I'm not an astronomer, wish I were, but I'm not. However, here's what I think:
In Einstein's day, everyone thought that the Milky Way "was" the Universe. They didn't understand that there are galaxies beyond ours. That's why the astronomers believed that the Universe was static and unchanging. General Relativity was telling Einstein that the Universe could not be static, that it had to be either expanding or contracting but not standing still. Einstein blinked! And he believed his astronomer friends instead of his equations, and got it wrong. Hubble came along and basically proved that the Universe was expanding.
"(Simply because there is a red shift does not mean it is caused by a Doppler effect from everything moving away from each other. We have many photographs of galaxies colliding with each other to show that is not the case.")
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I doubt that is the case and here's why: galaxies bound to each other by gravity is a rather complex thing, and if they are red shifted or blue shifted depends on your own frame of reference. Galaxies can be in the process of colliding with one another and still (as a complete system) be red shifted and moving away from us. Your statement doesn't prove a thing as far as I can tell.
("The red shift is simply caused by the wave lengths increasing as they travel father from the source, the way ripples do in a pond.")
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No, not even close. Water waves are mechanical by nature and require a medium to travel in, namely water. Light waves have no mass and do not require a medium to travel in, thus they cannot behave in a 'mechanical' fashion. As I understand it, light photon wavelength stretches because it follows a geodesics path through space-time continuum. It is the 'space-time that stretches due to cosmic inflation and expansion, not the photon wave itself; though, it does decrease in frequency, energy, and it wavelength does increase as a result.
("Therefore, if there was no "big bang", then there was no beginning.")
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No big bang, no beginning? However, you have to account for the cosmic microwave background radiation. It is a residual effect from the big bang itself. The big bang was a true black-body radiation effect. And the black-body radiation effect "is" the cosmic microwave background radiation. In short, the CMBR, the expanding Universe and red shifted galaxies, the 75% Hydrogen to 25% Helium ratio for the cosmos and the smooth uniformity of the Universe on the largest of scales, among many other pieces of evidence all suggest that there had to be a big bang origin of the Universe as we know it.
See: Cosmic microwave background radiation - Princeton University
http://www.princeton.edu/~achaney/tmve/wiki100k/docs/Cosmic_microwave_background_radiation.html
Best regards