians. Until then each to his last.
From: Virgil on 27 Aug 2006 13:08 In article <1156689385.395973.61570(a)74g2000cwt.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > Quantifier dyslexia is the assumption that there is a number that > counts all natural numbers, i.e. which is larger than all natural > numbers. Wrong, as usual. The most common form of quantifier dyslexia is mixing up " for all x \in S there is a y \in T such that ... " with "there is a y \in T such that for all x \in S ..." where the ellipses are both the same statement about x and y. > > My arguing holds for the unary numbers of my list. If you think not, > then give an example of a finite number n which indexes the digit > position n but does not cover all positions m =< n. One number indexes one number. What does "cover" mean? Until it is definided, it means nothing. > > Here is my example: The number 3 = 0.111 in my unary expression indexes > the digit position 3 of any unary number and it covers the digit > positions 1, 2, and 3 of any unary number. How does it "cover" them? If you mean that it contains substrings that exemplify them as unary numbers, so what? That is not an essential requirement of indexing. All indexing requires is a surjection from the index set to the set being indexed. It is even nicer if the index set is well ordered, in which case some well ordered subset of the index set can be shown to biject with the set being indexed. And, in fact, N is a well ordered set which injects to a proper subset of itself and, for any finite set S, can be surjected to N \/ S.
From: Virgil on 27 Aug 2006 13:15 In article <44f1b00c(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> Logic determines truth. Induction is more than just a form of proof. > > > > Logic determines VALIDITY. And induciton is indeed more than just a > > form of proof. But your concept of induction is uninformed, > > superficial, and confused. > > > > MoeBlee > > > > That second sentence refers to inductive logic Mathematical induction refers to what is known as the (finite) inductive principle of mathematics, which is far removed from inductive logic, and is strictly a form of deductive logic. > When you say my concept of induction is "uninformed, superficial, and > confused", I would disagree. Of course TO would. Anything revealing TO's true nature TO finds disagreeable.
From: MoeBlee on 27 Aug 2006 17:49
MoeBlee wrote: > However, as exception to the generalization just mentioned, in a > language with only finitely many non-logical primitive symbols, we can > state the identity of indiscernibles as a schema. CORRECTION: Darn. Posting too fast. Not thinking it through. I was trying to be generous, but I should have stuck with my original tightfistedness. Even for a language with finitely many primitive predicate symbols, I don't see how there can be a theorem schema (and, as I recall, it is a result mentioned in the literature that there cannot be). I can show you a formulation that comes close to being a theorem schema and I can explain why it is not a theorem schema, if you're interested. However, for certain theories such as set theory, we can specify equality conditions regarding each of the finitely many primitive predicate symbols. That will ensure the identity of indiscernibles. In other words, if x and y are indiscernible as to the membership relation, then x and y are the same set (and the axiom of extensionality makes the condition even weaker: x and y merely need to be indiscernible as to their members). MoeBlee |