• Shuffle
    Toggle On
    Toggle Off
  • Alphabetize
    Toggle On
    Toggle Off
  • Front First
    Toggle On
    Toggle Off
  • Both Sides
    Toggle On
    Toggle Off
  • Read
    Toggle On
    Toggle Off
Reading...
Front

Card Range To Study

through

image

Play button

image

Play button

image

Progress

1/17

Click to flip

Use LEFT and RIGHT arrow keys to navigate between flashcards;

Use UP and DOWN arrow keys to flip the card;

H to show hint;

A reads text to speech;

17 Cards in this Set

  • Front
  • Back
What are the components of the axiomatic method?
1. Primitive terms that are undefined.
2. Definitions of other terms using the undefined terms and other definitions.
3. Unproven axiom statements using the undefined terms and definitions.
4. Theorems that are proven using the axioms.
What are isomorphic models?
Models of axiomatic systems that contain a one-to-one corresondence between the interpretations of their undefined terms in such a way that any relationship between the undefined terms in one model is preserved under that one-to-one correspondence in the second model.
What is a consistent axiomatic sytem?
An axiomatic system is consistent when no theorem can be proven using the axioms of the system that contradicts another axiom or already proven theorem. Consistency is a necessary condition of an axiomatic system.
What is a concrete axiomatic model?
An axiomatic model in which the interpretation of the undefined terms is adapted from the real world.
What is an abstract axiomatic model?
An axiomatic model in which an interpretation of the undefined terms is taken from some other axiomatic system.
How can absolute consistency of an axiomatic model be established?
By producing a concrete model.
What does relative consistency of an axiomatic model mean?
The model is consistent with another abstract model that is considered consistent. Undefined terms of the first system are given definitions from a seond system such that the axioms in the first set are theorems in the second set.
What does it mean for an axiomatic system to be considered independent?
An axiom is independent if it cannot be deduced from the other axioms. An axiomatic system is independent if every axiom is independent (there is no redundancy among the axioms).
How is an axiom's independence established?
By producing a model of the system that is consistent with the other axioms, yet the model violates the axiom you are showing is independent.
Define the concept of a complete axiomatic system.
The set of axioms in the system is large enough to prove or disprove any statement concerning the undefined terms. Adding another consistent, independent axiom would require adding more undefined terms.
What is the primary characteristic of a categorical axiomatic system?
When all models of the system are isomorphic. Categoricalness implies completeness.
What is the description of a definition in mathematics?
Contains a GENUS, or broader category to which the definition belongs, and the SPECIFIC DIFFERENCE, or characteristics that distinguish this item's definition from others in that category.
What is an axiomatic model?
An interpretation of the undefined terms of an axiomatic system in which all the axioms are satisfied.
Describe the Fe-Fo axiomatic system.
Undefined terms: Fe, Fo, belongs to.

Axiom 1: There are 3 Fe's.
Axiom 2: Every 2 distinct Fe's belong to exactly one Fo.
Axiom 3: Not all Fe's belong to the same Fo.
Axiom 4: Every two distinct Fo's contain at least one Fe that belongs to both of them.
Describe the 4-point geometry
Undefined: point, line, on.

Axiom 1: There are 4 points.
Axiom 2: Every 2 distinct points have exactly 1 line on both of them.
Axiom 3: Each line is on exactly 2 points.
Name the undefined terms and 4 axioms of an incidence geometry
Undefined: point, line, on

Incidence Axiom 1: For each 2 distinct points there exists a unique line on both of them. (One line for every 2 points)

Incidence Axiom 2: For every line there exist at least 2 distinct points on it. (At least 2 points on every line)

Incidence Axiom 3: There exist at least 3 distinct points.

Incidence Axiom 4: Not all points lie on the same line.
Give an example of a model for the Fe-Fo geometry
Fe's are people (Bob, Ted, Carol) and Fo's are committees (Fun, Finance, Food).

Belongs to - means "is a member of"

Bob & Ted belong to Fun
Bob & Carol belong to Food
Carol and Ted belong to Finance