g-rh@cca.CCA.COM (Richard Harter) (07/04/87)
Some time ago I started a discussion in then net.math about denumerability which I would like to revive. Let us start with the Skolem-Lowenheim "paradox" which states that every formal system has a countable model. The paradox is that in ZF the continuum is supposed to be uncountable. How one deals with this "paradox" is, a matter of one's tastes in the philosophy of Mathematics. I propose to deal with it by rejecting the notion of absolute nondenumberability, and, in the process, by claiming that Cantor's treatment of countability is inadequate. As a preliminary let me define some terms: First of all I will use "set" uniformly for any aggregate which is assumed to exist in a particular theory at hand. I.e. I will use the word set for both the sets and classes of two sorted theories (ML, NB, my own H2, etc.). In reference to any multi-sorted theory an appropriate modifier will be used (e.g. Zermelo sets for ZF or NB). If I am discussing a particular theory and a meta theory about it and the meta theory has sets which are do not exist within the theory I will use meta-set or something equivalent. Secondly, I will be speaking of several different usages of the word "countability", so I will prefix each such usage with a qualifier, e.g. Cantor-set-countability: A set is Cantor-set-countable if there is a function (in the sense of a set of ordered pairs) which is one-to-one map from the integers to the set. Cantor-function-countability: A set is Cantor-function-countable if there is a logical function (in the sense of a two place wff) which is a one-to-one map from the integers to the set. Cantor-couability: A set is Cantor-countable if is Cantor-set- countable or Cantor-function-countable, or both. Subset-countability: A set is subset-countable if it is a subset of a Cantor-countable set. My interpretation of the Skolem-Lowenheim paradox is as follows: The specification of the Universe (of Logic and Mathematics) is Cantor-countable. The Universe and the Continuum are subset- countability; however the Continuum is not Cantor-countable. In consequence the entire Cantor programme of transfinite cardinals is, I believe, ill-conceived. Cantor cardinals do measure in some sense, a richness of structure. They do not measure "size". The notion of "large cardinals" rests on a fundamental confusion between different notions of "size". So much for the preliminaries. More to follow. -- Richard Harter, SMDS Inc. [Disclaimers not permitted by company policy.] [I set company policy.]
webber@brandx.rutgers.edu (Webber) (07/05/87)
In article <17406@cca.CCA.COM>, g-rh@cca.CCA.COM (Richard Harter) writes: > > Some time ago I started a discussion in then net.math about > denumerability which I would like to revive. Let us start with the Hmmm. Do you have a copy of that discussion that you could send me? If there is something interesting at the tail end of this discussion, I personally would rather know about it now before sufferring through all the messages about whether logic is philosophy or math and what the definitions of all the basic terms of logic are (just selfish, I guess). > ... My interpretation of the Skolem-Lowenheim paradox is as > follows: The specification of the Universe (of Logic and Mathematics) > is Cantor-countable. The Universe and the Continuum are subset- > countability; however the Continuum is not Cantor-countable. Since this is about the significance of a piece of math, it clearly will have some philosophic aspects. > In consequence the entire Cantor programme of transfinite > cardinals is, I believe, ill-conceived. Cantor cardinals do measure > in some sense, a richness of structure. They do not measure "size". > The notion of "large cardinals" rests on a fundamental confusion > between different notions of "size". Two good references here are: J. N. Crossley et al, What is Mathematical Logic?, Oxford University Press, 1972 (for an overview of what is up) and C. C. Chang and H. Jerome Keisler, Model Theory, North-Holland, 2nd Edition, 1978 (for getting one's hands wet -- particularly the open problems in the back). Also of interest is an intro article written by Abraham Robinson entitled ``Model Theory'', which appears in his collected works. The L\:{o}wenheim-Skolem-Tarski Theorem in this instance boils down to a statement that First Order Predicate Calculus cannot be used to describe systems that cannot be modelled by a subset of the integers. Thus, when you attempt to use FOPC to describe things like real numbers, you end up with a theory that has a `nonstandard' model. Regardless of what you think the `size' of the set of real numbers is, the LST theorem is going to claim that there is a one-to-one mapping between a subset of the integers and any FOPC formulation of real numbers. Thus, this does not seem like a reasonable place to start an analysis of the concept of `set size' from. On the other hand, it does strongly bring into question the signficance of FOPC to `naive set theory'. By this I do not mean to say that there is nothing wrong with the current mathematical formulation of `set size'. Only I see no indication that one really has to go as far as LST before one starts to run into problems with it. However, admittedly I have not yet seen the rest of your proposal and look forward to being surprised. --- BOB (webber@aramis.rutgers.edu ; rutgers!aramis.rutgers.edu!webber)