Re: Well Ordering the Reals

Tony Orlow wrote:
> Daryl McCullough said:
> > Tony Orlow says...
> >
> > >Induction is stated axiomatically by Peano and used without regard to the
> > >underlying nature of the method.
> >
> > That's true, but the axiom of induction is provable from the
> > principles of set theory. So if you disagree with the way people
> > do induction, then there must be some principle of set theory
> > that you reject. Which one, and why?
>
> The axiom of induction is part of Peano's axioms regarding natural numbers, but
> doesn't ultimately have that much to do with set theory itself. It is a proof
> construction where every implication leads to another implication recursively.
> What other pronciples of set theory do you think it is dervied from? It's a
> construction NOT unique to the natural numbers. The naturals are just a
> convenient way to explain that a fact proven it that way holds for all cases.

The induction principle for natural numbers follows from the definition
of the set of natural numbers and the axioms of set theory. Why do you
argue about that which you know nothing about?

And it is true that there are induction principles for certain sets
other than the set of natural numbers, since these are sets that are
inductive in ways like the set of natural numbers is inductive. But I
don't know what you think this contradicts.

> > >What do you think it follows from, that makes it only apply to finite
> > >iterations?

As as been explained to you already, the induction principle for
natural numbers, as well as the recursive defintion theorem for natural
numbers follows from the defintion of the set of natural numbers and
the axioms of set theory.

> > In set theory, one *defines* the naturals to be the finite
> > ordinals, and it is *provable* that induction holds for them.
> According to the inductive axiom, but the naturals are not *defined* to be
> finite. They are *proven* to be finite using induction, using a construction
> that only works for finite iterations. So, the inductive proof that all
> naturals are finite depends on assuming that very conclusion.

If you knew anything about this, you'd not be making such silly
statements. The naturals can be defined as the finite ordinals OR the
naturals can be defined as the members of omega. And the definitions
turn out to be equivalent. There is no circularity in any of the
proofs. You just don't know the axioms, the proofs, or the theorems, so
you are making false claims about them. And no one can properly explain
to you the mistakes you're making if you continue to refuse to simply
look at the axioms and read the proofs from them.

> I wouldn't mind seeing what it is that
> modern mathematicians think is acceptable

What mathematicians find acceptable are proofs from axioms such that
there is an algorithm to determine whether a formula is an axiom and
there is an algorithm to determine whether an argument is indeed a
proof from the axioms.

MoeBlee

.

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