As standardly presented in axiomatic form, set theory exhibits two fundamental features which together give rise to a number of foundational and philosophical problems. These features are so basic that they are seldom even isolated. The first is that the axioms are taken as categorical assertions (e.g. “There exists an empty set”, “There exists an infinite set”, “The power set of any set exists”, etc.). The second is that actualist quantifiers are employed in these axioms, so that they appear to be about actual objects; the domain of these quantifiers forms the fixed universe of sets (up to a choice of urelement basis), relata of a unique, fixed membership relation entering into the statement of the axioms.

Such presentations are associated with the philosophical positions of platonism and set-theoretic foundationalism.

Cantor was the founder of not one but two theories: an earlier theory of point-sets and a later theory of sets. How are they related? How much more about points and sets of points can be proved with the assumption of sets of sets of points, sets of sets of sets of points, and so on, than without? The present paper is a semi-popular (nontechnical except for presupposing some familiarity with first-order logic) survey of partial answers obtained by logicians during the last decades.

Tarski (1959) has shown how classical geometry (of any number of dimensions, say for definiteness three) can be formalized in a first-order language, here to be called L0, as a first-order theory, here to be called G0. L0 has variables x,y,z, and so on for points, and predicates for a couple of geometric relations among points. G0 has a dozen axioms and one scheme.

Government agencies and private risk assessors use (quasi) scientific “risk assessment” procedures to try to estimate or predict risks to human health or the environment that might result from exposure to toxic substances in order to take steps to prevent such risks from arising or to eliminate the risks if they already exist We might think of this as an aspect of “regulatory science”.

Using carcinogen risk assessment as a model I consider several aspects of risk assessment—use of epidemiology, animal bioassays, and extrapolation models in predicting risks—to show how scientific procedures and uncertainties may determine (in a way often unbeknownst to the practitioners) the public policy debates concerning the estimation of risks from carcinogens.

For one thing, actual and possible scientific uncertainties are large enough that two different researchers using exactly the same data could come to quite different conclusions; such uncertainties permit risk assessors in large measure to determine regulatory decisions even though this exceeds their authority.

The field of quantified risk assessment is a new field, only about 20 years old, and already it is considered to be in a crisis. As Funtowicz and J.R. Ravetz (1985) put it:

Risk assessments done by the Ford Foundation-Mitre Corporation and the Union of Concerned Scientists agree on the hazard probabilities and consequences associated with commercial nuclear fission, but make opposed recommendations regarding using atomic energy to generate electricity. The UCS risk analysis decided against use of the technology; the Ford-Mitre study advised in favor of it (Union of Concerned Scientists 1977; Nuclear Energy Policy Study Group 1977; Cooke 1982, p. 334).

The two studies agreed on the data but made different policy recommendations because they followed different methodological rules at the third, risk-evaluation, stage of assessment. (The first two stages are risk identification and risk estimation.) The Ford-Mitre research was based on the widely accepted Bayesian decision criterion that it is rational to choose the action with the best expected value or utility, where ‘expected value’ or ‘expected utility’ is defined as the weighted sum of all possible consequences of the action, and where the weights are given by the probability associated with each consequence.