Re: Definition of real number ℝ --infinitesimal--

Op 01.apr.2024 om 20:54 schreef olcott:

On 4/1/2024 1:39 PM, Fred. Zwarts wrote:

Op 01.apr.2024 om 16:33 schreef olcott:

On 4/1/2024 2:31 AM, Fred. Zwarts wrote:

Op 31.mrt.2024 om 21:42 schreef olcott:

On 3/31/2024 2:26 PM, Fred. Zwarts wrote:

Op 31.mrt.2024 om 21:02 schreef olcott:

On 3/31/2024 1:52 PM, Fred. Zwarts wrote:

Op 30.mrt.2024 om 21:27 schreef olcott:

On 3/30/2024 3:18 PM, Fred. Zwarts wrote:

Op 30.mrt.2024 om 20:57 schreef olcott:

On 3/30/2024 2:45 PM, Fred. Zwarts wrote:

Op 30.mrt.2024 om 14:56 schreef olcott:

On 3/30/2024 7:10 AM, Fred. Zwarts wrote:

Op 30.mrt.2024 om 02:31 schreef olcott:

On 3/29/2024 8:21 PM, Keith Thompson wrote:

olcott <polcott2@gmail.com> writes:

On 3/29/2024 7:25 PM, Keith Thompson wrote:

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What he either doesn't understand, or pretends not to understand, is
that the notation "0.999..." does not refer either to any element of
that sequence or to the entire sequence.  It refers to the *limit* of
the sequence.  The limit of the sequence happens not to be an element of
the sequence, and it's exactly equal to 1.0.
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In other words when one gets to the end of a never ending sequence
(a contradiction) thenn (then and only then) they reach 1.0.

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No.
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You either don't understand, or are pretending not to understand, what
the limit of sequence is.  I'm not offering to explain it to you.
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I know (or at least knew) what limits are from my college calculus 40
years ago. If anyone or anything in any way says that 0.999... equals
1.0 then they <are> saying what happens at the end of a never ending
sequence and this is a contradiction.
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It is clear that olcott does not understand limits, because he is changing the meaning of the words and the symbols. Limits are not talking about what happens at the end of a sequence. It seems it has to be spelled out for him, otherwise he will not understand.
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0.999... Limits basically pretend that we reach the end of this infinite sequence even though that it impossible, and says after we reach this
impossible end the value would be 1.0.

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No, if olcott had paid attention to the text below, or the article I referenced:

https://en.wikipedia.org/wiki/Construction_of_the_real_numbers

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he would have noted that limits do not pretend to reach the end. They

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Other people were saying that math says 0.999... = 1.0

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Indeed and they were right. Olcott's problem seems to be that he thinks that he has to go to the end to prove it, but that is not needed. We only have to go as far as needed for any given ε. Going to the end is his problem, not that of math in the real number system.
0.999... = 1.0 means that with this sequence we can come as close to 1.0 as needed.

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That is not what the "=" sign means. It means exactly the same as.

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No, olcott is trying to change the meaning of the symbol '='. That *is* what the '=' means for real numbers, because 'exactly the same' is too vague. Is 1.0 exactly the same as 1/1? It contains different symbols, so why should they be exactly the same?

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It never means approximately the same value.
It always means exactly the same value.

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And what 'exactly the same value' means is explained below. It is a definition, not an opinion.
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No matter what you explain below nothing that anyone can possibly
say can possibly show that 1.000... = 1.0
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I use categorically exhaustive reasoning thus eliminating the
possibility of correct rebuttals.

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OK, then it is clear that olcott is not talking about real numbers, because for reals categorically exhaustive reasoning proved that 0.999... = 1 and olcott could not point to an error in the proof.
It would have been less confusiong when he had mentioned that explicitly.
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Typo corrected
No matter what you explain below nothing that anyone can possibly
say can possibly show that 0.999... = 1.0
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0.999...
Means an infinite never ending sequence that never reaches 1.0

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Which nobody denied.
Olcott again changes the question.
The question is not does this sequence end, or does it reach 1.0, but: which real is represented with this sequence?

 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).

In the real number system it is incorrect to talk about a number immediately next to another number. So, this is not about real numbers.

If there is no Real number at that point then there is no Real number that exactly represents 0.999...

Again olcott is changing the meaning of the words and symbols. 0.999... represents a sequence x1 = 0.9, x2 = 0.99, x3 = 0.999, etc. That sequence is not a point. This sequence represents a real number namely exactly 1.0. It has nothing to do with the interval [0, 1). So, bringing up this interval is irrelevant.
If 0.999... ≠ 1.0, then tell us the value of a rational ε > 0 for which no N can be found such that |xn - 1| < ε for all n > N.

 

The answer is: This sequence represents one real: 1.
Therefore we can say 0.999... = 1.0. It follows directly from the construction of reals.
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If biology "proved" that cats are a kind of dog then no matter
what this "proof" contains we know in advance that it must be
incorrect.

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Similarly, if olcott 'proved' that 0.999... ≠ 1 then, no matter what this "proof" contains, we know that it must be incorrect. Most probably he is changing the question, changing the meaning of the words or the symbols, or is talking about olcott numbers instead of reals.
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Therefore, in the construction of reals it is defined how to determine whether two reals are 'exactly' the same. If one real X can be constructed with a sequence of xn and the other real Y with a sequence yn, then we can use X = Y if for every rational ε > 0 we can find an N so that for all n > N |xn - yn| < ε.
The consequence of this is that for each real we can use an infinite number of Cauchy sequences. E.g. the following sequences
a: 1/1, 1/1, 1/1, 1/1, etc.
b: 9/10, 99/100, 999/1000, etc.
c: 10/9, 100/99, 1000/999, etc.
d: 1/2, 2/3, 3/4, 4/5, etc.
e: 1/2, 3/2, 3/4, 5/4, 5/6, 7/6, etc.
are all sequences that are different representations of the same real which in decimal notation can be written as 1. So, a=b=c=d=e=1.
Olcott may not like it, but that is how the '=' is defined for reals.
One may try to create another number system with another meaning for '=', but then we are not talking about reals any more.
If I do not like that 3+4=7, then I can try to create another system for which 3+4=6 holds, which I like more, but I am no longer speaking of real numbers (and probably nobody is interested in my number system).

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For real numbers, a has exactly the same value as b, c, d, e, f and 1. That is how it is defined. If olcott has another definition of 'exactly the same value', then he is changing the meaning of the words. The meaning of '=' is exactly defined for reals.

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