The temporal foundation of the principle of maximal entropy

The principle of maximal entropy (further abbreviated as “MaxEnt”) can be founded on the formal mechanism, in which future transforms into past by the mediation of present. This allows of MaxEnt to be investigated by the theory of quantum information.MaxEnt can be considered as an inductive analog or generalization of “Occam’s razor”. It depends crucially on choice and thus on information just as all inductive methods of reasoning. The essence shared by Occam’s razor and MaxEnt is for the relevant data known till now to be postulated as an enough fundament of conclusion. That axiom is the kind of choice grounding both principles. Popper’s falsifiability (1935) can be discussed as a complement to them: That axiom (or axiom scheme) is always sufficient but never necessary condition of conclusion therefore postulating the choice in the base of MaxEnt. Furthermore, the abstraction axiom (or axiom scheme) relevant to set theory (e.g. the axiom scheme of specification in ZFC) involves choice analogically.

High-order metaphysics as high-order abstractions and choice in set theory

The link between the high-order metaphysics and abstractions, on the one hand, and choice in the foundation of set theory, on the other hand, can distinguish unambiguously the “good” principles of abstraction from the “bad” ones and thus resolve the “bad company problem” as to set theory. Thus it implies correspondingly a more precise definition of the relation between the axiom of choice and “all company” of axioms in set theory concerning directly or indirectly abstraction: the principle of abstraction, axiom of comprehension, axiom scheme of specification, axiom scheme of separation, subset axiom scheme, axiom scheme of replacement, axiom of unrestricted comprehension, axiom of extensionality, etc.

Inconsistency of the Axiom of Choice with the positive theory

The idea of the positive theory is to avoid the Russell's paradox by postulating an axiom scheme of comprehension for formulas without “too much” negations. In this paper, we show that the axiom of choice is inconsistent with the positive theory .

The Incompleteness of Peano Arithmetic with Exponentiation

We shall now turn to a formal axiom system which we call Peano Arithmetic with Exponentiation and which we abbreviate “P.E.”. We take certain correct formulas which we call axioms and provide two inference rules that enable us to prove new correct formulas from correct formulas already proved. The axioms will be infinite in number, but each axiom will be of one of nineteen easily recognizable forms; these forms are called axiom schemes. It will be convenient to classify these nineteen axiom schemes into four groups (cf. discussion that follows the display of the schemes). The axioms of Groups I and II are the so-called logical axioms and constitute a neat formalization of first-order logic with identity due to Kalish and Montague [1965], which is based on an earlier system due to Tarski [1965]. The axioms of Groups III and IV are the so-called arithmetic axioms. In displaying these axiom schemes, F, G and H are any formulas, vi and vj are any variables, and t is any term. For example, the first scheme L1 means that for any formulas F and G, the formula (F ⊃ (G ⊃ F)) is to be taken as an axiom; axiom scheme L4 means that for any variable Vi and any formulas F and G, the formula . . . (∀vi (F ⊃ G) ⊃ (∀vi (F ⊃ ∀vi G) . . . is to be taken as an axiom.

Natural deduction based set theories: a new resolution of the old paradoxes

The comprehension principle of set theory asserts that a set can be formed from the objects satisfying any given property. The principle leads to immediate contradictions if it is formalized as an axiom scheme within classical first order logic. A resolution of the set paradoxes results if the principle is formalized instead as two rules of deduction in a natural deduction presentation of logic. This presentation of the comprehension principle for sets as semantic rules, instead of as a comprehension axiom scheme, can be viewed as an extension of classical logic, in contrast to the assertion of extra-logical axioms expressing truths about a pre-existing or constructed universe of sets. The paradoxes are disarmed in the extended classical semantics because truth values are only assigned to those sentences that can be grounded in atomic sentences.

Solution to the P − W problem

Anderson and Belnap asked in §8.11 of their treatise Entailment [1] whether a certain pure implicational calculus, which we will call P − W, is minimal in the sense that no two distinct formulas coentail each other in this calculus. We provide a positive solution to this question, variously known as The P − W problem, or Belnap's conjecture.We will be concerned with two systems of pure implication, formulated in a language constructed in the usual way from a set of propositional variables, with a single binary connective →. We use A, B,…, A1, B1, …, as variables ranging over formulas. Formulas are written using the bracketing conventions of Church [3].The first system, which we call S (in honour of its evident incorporation of syllogistic principles of reasoning), has as axioms all instances of (B) B → C →. A → B →. A → C (prefixing),(B) A → B →. B → C →. A → C (suffixing), and rules (BX) from B → C infer A → B →. A → C (rule prefixing),(B’X) from A → B infer B → C →. A → C (rule suffixing),(BXY) from A → B and B → C infer A → C (rule transitivity).The second system, P − W, has in addition to the axioms and rules of S the axiom scheme (I) A → A of identity.We write ⊢SA (⊣SA) to mean that A is (resp. is not) a theorem of S, and similarly for P − W.

On existence proofs of Hanf numbers

This paper refines some results of Barwise [1] as well as answering the open question posed at the end of [1] about the Hanf number of positively. We conclude by showing that the existence of a Hanf bound for cannot be proved in the natural formally intuitionistic set theories with bounded predicates decidable of [3], [4] and [5].All notation not explained below is taken from [1]. In the Appendix, we give the axioms of ZF0, ZF1, and T in full. We remark that an important point about the axiom of foundation was not emphasized in [1]. This axiom was intended to be the axiom scheme (∀x)((∀y ∈ x)(A(y)) → A(x)) → (∀x)(A(x)), where y does not occur in A = A(x), instead of the more customary (∀x)(∀y)(y ∈ x → (∃z ∈ x)(∀w ∈ z) (w ∉ x)). This is of no consequence in the presence of full separation, but is vital when considering ZF0 and the T below, for with the customary form of foundation, these cannot even prove the existence of Rω+ω.In [1], a proof of the following is sketched.

Equality in

In [CLg. II, §15B6], the problem of representing equality in the system by means of an ob Q is discussed; it is shown there that if Q is taken to be a canonical atom of degree 2 and if the axiom schemeis postulated, then it follows by Rule Eq thatand, if no other properties are postulated for Q, then the converse of (1), Q-consistency, also holds. However, it is also desirable to have, in addition, the propertyat least for some obs Z, and the statement is made in [CLg. II, §15B6] that no known way of incorporating this principle existed (at the time this statement was written) in such a way that there was a proof of Q-consistency. In this paper it is shown that if Z is restricted to be a basic canob of degree 1, i.e., if Z is restricted to be a predicate of one argument, then a system in which (2) is postulated can be proved Q-consistent.The notations and conventions of [CLg. II] especially §15B, will be used throughout the paper.