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Originally Posted by Clodfobble
It is by definition possible. Infinity is not a number, infinity plus one (or multiplied by two, or whatever) always equals infinity.
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That is not exactly true.
∞ + 1 > ∞
Other strange things occur. For example a function divided by t does not become infinity as t approaches zero. It becomes an impulse of one. I don't remember exact details - this was many decades ago. But ∞ + 1 also is not same as ∞. ∞ + 1 is approximately ∞ which is good enough for calculations involving reality. But that is an approximation not valid for rigorous proofs or this algebraic solutions.
Of course, we can change an assumption. Same is accomplished in Euclidean geometry where two parallel lines never meet. We simply change some underlying principles (to create a different type of geometry) so that two parallel lines do meet at ∞. Suddenly the rules of that geometry change because we are using a completely new geometry (forgot the name of that geometry).
But we are using the domain of standard algebra.
So how does that S = -1 come about? Something in the equation before 2S + 1 = S is wrong because 2∞ + 1 = ∞ is wrong. There is apparently some restriction in algebraic rules used that I just don’t see. I just don't recognize the mistake - an overlooked restriction.