In fact, the practice for using quantifiers can be generalized by the following formula:
Q: O(I), where Q is a quantifier, “:” assigns some intensional expression (i.e. a complex operator) as a value of Q. Therefore, Q might be considered as a free variable, to which might be assigned any admissible (i.e. making sense) value O(I) for Q already considered as a bound variable.
One can approach the same problem from the viewpoint of set theory as follows: Always some set (S) is meant implicitly so that both Q and O(I) are defined on its set of subsets. S might be as finite as infinite. If S is finite, the equivalence of Q as a free variable defined implicitly on the set (S2) of all subsets of S and O(I) defining some subset (S3) of S2 is trivial as far as it means the domain, in which a variable is defined. Particularly, S3 might be the S2 itself, and then the corresponding complex operator O(I), where “I” are all properties defining one-to-one all subsets of S, would be the operator corresponding to the quantifier as to the set S.
If S is infinite, the fact that any finite expression Q(I) can define some infinite subset of S is obvious. Thus even if Q(I) is a finite expression the quantifier “Q: O(I)” understood as assigning a domain is always admissible. However, the correspondence of quantifiers and intesional operators in classic logic addresses also the problem whether Q might be exhaustedly explicated by some suitable O(I) if S is some infinite set, and O(I) is some finite expression. That problem is both nontrivial and fundamental for mathematics dealing first of all with finite definitions and proofs referring to infinite sets. That is the case properly discussed here.
Thesis: Any quantifier can be exhaustedly explicated by a relevant operator in classic logic iff “pure existence” (i.e. without any constructive fixing) is admitted. On the contrary accordingly, if constructiveness is required, some quantifier(s) cannot be exhaustedly expressed by any operator in classic logic.
A few comments of thesis:
1 The thesis concerns the way, in which (Peano) arithmetic and (ZFC) set theory should be subordinated in the foundation of mathematics. Kurt Gödel considerations (1930 and 1931) demonstrated that the way of application of classic logic to mathematics can be reduced to that of Peano arithmetic to mathematics, and thus to set theory in the final analysis.
2 A quite simple argument shows that Peano arithmetic does not and cannot introduce infinity in principle: 1 is finite; adding 1 to any finite natural number, one obtains a finite natural number again; consequently, all natural numbers are finite according to the axiom of induction.
3 Set theory, the axioms of which does not include the axiom of induction or equivalent to it, postulates (in ZFC for certainty) the existence of infinite sets by the inductive scheme in the so-called axiom of infinity consistently.
4 The equivalence to arithmetic in set theory is added only “at last” by the axiom of choice equivalent to the well-ordering principle. Thoralf Skolem was the first who paid attention immediately (1922) that the axiom of choice in set theory implies the “relativeness” of ‘set’ for any infinite set admits still another, nonintrinsic or unproper interpretation as some finite one, which cannot be constructively pinned at all and in principle, though. It is defined as existing only “purely”.
5 The concept of pure existence admits a consistent interpretation as some randomly chosen finite set (only by virtue of the axiom of choice) unlike that finite set, which is intentionally chosen by addressing constructively. Thus the pair of arithmetic and set theory in the foundation of mathematics seems to be equivalent to the pair of arithmetic and probability theory (e.g. by a relevant equivalent of the Kolmogorov axioms) therefore eliminating any explicit reference to infinity as reducing it to randomness and probability. Particularly, the quantifier "∃" can refer both to the arithmetic “part” as constructive and to the probabilistic “part” as nonconstructive in the foundation of mathematics.
The main argument for the thesis:
Admitting the “pure existence” in the sense above, one can assign some finite expression O(I) whether constructively known or constructively unknown (eventually even in principle) to any quantifier Q as to any ‘set’ S. Here ‘S’ can be understood in the meaning of set theory as usual and arithmetically as well, i.e. as a series of natural numbers therefore being able to encode some operator O(I). A series of natural numbers is always finite in the sense of arithmetic because of the comment "2" above. Furthermore, if S is infinite in the sense of set theory, this implies the “pure existence” of some equivalent finite set being therefore constructively inaccessible and unknown. In other words, the set of mathematical entities existing only “purely” is not empty.
An interpretation by the complex Hilbert space:
The triple of arithmetic, set theory, and probability theory in the foundation of mathematics appeals for a relevant interpretation demonstrating their relativeness. That is by means of the complex Hilbert space as follows:
1 Any probability distribution implies a characteristic function which is a “point” in complex Hilbert space.
2 Furthermore, the complex Hilbert space can be represented by a series of qubits, where a qubit is defined as usual, i.e. as the normed superposition of two orthogonal (1) subspaces of Hilbert space, which is isomorphic to a unit ball, in which are chosen two points, the one of which is on its surface. If the radius of the above unit ball is zero, the complex Hilbert space degenerates to Peano arithmetic.
3 Set theory can be modeled on a single qubit.
Bonus: Any point in the complex Hilbert space is interpretable as the state of some quantum system in quantum mechanics.
Footnote:
1 That is their intersection is empty: they are independent of each other.
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