The Maximality Principle mp is a scheme which states that if a sentence of the language of zfc is true in some forcing extension V , and remains true in any further forcing extension of V , then it is true in all forcing extensions of V . A modified maximality principle mpΓ arises when considering forcing with a particular class Γ of forcing notions. A parametrized form of such a principle, mpΓ...

The motivation for this paper is the following: In [Fuc08] I showed that it is inconsistent with ZFC that the maximality principle for closed forcings holds at unboundedly many regular cardinals κ (even only allowing κ itself as a parameter in the maximality principle for <κ-closed forcings each time). So the question is whether it is consistent to have this principle at unboundedly many regula...

I investigate versions of the Maximality Principles for the classes of forcings which are <κ-closed, <κ-directed-closed, or of the form Col(κ,< λ). These principles come in many variants, depending on the parameters which are allowed. I shall write MPΓ(A) for the maximality principle for forcings in Γ, with parameters from A. The main results of this paper are: • The principles have many conseq...

In this paper, by using relational notations, we improve and supplement a true particular case of an inaccurate maximality theorem Ra?l Fierro from 2017, which has to be proved in addition Zorn?s lemma famous principle H. Br?zis F. Browder.

where S = (St)t 0 is the maximum process associated with the one-dimensional time-homogeneous diffusion X = (Xt)t 0 , the function x 7! c(x) is positive and continuous, and the supremum is taken over all stopping times of X for which the integral has finite expectation. It is proved, under no extra conditions, that this problem has a solution, i.e. the payoff is finite and there is an optimal s...

We study the reverse mathematics of countable analogues of several maximality principles that are equivalent to the axiom of choice in set theory. Among these are the principle asserting that every family of sets has a ⊆-maximal subfamily with the finite intersection property and the principle asserting that if φ is a property of finite character then every set has a ⊆-maximal subset of which φ...

In this paper we review the most common forms of reflection and introduce a new form which we call sharpgenerated reflection. We argue that sharp-generated reflection is the strongest form of reflection which can be regarded as a natural generalization of the Lévy reflection theorem. As an application we formulate the principle sharp-maximality with the corresponding hypothesis IMH#. IMH# is an...

Infinite Time Turing Machines (or Hamkins-Kidder machines) have been introduced in [HaLe00] and their computability theory has been investigated in comparison to the usual computability theory in a sequence of papers by Hamkins, Lewis, Welch and Seabold: [HaLe00], [We00a], [We00b], [HaSe01], [HaLe02], [We04], [We05] (cf. also the survey papers [Ha02], [Ha04] and [Ha05]). Infinite Time Turing Ma...