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On 4/1/2024 6:11 PM, Keith Thompson wrote:Which is exactly the same number as 1.00olcott <polcott333@gmail.com> writes:IDK, probably not. I am saying that 0.999... exactly equals this number.
[...]Since PI is represented by a single geometric point on the number line[...]
then 0.999... would be correctly represented by the geometric point
immediately to the left of 1.0 on the number line or the RHS of this
interval [0,0, 1.0). If there is no Real number at that point then
there is no Real number that exactly represents 0.999...
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In the following I'm talking about real numbers, and only real
numbers -- not hyperreals, or surreals, or any other extension to the
real numbers.
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You assert that there is a geometric point immediately to the left of
1.0 on the number line. (I disagree, but let's go with it for now.)
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Am I correct in assuming that this means that that point corresponds to
a real number that is distinct from, and less than, 1.0?
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But aleph_0 - 1 = aleph_0, so they have the same number of points on them.More generally, does each real number correspond to a point on theThe line segment [0.0, 1.0] is exactly one geometric point longer than
number line, and does each point on the number line correspond to a real
number? (The real numbers can be formally defined without reference to
geometry, but let's go with your geometric model for now.)
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[0.0, 1.10), having all points in common besides the last point.
If so, let's call that real number (immediately to the left of 1.0) x.
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Consider ((x + 1.0)/2.0). Let's call that number y. (The intent is to
construct a real number that is exactly halfway between x and 1.0.)
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Is y a real number? (If not, the real numbers are, unexpectedly, not
closed under common arithmetic operations.)
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Is y less than, equal to, or greater than x?
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Is y less than, equal to, or greater than 1.0?
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Again, I am talking *only* about real numbers.
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Given your past history, I do not expect straight answers to these
questions, but I'm prepared to be pleasantly surprised.
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