Why is the ideal low pass filter not achievable?
General Background: Representing a signal numerically in a computer requires the signal to be discrete. That means it's periodic in the other domain. When you use something like a DFT you need the signal to be discrete in both domains: That means the signals are ALSO periodic in both domains.

If I apply an ideal low pass filter, I should get an infinite sequence of data points because the sinc function ...

If you want to do this numerically in a computer, it's NOT a sinc function. The lowpass filter MUST be periodic in both domains and so what you get in the time domain is a periodic repetition of a sinc function with a time domain aliasing.
It's still infinite it time. In fact it MUST be infinite in time since it's periodic. As long as all signals involved are actually periodic with a shared Period, than things work &quot;as expected&quot; .
At this point the term &quot;low pass&quot; filter becomes complicated. Since the spectrum is periodic, it will pass arbitrarily high frequencies. So it needs to be interpreted: it removes higher frequencies in the base band or something like this.

$y\left(m\right)\ =\ \displaystyle\sum_{i=1}^{N}\cos\left(m\alpha_{i}\right)$

Your missing phase or sine term here but you are re-inventing the DFT or cosine transform. You run into exactly the same problem. How to choose N ?

Also if I knew that the signal had exactly N points

That's a contradiction. The signal that you described in your original equation is infinitely long. As soon as you make it finite in time, you are windowing it and that changes the spectrum and the math behind considerably.

then the signal could be represented at most by N real frequencies

Correct. That's for example the Discrete Cosine Transform, which is a close cousin of the DFT. But it makes no material difference here.

Then why is the ideal low pass filter not realizable?

It is realizable as long as everything periodic &quot;the right way&quot;. The periodicity helps dealing with the infinite length in time: you need only to compute what's happening in one period and you have all information that can be had since all other periods are the same.
However most real world signals are NOT periodic &quot;the right way&quot;, so ideal low pass filtering is rarely feasible.