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That's partly why I'm so interested in these types of sequences. If they are an inconsistency, I'm not sure how much of the math we rely on would have to be sacrificed.
I think you answered that already. Like for my example, you concluded that in order to keep operating normally, we would have to limit ourselves (depending on the application) to a certain domain. We always have to do that in physics, for instance. Even with something as simple as the equation of the deceleration of a car stopping, you have to limit the domain to the moment it stops, as otherwise the equation predicts it would start accelerating in reverse.
Ah, but with this function, we know the inconsistency only involves a single point and nothing useful has been built downstream of assuming it is defined there.
We don't know exactly where the contradiction is (assuming there is one) with these kinds of series and there might be a lot of widely used results downstream of it.
But that is correct. The fact there is a contradiction do not makes the derived math incorrect. It's just part of the right conclusion. Maybe the paradox arises here from equating "contradiction" to "incorrect", when we have seen that some contradictions are structural to some math forms. Actually, an incorrect result will be obtained if the contradiction is pruned.
What would happen is that some proven theorems would be reduced to unproven conjectures (of which math has many) until they can be proven or disproven without relying on the pruned inconsistency.
It's a possibility. But if pruning an inconsistency introduces an error, maybe such an analysis is flawed itself from the start. Take for example again the simple form , now let's say we prune the inconsistency only, by limiting the domain of only to the positive side. Then would go from equating for any value of to equating for negative values of , which is incorrect. The inconsistency introduced by is needed for to yield the correct result. A different thing is to, instead of removing the inconsistency, limiting the domain for the whole expression.
I’m aware of some people who think infinite sets are an example of something that needs to be pruned, which would reduce the domain of sets to only having a defined number of elements.
I'm not following. My engineering mind interprets: shortening an infinite series to the first elements for which a value converges well enough for practical purposes. But I'm sure that's not what you mean
I'm sure of that, the thing is that every one of those combinations should reveal the construction of a different constant. That there are infinite ways to do so is the first hint that infinite different constants are obtainable, hence the differing results.