Here I want to address of the question if it is possible to make a sum over an uncontable set and discuss integration rules involving uncountably infinite constants.
I will provide introduction in very condensed form to get quicker to the essense.
Conservative part
First of all, let us...
If we have a straightangle, compass and an angle of Dottie number available, can we divide a disk into arbitrary number of parts of equal area with chords?
What if we have only interval of Dottie number and no angle?
Along with the problem of squaring a circle and trisection of an angle, there is one more great problem: quarisection of a disc.
You have a disk and have to dissect it into four parts of equal area with three chords coming from the same point on the disc's boundary (one of these chords is a...
1. It is described in the linked post what can be done with this extension.
2. It is a divergent integral, and it is well described what is meant. You can work with infinitely large values quite well.
3. Logarithmic singularity is not a pole.
4. Of divergent integral ##-\int_0^1 \frac1x dx##...
1. Investigation of extending reals
2. ##-\int_0^1 \frac1x dx##
3. Hmmm. This is not a pole.
4. ##-\gamma ## is its finite part. ##\lambda## is negatively infinite.
5. To be honest, ##\gamma ## is a constant that does not have a lot of nice properties (except being the real(finite)part of...
Also, from that same post we can see that in dual numbers ##\varepsilon^\varepsilon=1+\varepsilon(1+\lambda)##, and many other nice properties.
Including, for instance, representation of some divergent integrals:
##\int_0^\infty \frac{\ln t}{t} \, dt=\gamma\lambda+\gamma^2/2##
As you can see from the original post, while ##\gamma## is neither a period (member of ring of periods), nor an EL-number, ##\lambda## is both. Thus it has more nice properties.
In that quotation I wrote that we can represent ##\lambda=-\sum_{k=1}^\infty \frac1k##, yes. The regularized value (the finite part) of the right hand is ##-\gamma##.
I made a link to the post that has no comments. It describes another way to see how the regularized value of the integral ##\int_0^1 \frac1x dx## is ##\gamma##.
Obviously I have never defined ##\lambda=-\gamma##. If you read the post more carefully, you will see that ##-\gamma## is the finite part (which is the same as regularized value) of ##\lambda##.
Euler-Mascero
The Cauchy principal value of Zeta function at ##x=1## is ##\gamma##.
Also, take a look here: https://math.stackexchange.com/a/4083388/2513
That the Harmonic series has the regularized value ##\gamma## is a known mathematical fact. Take the Cauchy principal value for instance:
https://www.wolframalpha.com/input?i=Limit[(Zeta[1-h]+Zeta[1+h])/2,+h->0]
The Ramanujan summation gives the same result...