Semantics and CH
Overview
This document is concerned with methods for approaching the resolution of the continuum hypothesis is "classical" set theory. It is more a philosophical than a technical contribution, from an author with insufficient relevant technical competence to make a substantive technical contribution.
A brief statement of the problem and some relevant facts.
When set theorists talk of how CH might be resolved what they generally seem to be looking for is some conjecture which can be proven to entail either CH or its negation and which is intuitively more plausible than CH (or its negation). I begin by discussing some of the disadvantages of this approach.
A semantic approach is outlined, consisting first refining our understanding of the meaning of CH and then exploiting intuitions specific to that refined semantics.
"The" meaning of CH is refined. The method is to chose a set of models of ZFC and to take the refined "CH" to be the claim that CH is true in the nominated set of models. The set chosen is the set of "standard" models.
Some ideas on how the extra information available in refined statements of CH can be exploited in determining whether CH is true.
Some Technical Background
A brief statement of the problem and some relevant facts.
Cardinality
Cantor in his development of set theory refined the notion of size so that it is possible to talk about the size of infinite sets, and showed that not all infinite sets have the same size in this refined sense. The core of this refinement comes through defining "same size" as "can be put in one-one correspondence". There remains some difficulty in deciding from this what exactly a "number" is, this difficulty compounded by the antinomies in set theory (such as Russell's paradox) whose resolution prevents solution of the problem by taking equivalence classes of equinumerous sets as cardinals. However, what exactly a cardinal number is does not much concern us here. The important thing to know about cardinality is that if there is an injection from a set A into a set B then A is no larger than B, and that A and B have the same size iff neither is larger than the other.
CH
The Continuum Hypothesis, which originates with Cantor is the thesis that there are no sets intermediate in cardinality between the set of natural numbers and its power set (which is the same size as the continuum of real numbers). This is the same as saying that the cardinal of the real line or continuum is the smallest cardinal larger than that of the natural numbers.
CH is Independent of ZFC
Non-Semantic Approaches to the Resolution of CH
When set theorists talk of how CH might be resolved what they generally seem to be looking for is some conjecture which can be proven to entail either CH or its negation and which is intuitively more plausible than CH (or its negation). I begin by discussing some of the disadvantages of this approach.
Semantic Approaches to the Resolution of CH
A semantic approach is outlined, consisting first refining our understanding of the meaning of CH and then exploiting intuitions specific to that refined semantics.
Extracting Criteria from Godel
In his paper "The Continuum Hypothesis in Set Theory" (revised version in Benacerraf), Godel makes ststements about the resolution of CH which lend support to the idea that semantics is crucial and from which we can extract the following advice on resolving or perhaps even criteria for a convincing resolution of CH:
  • Chose interpretation

    There is more than one credible interpretation of set theory, and the truth value of the continuum hypothesis may differ according to the interpretation. It is therefore essential, for the problem to be well formulated that a choice of interpretation is made, znd desirable that this be explicit in the presentation of arguments. The identification of the chosen interpretation may involve information which cannot be formally captured, e.g. the notion of "full" power set.

  • The resolution of CH must then make essential use of information specific to the selected interpretation of set theory. Otherwise the argument would apply equally to interpretations in which CH takes a different truth value and must therefore be unsound. Any arguments not so based should be discarded.
From Interpretation to Semantics

I would like to mention here that the notion of and interpretation of set theory as this is normally understood can usefully be liberalised.

First note that one of the relevant "interpretations", Gödel's constructible universe comes really as a family of interpretations of various heights, one for each ordinal, of which infinitely many are models of ZFC.

Next note that the other interpretation mentioned by Gödel, sets as "arbitrary multitudes" regardless of definability, also comes in many versions of differing height. Its not clear that this is the view of Gödel who may well have thought it coherent to talk of a class of all such sets, but since any such class would itself be an arbitrary multitude, incoherence is too close for comfort.

Refining the Meaning of CH
"The" meaning of CH is refined. The method is to chose a set of models of ZFC and to take the refined "CH" to be the claim that CH is true in the nominated set of models. The set chosen is the set of "standard" models.
Exploiting Semantic Refinements
Some ideas on how the extra information available in refined statements of CH can be exploited in determining whether CH is true.

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Created:2002-10-15

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