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**Homo bibiens** I am very sorry you disagree with yourself:

Self-contradiction seems to be a very common property of many of the arguments I have seen in defense of one side of this argument at other boards, although this particular one, I have not seen before.

Different from what you refer to as 0.000....1? If that is what you mean, then I decline your offer to "fix" my number system, I prefer the one I have right now.

If 0.000...1 is different than zero, does it obey all the usual rules of arithmetic that the real numbers obey? (The reason I ask is related to the above discussion about self-contradiction; if someone invents a number system with contradictory properties, then I cannot agree that they are as right as everyone else.)

I am glad you say elsewhere "Clarify the definitions", because I think definitions are extremely important in this sort of thing. Can you clarify your definition of 0.000...1? Specifically, what is the "sequence" to which you refer? Is it 0, 0.0, 0.00, 0.000, 0.0000, etc., or 1, 0.1, 0.01, 0.001, 0.0001, etc., or something different from either of these?

But either way, my usual definition of the real number zero is a particular equivalence class of Cauchy sequences. There are other ways of doing it, but since all totally ordered complete fields are isomorphic, they get us to the same place.

Same as above. This is not my definition of the real number one. I am not sure what your definition is.

Real numbers are usually defined as limits. (As per above, there is at least one other way to do it, but it gets you the same end result.) Are you arguing that the real number one is not equal to the integer one? I have no problem defining equality between those two, and as per one of your two opinions on the matter, I am as right as everyone else.

Thank you once again for your kind offer, but the one hint about infinitesmal numbers causes me to want to stick with the number system that I have. If I have misunderstood Bruinekool-numbers, a very rigourous definition would be most helpful in setting things straight.

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