If you studied economics, you'd very quickly realize that many areas that people compete for and responsible for inequality are zero-sum (at least in the short-run and medium-run). It is more tricky for wealth, since once person owning something more esoteric, like a portfolio does not necessarily preclude anyone else from doing so.
The most notable example, of course, is real estate, especially in the areas where zoning laws prohibit from building upwards. You might also know it as a NIMBY suburban sprawl, which is one of the biggest factors in creating inequality and misery in the US.
Another example that I haven't explored fully is political influence. The attention and decision-making of policymakers is definitely zero-sum by virtue of there being a fixed amount of them.
Why would we move from a set of natural laws that govern allocation to a central authority based one? Ie, why should government take from the software developer some of his resource so that the laborer has more than what the market would allocate?
Do you seriously think this is an achievable goal ?
There will be always people that can't or won't work hard for a living. It might be due to their upbringing, education, DNA, personal choices, health conditions, personality. It doesn't mean that these people should be left to their own devices, for the "selfish" reasons mentioned by the OP, if not for more "noble" ones.
Resource redistribution does not aim to "solve" income inequality, but to lessen its side effects. It might be not the only way, it is definitely not perfect and you might say it is not fair, but at least it is a realistic way to tackle a difficult problem.
While life isn't exactly a zero-sum game, for certain moments it works that way. In any case, for now (nearly) all of us are Earth-bounded, so the limits of what our planet and our Sun can provide to us (over time) is what truly matters. There's only so much of Earth to go around, and only so much radiation to be received from the Sun. (And I am of course overlooking the occasional changes from meteors and comets and other whatnot from the heavens.)
This thing about income (and wealth) inequality is about more than mere discontents. There are plenty who fear that our global economic system (leaning heavily on capitalism with sprinkles of socialism) will soon arrive at a point of being unsustainable - a point that arrives prior to actual resource depletion.
The heart of it is in the ways that people can be so entitled to things, such as water, food, clothing, shelter, education, offspring, healthcare, etc. The worries are about the consequences of greater portions of the population being at or below subsistence levels. This is especially alarming when the statistics show it growing more significant within "First-World" nations.
The three big factors seem to be (to me, anyway): (1) Strong property rights which enables more rentiers and natural monopolies; (2) Rentiers/Monopolizers out-competing most or all others for political and economic influences and favors; (3) human labor becoming less needed over time due to the growing rise of adaptable and reliable automation.
The automation of item 3 would make a "Galt's Gulch" more tenable, but it would probably be realized as an artificial island nation. Well, one can dream of one, anyway (see http://theweek.com/articles/482427/libertarian-island-billio...).
One can easily google on "wealth inequality" and get some sober/reasonable/academic reads on this. A good primer is at https://www.washingtonpost.com/news/wonk/wp/2015/05/21/the-t.... Also, look up NBER Working Paper No. 20625.
Yes.
> I always believed we weren't playing a zero sum game.
That's true, so a redistribution can result in a net gain (or loss) of aggregate experienced utility. But you can't change the outcomes resulting from the current system of allocating resources without changing the allocation.
You're asking the wrong question. Why is income inequality a problem?