The Problem Expressibility of Properties. The Idea I Fraisse's Theorem. The Tool Ehrenfeucht-Fraisse-Method. Some Utilities Localities.
Math - Model Theory
Some Solutions for some Structures. Typical problem: Given a finite graph, can the property of being an acyclic be expressed in a first order language? From Wikibooks, open books for an open world. Wikipedia has related information at Finite model theory Basics Why? Motivation What is it? Definition and Background Why is it special? Typical Logics and Structures studied What is required?
Preliminaries What to start with? Expressive Power of Languages Typical problem: Given a finite graph, can the property of being an acyclic be expressed in a first order language? Descriptive Complexity Random Structures. This mathematics module is a stub.
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Views Read Edit View history. In common usage, the words hypothesis, model, theory, and law have different interpretations and are at times used without precision, but in science they have very exact meanings. Perhaps the most difficult and intriguing step is the development of a specific, testable hypothesis.
A useful hypothesis enables predictions by applying deductive reasoning, often in the form of mathematical analysis. It is a limited statement regarding the cause and effect in a specific situation, which can be tested by experimentation and observation or by statistical analysis of the probabilities from the data obtained. The outcome of the test hypothesis should be currently unknown, so that the results can provide useful data regarding the validity of the hypothesis.
Sometimes a hypothesis is developed that must wait for new knowledge or technology to be testable. The concept of atoms was proposed by the ancient Greeks , who had no means of testing it.
The Capital Asset Pricing Model: Theory and Evidence
Centuries later, when more knowledge became available, the hypothesis gained support and was eventually accepted by the scientific community, though it has had to be amended many times over the year. Atoms are not indivisible, as the Greeks supposed. A model is used for situations when it is known that the hypothesis has a limitation on its validity. The Bohr model of the atom , for example, depicts electrons circling the atomic nucleus in a fashion similar to planets in the solar system.
This model is useful in determining the energies of the quantum states of the electron in the simple hydrogen atom, but it is by no means represents the true nature of the atom.
Theories, Models, & Frameworks
A scientific theory or law represents a hypothesis or group of related hypotheses which has been confirmed through repeated testing, almost always conducted over a span of many years. Generally, a theory is an explanation for a set of related phenomena, like the theory of evolution or the big bang theory. The word "law" is often invoked in reference to a specific mathematical equation that relates the different elements within a theory.
In the overall theory of universal gravitation developed by Sir Isaac Newton , the key equation that describes the gravitational attraction between two objects is called the law of gravity.
Analysis, geometry and dynamical systems
These days, physicists rarely apply the word "law" to their ideas. In part, this is because so many of the previous "laws of nature" were found to be not so much laws as guidelines, that work well within certain parameters but not within others. Once a scientific theory is established, it is very hard to get the scientific community to discard it.
In physics, the concept of ether as a medium for light wave transmission ran into serious opposition in the late s, but it was not disregarded until the early s, when Albert Einstein proposed alternate explanations for the wave nature of light that did not rely upon a medium for transmission.
The science philosopher Thomas Kuhn developed the term scientific paradigm to explain the working set of theories under which science operates.