The physics simply doesn't work that way. Aliasing irreversibly transforms frequencies past the Nyquist limit to appear as frequencies below the limit, although those "aliases" aren't really there. No amount of processing a aliased signal can recover the original signal in the general case. The fancy mathematical explanations are rather long to get into unless you've had a class in sampling theory and digital signal processing. If you had though, you wouldn't be asking the question. Unfortunately then the best answer is simply "That't not how the physics works. Sorry, but you're going to have to trust me on this.".
To try to give some rough feel that the above might be true, consider the case of a picture of a brick wall. Without a AA filter, there will be moire patterns (which are actually the aliases) making the brick lines look wavy. You have never seen the real building, only the picture with the wavy lines.
How do you know the real bricks weren't laid down in a wavy pattern? You assume they weren't from your general knowledge of bricks and human experience of seeing brick walls. However, could someone just to make a point deliberately make brick wall so that it looked in real life (when viewed with your own eyes) like the picture? Yes they could. Therefore, is it possible to mathematically distinguish a aliased picture of a normal brick wall and a faithful picture of a deliberately wavy brick wall? No, it is not. In fact you can't really tell the difference either, except that your intution about what a picture probably represents may give you the impression that you can. Again, strictly speaking you can't tell whether the wavies are moire pattern artifacts or are real.
Software can't magically remove the wavies because it doesn't know what is real and what is not. Mathematically it can be shown that it can't know, at least by only looking at the wavy image.
A brick wall may be a obvious case where you can know that the aliased picture is wrong, but there are many more subtle cases where you really don't know, and may not even be aware that aliasing is going on.
Added in response to comments:
The difference between aliasing a audio signal and a image is only that the former is 1D and the latter 2D. The theory and any math to realize effects is still the same, just that it is applied in 2D when dealing with images. If the samples are on a regular rectangular grid, like they are in a digital camera, then some other interesting issues come up. For example, the sample frequency is sqrt(2) lower (about 1.4x lower) along the diagonal directions as apposed to the axis-aligned directions. However, sampling theory, Nyquist rate, and what aliases really are is not different in a 2D signal than in a 1D signal. The main difference seems to be that this can be harder for those not used to thinking in frequency space to wrap their mind around and project what it all means in terms of what you see in a picture.
Again, no you can't "demosaic" a signal after the fact, at least not in the general case where you don't know what the original is supposed to be. Moire patterns caused by sampling a continuous image are aliases. The same math applies to them just as it applies to high frequencies aliasing into a audio stream and sounding like background whistles. It's the same stuff, with the same theory to explain it, and the same solution to deal with it.
That solution is to eliminate the frequencies above the Nyquist limit before sampling. In audio that can be done with a simple low pass filter you could possibly make from a resistor and capacitor. In image sampling, you still need a low pass filter, in this case it's taking some of the light that would hit only a single pixel and spreading it out to neighboring pixels. Visually, this looks like a slight blurring of the image before it is sampled. High frequency content looks like fine detail or sharp edges in a picture. Conversely, sharp edges and fine detail contain high frequencies. It is exactly these high frequencies that get converted to aliases in the sampled image. Some aliases are what we call moire patterns when the original had some regular content. Some aliases give the "stair step" effect to lines or edges, especially when they are nearly vertical or horizontal. There are other visual effects caused by aliases.
Just because the independent axis in audio signals is time and the independent axes (two of them since the signal is 2D) of a image are distance doesn't invalidate the math or somehow make it different between audio signals and images. Probably because the theory and applications of aliasing and anti-aliasing were developed on 1D signals that were time-based voltages, the term "time domain" is used to contrast to "frequency domain". In a image, the non-frequency space representation is technically the "distance domain", but for simplicity in signal processing it is often referred to as the "time domain" nonetheless. Don't let that distract you from what aliasing really is. And no, it's not evidence at all that the theory doesn't apply to images, only that a misleading choice of words is sometimes used to describe things due to historical reasons. In fact, the shortcut "time domain" being applied to the not-frequency domain of images is actually because the theory is the same between images and true time-based signals. Aliasing is aliasing regardless of what the independent axis (or axes) happen to be.
Unless you are willing to delve into this at the level of a couple college courses on sampling theory and signal processing, in the end you're just going to have to trust those that have. Some of this stuff is unintuitive without a significant theoretical background.
