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Thread: Rational numbers == infinitely repeating sequences of digits

  1. Top | #261
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    Quote Originally Posted by untermensche View Post

    The symbols refer only to a specific value within the scheme.
    As per the definition of the decimal system, any wellformed numeral within it refers to a unique value within the scheme of numbers, presuming that they have been defined. Those symbols are however not part of the scheme of numbers, they live in an extra layer atop the numbers. You should know, you have us an algorithm for the numbers that doesn't make reference to strings.

    The value is created by the scheme.

    It does not exist out there somewhere before the scheme exists.
    This doesn't contradict anything I said.


    The scheme of positive integers doesn't contains symbols. It's a scheme of values.
    Of course it does. A value created is represented by a specific symbol or grouping of symbols. The symbols were already assigned by the time you showed up so you don't think they were assigned I suppose.
    You got this exactly backwards. I'm well aware the mapping from strings to values is arbitrary. You seem to have your reservations.

    You should tell untermensche he keeps irrelevantly bringing up symbols of the decimal notation scheme in the middle of discussing the abstract conceptual scheme of numeric values.
    Nope.

    The symbols refer to the abstract values that do not exist until a value system is created.
    True, but that reference relation is not part of the world of values. The value defined as the result of applying the Addition operation to the Unit value is literally by definition the same whether you use the English word "two", Russian "dva" or Arabic "ithnaan", the decimal "2", binary "10", or Roman "II". If it makes you happy, you could call it "Gandalf" or "Captain Kirk" and still be talking about the same abstractly defined value.

    The symbols refer to values.
    They do, but not vice versa. Their relationship is asymmetric.

    0.3 refers to a specific value in the scheme. 0.33 refers to a different value.
    They refer to the values resulting from dividing 3 by 10 and 33 by 100, respectively. In any Positional notation that doesn't have a multiple of ten as its base, they'd have to be represented with a non-terminating string. That's a property of notations, not of values. What was that about not making reference to entities from other schemes?
    0.333... refers to some imaginary entity that never has a final value.
    It refers to the value resulting from dividing 3/9 or 1/3. If this couldn't be proved - though it can - we could stipulate it and still be good. We are, after all, talking about definitional schemes - you said it first. Nothing in the scheme of NUMBERS makes 1/3 less final than 1/10, that's a known and inevitable bug of the decimal system, with well-known workarounds.

  2. Top | #262
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    Quote Originally Posted by Jokodo View Post
    As per the definition of the decimal system, any wellformed numeral within it refers to a unique value within the scheme of numbers, presuming that they have been defined. Those symbols are however not part of the scheme of numbers, they live in an extra layer atop the numbers. You should know, you have us an algorithm for the numbers that doesn't make reference to strings.
    By what value does 1 increase if 3 is added to it?

    How do you claim the symbol has nothing to do with the value?

    True, but that reference relation is not part of the world of values. The value defined as the result of applying the Addition operation to the Unit value is literally by definition the same whether you use the English word "two", Russian "dva" or Arabic "ithnaan", the decimal "2", binary "10", or Roman "II". If it makes you happy, you could call it "Gandalf" or "Captain Kirk" and still be talking about the same abstractly defined value.
    Values only exist in a scheme and a scheme is only in one language at a time.

    The symbols refer to values.
    They do, but not vice versa. Their relationship is asymmetric.
    There is nothing there without a symbol. The symbol refers to an individual value that exists because of other symbols.

    The values exist because there is a symbol "1" creating the concept of the value "1".

    0.3 refers to a specific value in the scheme. 0.33 refers to a different value.
    They refer to the values resulting from dividing 3 by 10 and 33 by 100, respectively.
    It is the same string as dividing 33 by 100 but you don't need to divide 33 by 100 to have it.

    But once you have it it designates a separate value from all other values in the scheme.

    0.333... refers to some imaginary entity that never has a final value.
    It refers to the value resulting from dividing 3/9 or 1/3.
    In the value scheme it refers to something that has no final value.

  3. Top | #263
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    Quote Originally Posted by untermensche View Post

    By what value does 1 increase if 3 is added to it?
    It doesn't increase, it's a constant, immutable value. If you meant to ask what value results: the same whether you call them "one" and "three" or "waahid" and "thalaatha".

    How do you claim the symbol has nothing to do with the value?
    I'm following the Definition for the values you gave us. Are you now saying it's wrong?


    Values only exist in a scheme and a scheme is only in one language at a time.
    That sounds like some superduper strong variant of Sapir-Whorf. Someone who Likes to quote Chomsky can be expected to be aware the strong version of the Sapir-Whorf Hypothesis has been discredited before they were born. Plus, no part of the definition you gave makes reference to elements of any language. You just want it to be so but can't give any reason.
    The symbols refer to values.
    They do, but not vice versa. Their relationship is asymmetric.
    There is nothing there without a symbol. The symbol refers to an individual value that exists because of other symbols.
    I think you meant to say "other values"

    The values exist because there is a symbol "1" creating the concept of the value "1".

    0.3 refers to a specific value in the scheme. 0.33 refers to a different value.
    They refer to the values resulting from dividing 3 by 10 and 33 by 100, respectively.
    It is the same string as dividing 33 by 100 but you don't need to divide 33 by 100 to have it.

    But once you have it it designates a separate value from all other values in the scheme.

    0.333... refers to some imaginary entity that never has a final value.
    It refers to the value resulting from dividing 3/9 or 1/3.
    In the value scheme it refers to something that has no final value.
    More Map-Landscape. Literally no more sensible than "the domestic dog is actually two species: domestic and dog"

  4. Top | #264
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    A simple question:

    In what sense is the value resulting from dividing ONE by THREE less final than the value resulting from dividing ONE by FIVE? Can you answer this without leaving the world of VALUES and irrelevantly bringing up strings of the decimal notation?

  5. Top | #265
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    Quote Originally Posted by Jokodo View Post
    By what value does 1 increase if 3 is added to it?
    It doesn't increase, it's a constant, immutable value. If you meant to ask what value results: the same whether you call them "one" and "three" or "waahid" and "thalaatha".
    There is a starting value labeled "1" and you add the value labeled "3" to it.

    You have a final value.

    It is not the same starting "immutable" value. It is a new value.

    By what value is the new value greater than the old value?

    - - - Updated - - -

    Quote Originally Posted by Jokodo View Post
    A simple question:

    In what sense is the value resulting from dividing ONE by THREE less final than the value resulting from dividing ONE by FIVE? Can you answer this without leaving the world of VALUES and irrelevantly bringing up strings of the decimal notation?
    I can give you 1/3 of 90 cents.

    I can't give you 1/3 of a dollar.

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