My question concerns restrictions on the exercise of normal algebraic rules. The most well known restriction is the prohibition on division by zero (PDZ). This is justified by various 'proofs' of incorrect results, for example:
$$a = b$$
$$ab = b^2$$
$$ab - a^2 = b^2 - a^2$$
$$a(b - a) = (b - a)(b + a)$$
$$a = b + a$$
$$a = 2a*$$
$$1 = 2$$
*From line 1.
In this 'proof' the zero division is not immediately obvious. Another possible restriction would be a prohibition on multiplication by infinity (PMI), which is at least conceptually similar to PDZ. But is there any need for it? Consider the following series:
$$x = 0.999...$$
$$10x = 9.999...$$
$$10x = 9 + x$$
$$9x = 9$$
$$x = 1$$
And:
$$x = 1 - 1 + 1...$$
$$x = 1 - (1 - 1 + 1...$$
$$x = 1 - x$$
$$2x = 1$$
$$x = 1/2$$
And finally:
$$x = 1 + 2 + 4...$$
$$x = 1 + 2(1 + 2 + 4...$$
$$x = 1 + 2x$$
$$-x = 1$$
$$x = -1$$
Debate about this issue has previously got stuck on whether the second series makes sense, but since the third series is obvious nonsense (the sum is increasing - it cannot be negative) the debate should be about how to disallow series such as the third using an uncontroversial restriction. The third series introduces a multiplication by infinity in line 2 (infinity can be defined as the sum of an additive sequence with equal or increasing terms) so disallowing such a step invalidates that series without affecting the other two. Is this a good way to treat the issue?
Edit: The above definition of infinity should include series with decreasing sequences which can be proven to be divergent because the last step in such proofs should explicitly show that they comply with it. Also note, nothing in this post has implications for conditionally convergent series.
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