The infinite idealizations in science imply an underlying mathematical model containing explicitly at least one infinite subset. The interpretation of that subset in terms of the modeled theory is what is meant as usual as an “infinite idealization in science”.
Thus the general setting of the problem allows of being presupposed a fundamental way of how finiteness and infinity are combined in the mathematics itself by its fundament in (Peano) arithmetic and (ZFC) set theory:
1 Peano arithmetic does not admit infinity: Indeed: 1 is finite; adding 1 to any natural number being finite, one obtains a finite natural number again; then, all natural numbers are finite according to the axiom of induction. If one needs infinity, at least one infinite number should be postulated, however there is not that axiom in Peano arithmetic. That seems to be consistent to Peano arithmetic, but that complementation (Peano arithmetic + an axiom of infinite natural number) does not seem to admit any model in the proper Peano arithmetic (because of 2):
2 Set theory (meaning ZFC axioms only for certainty) elucidates the same problem differently enough: an “axiom of infinity” postulates one interpretation of the arithmetical successor function as unlimited. For any axiom of induction is absent, that axiom can introduce infinity consistently. Arithmetic is anyway added “at last” by the axiom of choice and the concept of well-ordering for the axiom of choice is equivalent to the well-ordering principle (or “theorem”). Indeed, any well-ordering is equivalent to an initial segment of natural numbers and can be considered as the definition of the class of equivalence of well-orderings therefore generating the Cantor – Russell concept of ‘ordinal’.
3 Skolem (1922) is the first who paid attention that arithmetic, added in thus to set theory by the axiom of choice as above, relativizes the concept of set complementing it by an arithmetical (“unproper” or “nonintrinsic”) interpretation, which (for the argument “1” above) is finite. Thus the “relativization of ‘set’” for the axiom of choice in Skolem’s manner implies first of all the relativization of infinity by some finite interpretation existing only “purely”, though for any infinite set.
Thesis:
Any infinite idealization in science can be interpreted as finite but uncertain, or only “purely” existing. Particularly, that uncertainness of the finite set corresponding to an infinite one can be interpreted as randomness including the experimental and exact sciences such as quantum mechanics.
A few main arguments for the thesis:
4 The arguments 1-3 above referring to the orthodox ground of mathematics in set theory and arithmetic therefore can be further referred to any theory utilizing any mathematical model including explicitly any infinite set. The interpretation of that infinite set in terms of the modeled theory can be reversed to an infinite idealization in the opposite direction: from the scientific theory to the mathematical model.
5 That finite meaning of any infinite idealization as above can be formulated as any finite set (“F”) satisfying the property generating the infinite set (“I”) at issue. If one considers the set of all sets of kind “F” (i.e. all Fs or I), the relation (“R”) of Fs (I) and any given F is uncertain since it is identical for any F. In other words, R is a class of equivalence of relations.
6 The meta-relation of R (i.e. as a class of equivalence of relations including any F) and a given F can be interpreted as a “pure existence” of F guaranteed just by R and therefore by the axiom of choice in general, in the ground of mathematics in both set theory and arithmetic. Indeed, the choice of any finite F from I though sharing one and the same generating property, “P” needs the axiom of choice. Thus the axiom of choice guarantees furthermore that P and therefore I might be defined otherwise and independently, namely by enumerating all elements being a finite number of some F though unknown and therefore existing only “purely”.
7 One can add in the modeled theory that a certain F is meant as I in the model, i.e. infinity from the model can be always interpreted finitely in the modeled theory by a choice, the option of which is guaranteed by the axiom of choice in the final analysis.
8 The meta-relation of R to a given F can be furthermore interpreted as random as substituting “any F” in R with “a F” in the meta-relation. This allows of the meta-relation and relation in question to be considered as two different relations from one and the same level (unlike one and the same relation but in two different levels in the former case). Thus the concept of randomness is able to express the relation of finite and infinite sets in general.
An example from quantum mechanics:
9 Any quantum system before measurement is unorderable in principle for the theorems about the absence of hidden variables in quantum mechanics (Neumann 1932; Kochen, Specker 1968). Nevertheless, it is always represented as well-ordered after measurement (e.g. by the parameter of registration time). This implies the well-ordering principle for the states before and after measurement to be able to be equated to each other.
10 The above argument (9) can be interpreted in terms of both arithmetic and set theory as follows: The state before measurement (i.e. “by itself”) should be described in terms of set theory and thus involving actual infinity for any coherent state. The state after measurement (i.e. “for us”) should be done in terms of arithmetic involving only well-ordered and thus finite sets. Only the well-ordering principle equivalent to the axiom of choice is power enough to maintain the fundamental epistemological identity of the system before and after measurement.
11 All exhibited in 10 for the argument 8 implies that the choice of any finite set after measurement to represent the infinite set before measurement is fundamentally and initially random therefore grounding the probabilistic interpretation of quantum mechanics by the arguments even yet from the foundation of mathematics.
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