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The sources is this notes http://www.maths.manchester.ac.uk/~jeff/lecture-notes/MATH33001.pdf

The thing is I'm confused about the proof of MON.

We know that if $\Gamma | \theta$, then $\Gamma \cup \Delta | \theta$

So to prove it you need to say let $\Gamma \vdash \theta$

So exists a proof $\Gamma_1 | \theta_1, \Gamma_2 | \theta_2 ,...,\Gamma_k | \theta_k$ where $\Gamma_i \subset \Gamma$ and $\theta_k=\theta$

Hence

$\Gamma_1 | \theta_1, \Gamma_2 | \theta_2 ,...,\Gamma_k | \theta_k, \Gamma_k \cup \Delta_1 | \theta$

But, does $\Delta_1$ have to be finite. Also, what the point of this anyway? Having a discussion that says the point is extending it to infinity. However, I think just think you can give a proof in the form of finite subset.

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    @SamuelReid But, the condition to be a proof is that it follows from finite subset. However, for MON you can add infinite stuff. I'm just worried does this proof MON. I suppose it's really pedantic through. It's hard to see what the lemma is doing. Like don't really understand it so more wanted clarification on what the hell is happening.2012-01-13

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