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- Thread starter kurushio95
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Hurkyl

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Why do you think there is a closest?What is the closest positive number to zero that you can have?

That cannot a decimal number. The places in a decimal number are indexed by the integers; therefore, each place must be at a finite distance from the decimal point.it would be 0.000...01, right? infinite zeros plus one.

Also, your notation is poor. It suggests that you have a consecutive sequence of zeroes, which must necessarily be finite. If it were infinite, then it must consist of two or more disconnected sequences. I use a pipe (|) to denote these breaks -- it is, in fact, possible to have an infinite connected sequence of zeroes:

000...

and it is possible to have an infinite connected sequence of zeroes followed by a 1:000... | 1

however, there is no number to the immediate left of the 1. If you want every place (aside from the left endpoint) to have a number to its immediate left, you can build a sequence like000... | ...0001

but these two components do not connect in the middle. Of course, this and the previous sequence cannot denote a decimal number.
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Ok, so you think infinite isn't real, and yet you think that you can have 0.[infinite 0s]1?? How can you have an infinite number of 0s if infinity isn't real? A self-contradictory stance that proves itself is interesting, but of little philosophical or scientific value, IMO.

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Hurkyl

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What does any of this have to do with reality?I think infinite is not real.

This is not what you meant. The zeroes are not infinite; it's the quantity of zeroes that is infinite. You meant to say something like "infinitely many zeroes".infinite zeros plus one.

In fact, zero has the

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Hurkyl

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Disconnected is an awkward word; to use here. I think I implied what I meant, but shame on me for not explicitly defining it!it must consist of two or more disconnected sequences.

What I mean is that if you have a zero in one sequence, you cannot reach the other sequence by iteratively performing the operation "move from this zero to an adjacent zero". Maybe I should have called this "stepwise disconnected".

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P.S. I'm only in high school math, so I wasn't expecting to be 100% correct in my notation. thanks for pointing it out though and showing me the correct way.

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russ_watters

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Hurkyl

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The problem may be that you don't realize you have made an assumption. When you askI guess the best way i could explain it is to ask what the smallest positive number is that is not zero. I'm having a hard time phrasing my thoughts into words, in case you couldn't tell.

what is the smallest positive number that is not zero

you have The truth is that such a thing does

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After learning that there is also different extensions to the real numbers, I've started to think if such extension could exist also, where the smallest positive number would exist. It couldn't be a field extension at least, but perhaps some other extension, that would still contain the field of real numbers as a subset.

kurushio95, want to proceed with you philosophy? Then get rigor!

(... although it would probably be better to first understand real numbers well)

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