Re: Well Ordering the Reals

imaginatorium@xxxxxxxxxxxxx wrote:
> Give him credit where credit is due. He's doing an "exercise" I set, to
> describe the difference between an enumeration (by which I hope is
> normally meant a mapping from the naturals) and a well-ordering. By
> Tony's standards the above is quite good IMHO.

Fair enough. But then, by ordinary standards I give his answer a grade
of D. Anyway, he can't state a proper distinction between an
enumeration and a well ordering if he doesn't even know what an
ordering is, what a relation is, what an ordered pair is, what an
unordered pair is, what a singleton is, ... or what a FORMULA is.

I wrote him a long post a few months ago about the method of formal
axiomatics and encouraged him to find out more about it. Nothing took.
Then I held my tongue about his foolishness for all these months and
thousands of posts. Finally, I decided to speak up. He needs an
intervention. I guess that's what these threads are, in a way. If he
starts improving (and improvement here is not just a matter of learning
a course of study), then I'm glad and give him my congratulations. But
he's got a long way to go, yet he still refuses to open a book. As I've
already harped enough, he's in dire need to study a book on
mathematical logic.

> > You don't know what an enumeration is nor what an ordering is (let
> > alone a linear ordering or a well ordering). You don't know what
> > ANYTHING is in this subject. Really, that is not an exaggeration. You
> > don't know what membership is, what a set is, what a relation is, what
> > a function is, what a bijection is, what a natural number is, what a
> > real number is, on and on, so you can't possibly comment intelligently
> > on enumerations or well orderings or well ordering the reals or Godel's
> > relative consistency result, or ANYTHING about set theory.
>
> Actually your list is a bit too sweeping, I think, but in a way the
> problem is much deeper. Tony doesn't really understand what a
> "definition" is. That's why I (and various others) end up writing funny
> words like "pofnats", in an attempt to refer to the normal concept in
> distinction to Tony's imagined alternative. Most basically, of course,
> Tony has never managed to give even a clear but informal definition of
> what he means by (in)finite, so most of the discussion is totally
> futile.

I don't think the list is too inclusive. Ask him to tell you what
membership is (that will be a good exercise, since it will force him to
explain what it means for a predicate symbol to be primitive but to
have meaning through the narrowing of the class of models of the
axioms). Ask him to tell you what a natural number is, etc. You're
right that he doesn't know what a definition is, but the ignorance is
even more basic: He doesn't know what a valid mathematical-deductive
inference is. By that I don't mean that he commits numerous fallacies
(which he does) but rather that, literally, he does not know the
definition, nay, he doesn't even know that there ARE, definitions of
validity.

> I at least have learned something though.

What have you learned?

Thanks,

MoeBlee

.

Relevant Pages

  • Re: An uncountable countable set
    ... If an enumeration rule exhausts the naturals, ... That does not make their sizes infinite. ... of transpositions you give has order type w * w. ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... > Tony Orlow wrote: ... It has no definition on the Enumeration page. ... I take it you mean Cantor's diagonal proof of the uncountability of the reals? ... shows we need infinite digits to represent the elements of the infinite set. ...
    (sci.math)
  • Re: An uncountable countable set
    ... If an enumeration rule exhausts the naturals, ... of transpositions you give has order type w * w. ...
    (sci.math)
  • Re: Why?
    ... A provided that there is a surjection, ... naturals to that set) and being subcountable: ... It is not consistent to ... surjection e from N to 1 + A, called an enumeration of A. ...
    (sci.logic)
  • Re: abundance of irrationals!)
    ... > The the set of natural numbers is NOT defined by enumeration? ... >> There is no aleph_0 case which comes up in the induction. ... > infinite set of naturals, and not all of them, despite Peano? ... Peano said it did. ...
    (sci.math)