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 >
The other links are certainly helpful.
What I am curious about is your particular 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 nonstandard views as arising from that understanding of mathematics.
[remember, I am ambivalent concerning the axiom of infinity]
In fact, my settheoretic 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(1a) + 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 2sphere 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 2sphere. 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?

