This paper presents various forms of induction formulas and at the same time proves equivalence these by proving that some are equivalent to Well-Ordering Principle which applies natural number set.

Independent component analysis (ICA) has been recently used as a model of natural image statistics and V1 simple cell receptive "elds. Here we show how to extend the ICA model to explain V1 topography as well. This is done by relaxing the independence assumption and ordering the basis vectors so that vectors with strong higher-order correlations are near each other. This is a new principle of t...

The aim of this paper is to present a general framework for comparing fuzzy sets with respect to a general class of fuzzy orderings. This approach includes known techniques based on generalizing the crisp linear ordering of real numbers by means of the extension principle, however, in its general form, it is applicable to any fuzzy subsets of any kind of universe for which a fuzzy ordering is k...

In this note we answer a question raised by John Truss. He asked if every definable partial ordering in an o-minimal structure can be extended to a definable total order. His question is motivated by analogy with the Order Extension Principle, a weak choice-like axiom of interest to set theorists, which asserts that every partial ordering of a set can be extended to a total order of the set (se...

In this note we prove the equivalence between the axiom of choice, the well ordering principle and Zorn’s lemma, and discuss to some extent how large fragment of ZF we need in order to prove the individual implications.

Combinatorial functions are used to replace indecomposable cardinals in certain types of set theoretic arguments. This allows us to extend decidability results from nonchoice set theories to set theories with a principle of linear ordering.