Your algebraic conclusions are essentially off-target for the same reason that you have to be careful about how you apply the law of associativity to infinite sums. We all should have seen at one point or another the "proof" that 1=0 that goes:
1=1+(-1+1)+(-1+1)...=(1-1)+(1-1)+...=0
This is not a proof of anything precisely because the associative law doesn't necessarily apply to infinite sums. Infinities of any sort are strange in that manner, and so basically all of your algebra is.
Another good example is limits of sequences, which anyone that has ever taken a basic calculus course is familiar with. Suppose we have a sequence a_n=f(n)/g(n), then while lim as n->infinity might look like 0/0, but we can in fact take the derivative of both f(n) and g(n) and find something that isn't of the indeterminate form, and find the true limit.
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How do those of us living in a Republican form of government handle problems like this?
1=1+(-1+1)+(-1+1)...=(1-1)+(1-1)+...=0
This is not a proof of anything precisely because the associative law doesn't necessarily apply to infinite sums. Infinities of any sort are strange in that manner, and so basically all of your algebra is.
Another good example is limits of sequences, which anyone that has ever taken a basic calculus course is familiar with. Suppose we have a sequence a_n=f(n)/g(n), then while lim as n->infinity might look like 0/0, but we can in fact take the derivative of both f(n) and g(n) and find something that isn't of the indeterminate form, and find the true limit.