From: cbrown on
stephen(a)nomail.com wrote:
> cbrown(a)cbrownsystems.com wrote:
> > Tony Orlow wrote:
>
> > <snip>
>
> >> I did give another curve with the same "Tlimit" as the staircase in the
> >> limit, which produced an interesting result, giving weight to the notion
> >> that an infinitesimal is something distinct from 0, whose square is not
> >> distinct from 0.
>
> > Suppose we let B represent Big'un; then B*1/B = 1, where 1/B is an
> > infinitesimal. Then what you are saying means
>
> > 1/B = 1/B
> > 1*1/B = 1/B
> > (B*1/B)*1/B = 1/B
> > B*(1/B*1/B) = 1/B
> > B*(1/B^2) = 1/B
>
> > Since 1/B is infinitesimal, its square is not distinct from 0; so...
>
> > B*(0) = 1/B
> > 0 = 1/B
>
> > So 1/B is identical to 0. Where is my error?
>
> Assuming that Tony's definition of infinitesimal has anything
> to do with any standard definition of infinitesimal. :)

If TO claims that every T-infinitesimal, when T-squared, is not
T-distinct fom T-0, I'm T-willing to T-take him at his T-word :).

Cheers - Chas

From: Virgil on
In article <45119c60$1(a)news2.lightlink.com>,
Tony Orlow <tony(a)lightlink.com> wrote:

> MoeBlee wrote:
> > Tony Orlow wrote:
> >>> It isn't. What does "successor to N" even mean?
> >> Ask von Neumann. It is the set of all naturals,
> >
> > What are you talking about? The successor of w is not w. The successor
> > of w is wu{w}.
>
> The size of the set x is the value
What "value"


> and it contains every natural less
> than x.
Cardinalities, as cardianlities, do not contain anything.

> Is this not the generalizable scheme?

No!
>
> >
> >> and any ordinal being
> >> the set of all preceding naturals,
> >
> > What are you talking about? An ordinal is the set of all preceding
> > ordinals, not necessarily the set of all preceding naturals.
>
> Aren't finite ordinals and cardinals and naturals the same thing? Perty
> much.
>

> Really no
> need to pretend there's any smallest infinity, is there?

And where does TO propose to find one smaller than Aleph_0.
From: Virgil on
In article <45119eae(a)news2.lightlink.com>,
Tony Orlow <tony(a)lightlink.com> wrote:

> MoeBlee wrote:
> > Tony Orlow wrote:
> >> Omega is successor to N
> >
> > No, it is not not. Why do you keep repeating your error?
>
> In the limit ordinal sense, it's what comes after the set of finite
> naturals.

The "limit ordinal sense" says that a limit ordinal is not "the
successor" of any ordinals. They are, in fact, the only ordinals, other
than {}, which are not successors of other ordinals.

So That TO has it backwards.

AS usual.
From: Han de Bruijn on
Tony Orlow wrote:

> Virgil wrote:
>
>> The axioms system of Euclidean geometry is inconsistent with that of
>> various non-Euclidean geometries. and there are a lot of other places
>> where one system contradicts another.
>
> What happened is that the parallel postulate was downgraded to a
> postulate, rather than a law. Whether two parallel lines meets at 0, 1
> or two places determines the type of space we are discussing. To say
> that the parallel postulate applied and also DIDN'T apply to a GIVEN
> space would contradictory.

I would like to add that these non-Euclidian geometries have a _model_
inside Euclidian geometry. (If I remember well, the hyperbolic case is
covered by Euclidian geometry inside a circle).

It's a bit off-topic but I have the following question. The mathematical
discipline called 'analytical geometry' - as teached in our schools - is
the algebraic equivalent of Euclidian geometry. OK ? Parallellism of two
straight lines can be detected algebraically. And moreover, two straight
lines CAN be parallel. How then can 'analytical geometry' translate into
an algebraic equivalent for non-Eulcidian geometries?

Han de Bruijn

From: Han de Bruijn on
imaginatorium(a)despammed.com wrote:

>>>>Han de Bruijn wrote:
>>>
>>>>>Precisely! Mathematicians get confused by the idea of a "bijection",
>>>>>which is an Equivalence Relation, which in turn is a "generalization"
>>>>>of "common equality" (yes: the one in a = b). But the funny thing is
>>>>>that EQUALITY HAS NEVER BEEN DEFINED.
>
> Idiot. Do you know the definition of an equivalence relation? Do you
> claim that bijection is not an equivalence relation?

I'm not an idiot. I *know* the definition of an equivalence relation.
And I know that a bijection is an equivalence relation. Mind the ""s.

> (I think that was a different crank - here's Tony...)

And wash your mouth.

Han de Bruijn