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Ultrafilters are standard tools in mathematical logic and set theory that can be used in combinatorics and topology. Three problems providing finite situations with similar conclusions using ultrafilters are presented to help understand the concept better.

The article reviews the foundations of positive model theory through the introduction of h-inductive sentences.

Abstract: Logic should have been defined as the unity of contradiction between logic director and logic implementation. Chinese Daoism asserts that everything is defined in the unity of opposites, namely [...]

This department recognizes the importance of children's exploring hands-on and minds-on mathematics and presents teachers with open-ended explorations to enhance mathematics instruction. These tasks are designed to evolve as they [...]

The surprise examination paradox is characterized by mathematical ideas with pedagogical values in teaching elementary logical reasoning. The paradox focuses on the holding of a surprise exam on some day during a following week. It embodies incompatible facts that seemed easy to resolve, and yet there is still no consensus on its correct resolution. It has often been called a significant problem for philosophy and has also been criticized as a ridiculous problem.

A class pizza party is used to introduce the concept of data analysis by testing the students' capability to determine how many medium-sized pizzas to order from the neighborhood pizza parlor. Knowledge about graphs and charts is also imparted to the students' by asking them to vote for their favorite pizza types. The voting results are then represented in graphs and charts.

The article demonstrates that the set of tautological quantum logical propositional formulas for a finite dimensional vector space [C.super.n] is different for every n.

A demonstration of the use of two well-known examples, the Map Coloring problem and the traveling Salesperson problem, to teach logical concepts and methods of proof is presented. The scenarios described are based on a mathematics course for first-year college liberal arts students.

The complexities involved in writing mathematics problems are discussed. The article discusses a cooperative learning activity which gives students the challenge of writing mathematical logic problems for peers. Making students construct mathematics problems taps into several important skills.

A study puts forth a contrast between the idealized model of knowledge and a pragmatic model. The result showed that while the idealized model of knowledge cannot deal very effectively with the argument from ignorance or lack-of-evidence argument as it might less prejudicially be called, the pragmatic model sees knowledge as a matter of evidence based on practical acceptance rather than on abstract and idealized notions of truths and absolute proof.

A study of the relationship between ideals I of Turing degrees and groups of I-recursive automorphisms of the ordering on rationals is presented. The differences between such groups and groups of all automorphisms which prove that the isomorphism type of such groups completely define the ideal I as well as outline a general correspondence between principal ideals of Turing degrees and the first-order properties of such groups are also discussed.

A study of the basic new result which says that for any completion T of PA, there is another completion S and the variant which says that natural upper bounds for sets of degrees associated with theories and models of arithmetic cannot be minimal is presented. The two results, which were proved using some ideas from the proof of Solovay's theorem basing the applications on finite-injury constructions are also discussed.

A study considering the set (script) ML prefix 1 of first-order polymodal formulas, the modal operators which can be applied to subformulas of at most one free variable and prove the general satisfiability criterion for formulas in (script) ML prefix 1 using a mosaic technique is presented. The modal satisfiability is reduced to the classical one and the criterion is then used to single out a number of new, optimal and decidable fragments of various modal predicate logics.

A proof of the theorem of Zermelo that any set possessing a choice function for its set of inhabited subsets can be well-ordered, which is valid in any elementary topos is presented. The fact that this proof is considerably simpler than existing proofs in the literature and is seen as a direct generalization of Zermelo's own 1908 proof of his theorem is also outlined.

Study of a new large cardinal concept called 'remarkability' which measure the power of proper forcing to change in certain aspect the shape of the universe is discussed and proofs of main results previously presented are mentioned. Characteristics of these remarkable cardinals are also discussed.

A study to show that two abelian-by-finite groups are elementarily equivalent if and only if they satisfy the same sentences with two alternations of quantifiers is presented. It is also proved that abelian-by-finite groups satisfy a quantifier elimination property giving examples of nilpotent groups for each integer n, which satisfy the same sentences with n alterations of quantifiers but not with n + 1 alterations of quantifiers.

Infinite p-adic fields, which admit only tamely ramified algebraic extensions as well as answer the question whether quantifier elimination is possible in some infinite algebraic extensions of Q prefix p using a natural language extension are presented. It is also seen that for tame infinite p-adic fields with algebraically closed residue fields, which satisfy Kaplansky's condition, an extension by the power predicate P prefix n introduced by Macintyre is sufficient.

A study of the problem of the existence of a universal structure omitting a finite set of forbidden substructures which is reducible to the category of graphs with a vertex coloring by two colors based on an extensive graph theoretic literature is presented. The problem of the existence of universal graphs is generalized further to the category of ordinary graphs or whether these problems are decidable is still not known and proofs to support these results are also discussed.

A study to prove that for any model M of a weakly o-minimal theory T, any expansion of M suffix + of M by a family of unary predicates has a weakly o-minimal theory if the set of all realizations of each predicate is a union of a finite number of convex sets is presented. This solves the problem of Cherlin-Macpherson-Marker-Steinhorn [MMS] for the class of weakly o-minimal theories.

Theorems to prove the relation of forcing or 'filtering through' many positive polarized partitions between a cardinal and its powerset are presented. Results derived are also discussed.

The project of determining the set existence axioms needed to prove various theorems of analysis is presented. It is shown that the Mean Value Theorem can be proven in RCA (suffix 0) where RCA stands for Recursive Comprehension Axiom with all reasoning in the theorems presented taking place in RCA (suffix 0) working in a weak subsystem of second order arithmetic.

Heather and Lindsay were planning to decorate their room for the holidays. Their mom gave them a bowl of popcorn, a bowl of cranberries, and a bowl of green peas. [...]