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The Forum > General Discussion > Logic proves: All opinions -for and against are equally valid

Logic proves: All opinions -for and against are equally valid

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Dear Thinkabit,

In my first year of university, many years ago, we studied Dedekind cuts and yes, I missed the "no greatest element" condition. We then had it as exercise to prove that Dedekind cuts carry all the field and ordering properties of rational numbers, which allowed us to seamlessly map the rational numbers onto the real numbers.

According to this mapping, "1/1" is mapped to the Dedekind cut containing all rational numbers that are less than 1. Next we can also map all countable-infinite decimal fractions onto Dedekind cuts and if we do so for "0.99999..." then it is easy to show that no elements occur in "1/1" that do not occur in "0.99999..." and vice-versa, thus the two sets are identical, one and the same.

This was in algebra class, independent of the the study of Cauchy sequences in "infinitesimal" class.
Posted by Yuyutsu, Sunday, 6 January 2019 11:12:31 PM
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wow, thnkabit, I must congratulate you – your obfuscation skills are truly astounding!

Nevertheless, your argument is wrong. You seem to be relying on the false assumption that a number constructed one way can't be the same as a number constructed another way. But in reality there is usually, if not always, infinite scope for needlessly complicating mathematics by including unnecessary trivialities!

'Tis a pity that while you were checking Wikipedia you forgot to check the Integer page http://en.wikipedia.org/wiki/Integer which concurs with my earlier claims.

BTW I'm surprised nobody pointed out the mistake in my last line yesterday. I should have said "parseFloat" as parseInt would be unsuitable for comparison unless I knew the number I wanted translated into integer format was already mathematically an integer.
Posted by Aidan, Monday, 7 January 2019 1:40:22 AM
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All logic is not equal. For instance let's look at two logical sequences of numbers. An infinite set of numbers ordered numerically, (... -3,-2,-1,0,1,2,3 ...), then compare that to the same numbers ordered alphabetically. Not as easy to order in that fashion because it's not about counting and knowing which value is greater or lesser numerically, but is reordered based on a different form of logic based on language and which letter comes after an other in the alphabet.

These two forms of logic are not equal, and when applied outside of the scope that that logic is used for makes one set of logic illogical.

For instance in the numerically ordered set the numbers greater then other numbers are the larger one thus the last numbers in this sequence would be greatest numbers in the sequence, the lesser numbers both the earlier numbers, and zero is in the middle of this infinite set of numbers.

Alphabetically, 5 (five) comes before 3 (three) because 5 starts with an "F" while 3 starts with a "T." All negative numbers would be placed in the "N" section of the sequence between "million" and "one." (Assuming you represent million in that way instead of "one-million," or even "a million"). With this sequence and logic Zero is one of the last numbers in this infinite sequence followed only by "zillion" which isn't a qualifiable number just an expression of a very high number.

Take the logic of greater and lesser applied to first and last of a numerical sequence, and try to apply the same logic to numbers alphabetically and it is illogical and no longer valid math or valid logic. Three being later in the sequence is not greater then five.

Not all logic is equal and therefore interchangeable as it is with equations equaling each other are interchangeable in mathematical logic.
Posted by Not_Now.Soon, Monday, 7 January 2019 1:49:23 AM
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he paper proves

1 is a finite number it stops
A finite decimal is one that stops, like 0.157
A non-finite decimal like 0.999... does not stop
when a finite number 1 = a non-finite number 0.999.. then maths ends in contradiction

another way

1 is an integer a whole number
0.888... is a non-integer it is not a whole number
0.999... is a non-integer not a whole number
when a integer 1 =a non-integer 0.999... maths ends in contradiction
Posted by sara242, Monday, 7 January 2019 1:45:42 PM
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Dear Sara,

"1" is only a representation, not a number.

It can be designated to represent an integer or it can be designated to represent a real number.

"0.999..." is a common representation of a real number. Again, it is not itself a real number, only a representation thereof.

When "1" represents a real number, it represents the same real number that "0.999..." represents. These are just two different ways of writing the same.

Any number multiplied by "1" remains the same.
Similarly any number multiplied by "0.999..." remains the same.

Similarly, any number multiplied by "8/9" and then divided by "0.888..." remains the same: "8/9" and "0.888..." represent the same real number.
Posted by Yuyutsu, Monday, 7 January 2019 2:00:48 PM
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this is waaaaaay out of my comprehension (only went to year 8 primary school then off to work it was at 14 until 54 years later, last year) but in my life it was the decimal . that made all the difference.
Posted by individual, Monday, 7 January 2019 2:36:50 PM
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