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Topic: Summing Integers to infinity
Replies: 40   Last Post: Jan 25, 2014 9:02 PM

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 fom Posts: 1,968 Registered: 12/4/12
Re: Summing Integers to infinity
Posted: Jan 22, 2014 9:39 PM

On 1/22/2014 3:34 PM, WM wrote:
> Am Mittwoch, 22. Januar 2014 19:28:05 UTC+1 schrieb fom:
>
>

>> Do you have this in any other format?
>
> Unfortunately not. Bur you will find it certainly at many places in the internet.
>
> Regards, WM
>

claim.

Last weekend I had reviewed Bolzano's proof
of the intermediate value theorem. It is
based on fixed point logic. And, I have now
identified many of my non-standard views as
arising from that understanding of mathematics.

[remember, I am ambivalent concerning the
axiom of infinity]

In fact, my set-theoretic axioms reflect the
notion of a "located subset" as described in
"Constructive Analysis" in Bishop and Bridges.

Since they are doing analysis, they take a
metric for granted. My logic representations
cannot do that. So, they must represent
"subset" and "point" separately.

The core statements are

AxAy(x psubset y <-> (Az(y psubset z -> x psubset z) /\ Ez(x psubset z
/\ ~(y psubset z))))

AxAy(x in y <-> (Az(y psubset z -> x in z) /\ Ez(x in z /\ ~(y psubset z))))

When a metric is available, one can imagine
"topological separation" in terms of the
standard convex segment

p = x(1-a) + y(a), 0 <= a <= 1

Without a metric -- and in the context of
Bolzano's proof -- one must imagine a
point on that line segment different from
x or y. The axioms above permit me to
express this in the sense that every "object"
of the domain is also a "separating surface".

To imagine this, consider a 2-sphere with a
right conic. Let the conic be situated so
that its vertex intersects the sphere (a "point
object") and so that it also intersects the
sphere as a circle (a "separating surface
object"). If one were to add a tangent plane
to the antipodal point of the vertex intersection,
the antipodal point would be the center of a
circle forming a base for a right conic having
the diameter of the sphere for its height.

With this construction, the axioms above provide
the ground for a "fixed point" in the sense of
Bolzano. Separating surfaces partition the
2-sphere. The corresponding points provide an
"extensional interpretation" of the system.

There is still a problem of definiteness. Families
of conics are determined by great circles. However,
if situated on the Riemann sphere, most of the
antipodal points can be associated with definite
circles. Except for the equator and the great
circles passing through the point at infinity,
there will be a circle in the opposed hemisphere
that will either intersect the point at infinity
or intersect the origin.

How one interprets this to justify set theory
is irrelevant to the fact that it begins
with analysis. So, the potential infinity
of limits and continuity apply. I am interested
in your statement concerning this equation
because it might assist me.

I do not expect your account to differ greatly
from your general agenda. But, constructive
views are hard to come by. You claim Euler
made a mistake. Can you explain what you mean
in a couple of sentences?

Date Subject Author
1/22/14 J.B. Wood
1/22/14 FredJeffries@gmail.com
1/22/14 Richard Tobin
1/22/14 wolfgang.mueckenheim@hs-augsburg.de
1/22/14 fom
1/22/14 wolfgang.mueckenheim@hs-augsburg.de
1/22/14 Port563
1/22/14 fom
1/25/14 wolfgang.mueckenheim@hs-augsburg.de
1/25/14 Port563
1/25/14 Roland Franzius
1/25/14 Port563
1/25/14 Tanu R.
1/25/14 wolfgang.mueckenheim@hs-augsburg.de
1/25/14 Port563
1/25/14 wolfgang.mueckenheim@hs-augsburg.de
1/25/14 Port563
1/25/14 Virgil
1/25/14 Virgil
1/25/14 Port563
1/25/14 Virgil
1/25/14 Tanu R.
1/25/14 Port563
1/25/14 Tanu R.
1/25/14 wolfgang.mueckenheim@hs-augsburg.de
1/25/14 Virgil
1/25/14 Virgil
1/22/14 Virgil
1/23/14 David C. Ullrich
1/22/14 magidin@math.berkeley.edu
1/22/14 J.B. Wood
1/22/14 FredJeffries@gmail.com
1/23/14 Tanu R.
1/22/14 Wizard-Of-Oz
1/22/14 Wizard-Of-Oz
1/22/14 Virgil
1/23/14 David C. Ullrich
1/23/14 gnasher729
1/24/14 ross.finlayson@gmail.com
1/23/14 Tanu R.
1/23/14 Tanu R.