Set theory, models, and Skolem

Models of ‘Reality’

References:

‘Mathematical Undecidables, Metaphysical Realism, and Equivalent Descriptions’, Hartry Field

‘The Gibbs Paradox’, E T Jaynes

The whole discussion of set theory, models and reality on pages 8 and 9 of the first reference is best understood the way E T Jaynes deals with entropy in his work referenced above, in section 3, on page 5:

QUOTE

This has some interesting implications, not generally recognized. The thermodynamic entropy S(X) is, by definition and construction, a property not of any one microstate, but of a certain reference class C(X) of microstates; it is a measure of the size of that reference class. Then if two different microstates are in C, we would ascribe the same entropy to systems in those microstates. But it is also possible that two experimenters assign different entropies S, S’ to what is in fact the same microstate (in the sense of the same position and velocity of every atom) without either being in error. That is, they are contemplating a different set of possible macroscopic observations on the same system, embedding that microstate in two different reference classes C, C’.

Two important conclusions follow from this. In the first place, it is necessary to decide at the outset of a problem which macroscopic variables or degrees of freedom we shall measure and/or control; and within the context of the thermodynamic system thus defined, entropy will be some function S(X1 ;:::;Xn) of whatever variables we have chosen. We can expect this to obey the second law (TdS is greater than or equal to dQ) only as long as all experimental manipulations are confined to that chosen set. If someone, unknown to us, were to vary a macro variable Xn+1 outside that set, he could produce what would appear to us as a violation of the second law, since our entropy function S(X1 ;:::;Xn) might decrease spontaneously, while his S(X1 ;:::;Xn,Xn+1 ) increases.

END QUOTE

Thus probability as developed by Jaynes is related to the cardinality of a ‘set’, and that cardinality depends on both the domain of definition for that set and the chosen ‘boundary’ of the set implied by its members…in Jaynes’ case, the thermodynamic variables chosen for measurement.

There is here a clear parallel to issues of cardinality in a set theory founded on a chosen set of axioms as argued by Skolem…the issue is with the choice of models and (since in set theory we are not even talking about a ‘reality’ that is measurable) the choice of domain for those sets.

Skolem’s critique of a set theory founded on axioms is even stronger in its logic than Jaynes’ criticism of those who argue for a unique extension for ‘entropy’ where there is no uniqueness to be found. It is ‘stronger’ simply because the arbitrariness of choice extends not only to the boundaries of the sets chosen, but rather beyond to the domain of definition of those sets.

At least with entropy we are talking about a domain whose structure is reasonably understood as determined (assuming that we don’t deny that there is a real world that we are taking measurements of). With set theory the arbitrariness of set boundaries extends to the domain from which the members of those sets are ‘chosen’.

Thus the Skolem Paradox only asks that mathematicians acknowledge the self-referential, and therefore, conditional nature of their foundations. ‘There are no ABSOLUTES’, beyond exactly that.