I quit after the first paragraph.

If you assume "proof by contradiction" to be a valid form, then you also assume that mathematics is consistent. Nothing to see here... Move along...

Edit: In case some people missed it... In our logical framework, an inconsistent theory can prove both Φ and ¬Φ. Notice that you are using our logical framework; you are assuming it to be consistent. Yes. Assuming the framework to be consistent, you can prove that the framework is consistent. Shocking.