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  • Re: Another proof: The Halting Problem Is Undecidable.

    From Mikko@21:1/5 to wij on Fri Oct 11 12:38:50 2024
    On 2024-10-10 14:43:52 +0000, wij said:

    Axiom: Part is smaller than the whole.

    Not always. A half-line is infintely long and has only one endpoint.
    Any other point of the half-line divides it to two parts: a finite
    line segment and an infinite half-line. The former part is obviously
    smaller than the original half-line but the latter is not.

    --
    Mikko

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  • From Andy Walker@21:1/5 to wij on Fri Oct 11 11:42:21 2024
    On 10/10/2024 16:26, wij wrote:
    This "0.999...!=1" proof [...].

    Any such proof would breach the Archimedean [Eudoxus] axiom, so
    is not a proof about the real numbers. You have been told that before.
    If you propose to repeat this or to take it further, you /really, really/
    do need to tell us what axioms you are using instead of those of the real numbers. Without that, your claims, whatever they may be, are worthless,
    and no-one qualified to do so can comment more usefully. WIYF.

    --
    Andy Walker, Nottingham.
    Andy's music pages: www.cuboid.me.uk/andy/Music
    Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Valentine

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  • From Andy Walker@21:1/5 to wij on Fri Oct 11 21:32:23 2024
    On 11/10/2024 18:11, wij wrote:
    Archimedes likely believes that all (real) numbers, including pi, sqrt(2), are
    p/q representable. Is that what you suggest?

    By the time of Archimedes it had been known for several hundred
    years that "sqrt(2)" is irrational. [The status of "pi" remained unknown
    for a further ~2K years.] So no, Archimedes did not believe that, not
    least when he laid some of the foundations of calculus.

    Archimedean axiom is an *assertion* that infinitesimal does not exist without knowing the consequence (violating Wij's Theorem which is provable from the rules
    stronger than 'assertion').

    If "Wij's Theorem" is inconsistent with the axioms of real numbers, then it is not a theorem of real numbers. Try one of the other systems of numbers, which you would probably find more to your taste, given the other things you say in this group.

    --
    Andy Walker, Nottingham.
    Andy's music pages: www.cuboid.me.uk/andy/Music
    Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Valentine

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  • From Andy Walker@21:1/5 to wij on Sat Oct 12 17:16:54 2024
    On 12/10/2024 02:53, wij wrote:
    Archimedes likely believes that all (real) numbers, including pi, sqrt(2), are
    p/q representable. Is that what you suggest?
    By the time of Archimedes it had been known for several hundred
    years that "sqrt(2)" is irrational.

    Established, at the latest, by the Pythagoreans [~500 BCE, nearly 300y before Archimedes]. Possibly known earlier. ...

    [The status of "pi" remained unknown
    for a further ~2K years.]

    First proved by Lambert, in 1761; 1973y after Archimedes died. ...

      So no, Archimedes did not believe that, not
    least when he laid some of the foundations of calculus.

    Euclid's "Elements" book 10 is about irrational numbers, and
    Archimedes referred to that book in his writings, so he was certainly
    aware that root(2) is irrational. ...

    That is a fabrication (there are many, but... accepted, as a fabrication)

    ... So everything above can be verified by anyone able to google or wiki. Accusing me of lying is not a good way to get the mathematical help
    that you clearly need, esp when the accusation is so clearly false.

    Archimedean axiom is an *assertion* that infinitesimal does not exist without
    knowing the consequence (violating Wij's Theorem which is provable from the rules
    stronger than 'assertion').
    If "Wij's Theorem" is inconsistent with the axioms of real numbers,
    then it is not a theorem of real numbers.  Try one of the other systems of >> numbers, which you would probably find more to your taste, given the other >> things you say in this group.
    Are you kidding? "x>0 iff x/n >0, where n∈ℤ⁺" is inconsistent?

    It was /your/ claim that "Wij's Theorem" is "violated" by the Archimedean axiom. /If/ you are right, /then/ your theorem is not true
    in the standard reals, as used by all numerate scientists and engineers
    and as understood by all mathematicians [even if they /also/ use NSA or surreals or other systems]. FWIW, /I/ think your theorem is correct
    in standard analysis, but you seem to object to that.

    With your real, yes. My real is based on the abacus that can be physically modeled. Tell me, how can
    it be inconsistent?

    Perhaps you should first explain how you represent infinity and infinitesimals on a standard abacus? "Your" reals can, of course, be inconsistent if you insist on axioms that are inconsistent with them.

    [If you persist in insulting those who are trying to help you,
    then you will not get any further reply from me. I don't intend to play "Fetch" with you.]

    --
    Andy Walker, Nottingham.
    Andy's music pages: www.cuboid.me.uk/andy/Music
    Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Necke

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  • From Ben Bacarisse@21:1/5 to wij on Sun Oct 13 22:01:02 2024
    wij <[email protected]> writes:

    If 0.999..=1, you have to explain your arithmetic system.

    Almost. First you have to explain the notation. That's easy (but
    relatively advanced) as 0.999... denotes the limit of a sequence of
    partial sums (sometimes called a series limit). The arithmetic system
    (the reals, where all such sums converge) comes after saying what the
    ... denotes.

    When *you* say that 0.999... =/= 1 you always avoid saying what the
    notation (specifically the ...) means in formal terms.

    --
    Ben.

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  • From Ben Bacarisse@21:1/5 to wij on Mon Oct 14 13:41:04 2024
    wij <[email protected]> writes:

    On Sun, 2024-10-13 at 22:01 +0100, Ben Bacarisse wrote:
    wij <[email protected]> writes:

    If 0.999..=1, you have to explain your arithmetic system.

    Almost.� First you have to explain the notation.� That's easy (but
    relatively advanced) as 0.999... denotes the limit of a sequence of
    partial sums (sometimes called a series limit).� The arithmetic system
    (the reals, where all such sums converge) comes after saying what the
    ... denotes.

    When *you* say that 0.999... =/= 1 you always avoid saying what the
    notation (specifically the ...) means in formal terms.

    What I would say now is probably not different from https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber2-en.txt/download

    "..." conventionally means "so on,..,etc.", likely infinitely. I use
    it like that.

    So it might even be finite and it has some element of likelihood about
    it? I'll stick with the actual conventional meaning, thanks.

    --
    Ben.

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