Re: Ultimate debunking of Cantor's Theory

Peter Webb wrote:


...set theory minus the Axiom of infinity is a perfectly valid set theory. The advantage for the poster is that Cantor's diagonal construction of the Reals doesn't exist, or indeed any form of the diagonal argument applied to infinite sets. Indeed, if you consider that the existence of more than one type of infinity to be "absurd", you can use this to show that the Axiom of infinity must be false through a reductio-ad-absurdum argument.


One can also avoid Cantor's contruction by keeping the Axiom of Infinity but dropping the Power Axiom. I don't know if there is anything else interesting about such a system.

--
Stephen J. Herschkorn sjherschko@xxxxxxxxxxxx
Math Tutor on the Internet and in Central New Jersey and Manhattan

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Relevant Pages

  • Re: Cantors "diagonal argument". My Objection.
    ... Infinity is NOT relevant to the proof ... you DENY the axiom of infinity (if you replace it with an axiom ... It DOESN'T MATTER whether you restrict these reals to being between 0 ... This IS NOT specific to Cantor IN ANY way except that he got there ...
    (sci.logic)
  • Re: Is continuum completely filled up?
    ... compactification of the reals, as used in analysis. ... the surreals do not satisfy the least upper bound ... If the axiom of power set implys uncountable reals, ... The axiom of infinity guarantees that an infinite set exists. ...
    (sci.math)
  • Re: abundance of irrationals!) - rectangles of area 1.bmp [0/1]
    ... >>> But the axiom of infinity and the principle of induction do get you to ... >>> infinity in finitely many steps. ... > rationals all of different sizes but the ratioals and the reals of the ... that it therefore constitutes an enumeration of the reals. ...
    (sci.math)
  • Re: Cantor Confusion
    ... rationals that comes arbitrarily close to other such sequences and so ... diagonal proof was not about the reals at all. ... *based* on the axiom of infinity, ...
    (sci.math)
  • Re: Zenkins paper on Cantor (reply of Dr. Zenkin)
    ... > theorem is proven for you without needing any infinitary reasoning. ... proof are really objections to the axiom of infinity. ... reals are not even defined. ...
    (sci.math)
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