I appreciate the comments by JS Princeton. He stated that I made a common math error, and I could not find such an error.

I would like to apologize to JS Princeton, because when I created my web site and made it available to bad astronomy, there was typographical error in some of the equations. I immediately corrected the errors, but before I could correct it, JS Princeton saw the web site and made his comments. The typographical errors did not affect my conclusions, and the corrections were made very rapidly, so I would be astonished if anyone else reviewed the web site with the typos.

I agree with GrapesOfWrath, Bad PhD, "I don't see where he does that directly, but as you point out below, it is a consequence at non-relativistic velocities." I cannot find where I made any math errors.

Back to JS Princeton Bad Intern. He establishes the point I am trying to make, rather than contradicting it.

The point in my mathematical exercise was NOT to find how to add velocities under relativity, but rather to show a mathematical contradiction.

When you want to prove that something is false in mathematics, you start first by assuming it is true. Then you work through the math and arrive at equations or results that cannot be true. When you do that, you have established that your original assumptions were wrong. This is a very common approach in mathematical proofs.

I used this approach, and I started with the assumption that all frames of reference are equally valid. I proceeded with the mathematics and arrived with an equation for adding velocities. I took the position that my equation for adding velocities was not correct, and therefore, the original assumption that all frames of reference are equally valid is not correct.

JS Princeton Bad Intern also took the position that my equation that for adding velocities is incorrect, which is exactly the point that I am trying to make.

I started by working with the effect that time has on multiple observers, and the assumption that all frames of reference are equally valid. The math developed into an equation for adding velocities, which both JS Princeton Bad Intern and I state is an incorrect conclusion.

JS Princeton Bad Intern gave an alternate equation for adding velocities, and I would like to compare my equation with his.

Both give relatively accurate results at velocities close to the speed of light. Both are correct when one of the velocities is zero. Both show the effects of relativity being realized only at higher velocities. My equation shows the effects of relativity being realized at much lower combined velocities than it should be realized, from an intuitive standpoint. From an intuitive standpoint, I would expect JS Princeton to be a more accurate depiction.

GrapesOfWrath Bad PhD also agrees with us, which helps prove my point, that my equation for adding velocities is incorrect.

GrapesOfWrath Bad PhD did a good job of articulating my assumption: "I think the basic error is his assumption that the dilations compound by multiplication. He assumes that if A sees B as twice as slow, and B sees C as twice as slow, then A will see C as four times as slow. That's obviously not true--just let C be the first observer A. A would see A as four times as slow?"

My assumption is that time dilations compound by multiplication. GrapesofWrath stated the assumption obviously is not true, but I do not follow the reasoning of GrapesofWrath.