Critiques and Answers

    Some criticisms of the idea that the traditional method of infinite
set-subset size comparison creates experimental artifacts and is, therefore,
invalid as well as counterresponses are described below.

1. Critique: One math professor actually seemed to agree with me that the
   set-subset one-to-one pairing off method actually does destroy the
   relationships present in the single set.  He suggested that if one wants to
   maintain the intuitive result that there are one-half as many even integers
   as total positive integers, ratios should be used instead of the "counting"
   method (pairing off).  He went on to add that while a ratio of 1:2 for the
   number of evens relative to the total number of positive integers would be
   found over a finite (local) interval, unfortunately, this ratio would
   become undefined over the entire set because the ratio of two infinities is
   undefined.

   Counterresponse: I am grateful to this person for at least fully
   considering my arguments instead of just saying it's different for
   infinities, etc. (see below).  This is far more than most people have
   offered.  In regard to his criticism, I would agree that the pairing off
   method destroys the relationships in the single set system (the original
   hypothesis) and would be happy to use ratios as he suggests. My only 
   disagreement is withe the view that the ratio of two infinities that start
   out in the same single set is undefined.  The reasoning for this is
   discussed in the next section.



Is the ratio of two infinities undefined when the two infinities start out in the same single set?

  
   My only disagreement is with the idea that the ratio of two infinities is
   undefined.  If the two infinities start out in separate sets, then I agree
   that their ratio is undefined because there is no fixed  relationship
   between the elements in each set.  However, if the two infinite amounts
   (even integers and total positive integers) start out in the same, single,
   original set where every integer has a fixed relationship with every other
   integer throughout the entire infinite set and where all the integers
   maintain this relationship while marching lockstep towards a single
   infinite endpoint, then I would suggest that the ratio of these two
   infinite amounts is not undefined.  The ratio is 1:2 throughout the entire
   infinite set.  This is true only when the two infinite amounts start out in
   a single set, which is the experimental system being studied.


2. Critique: Infinities are different/weird and don't follow the same
   intuitive rules that finite sets do.

   Counterresponse: It is true that infinities are different/weird, but:

   A. Just saying this with no evidence to back it up is not a valid
      critique.  Also, evidence based on the results of the traditional
      set-subset pairing off method is not valid for use in critiquing this
      method.

   B. While infinities are different/weird, the techniques of experimental
      methodology and the scientific method should not change just because one
      is studying infinities.  And, one of the main principals of performing
      valid experiments is that one shouldn't change the experimental system
      being studied in such a way as to change the results obtained relative to
      what would be obtained in the original system (in this case, the single
      set of all positive integers)


3. Critique: Infinite subsets are abstract constructs (studied in the mind
   only) and experimental methodology doesn't apply to mental things.

   Counterresponse: Why not?  If certain rules don't apply to doing thought
   experiments, then the results of all thought experiments, and indeed all
   reasoning of any type, are useless and invalid.  Additionally, whenever
   one is methodically trying to figure out what is occuring within a system,
   one is performing experiments whether the system is in the "real" world or
   in the mind.  Furthermore, even when we do "real world" experiments, all
   knowledge of these experiments is perceived in the mind and so, in a way,
   even "real world" experiments are abstract constructs.  Do the rules of
   experimental methodology (ie, artifacts are bad) not apply to them either?


4. Critique: Infinities aren't found in nature where experiments are done, so
   the rules of experimental methodology don't apply to them.

   Counterresponse: I don't think it's yet determined that infinities do not
   occur in nature.  Our universe may be infinite in size.  Space may be
   infinitely divisible.  Our universe may be one of an infinite number of
   universes in a multiverse (multiple universe interpretations of quantum
   physics).  Additionally, the rules of experimental methodology should apply
   to any system being studied.







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Copyright, 2001