Essential Information & explanations, latest texts & monographs on
Axiom.
Armey's Axioms: 40 Hard-Earned Truths from Politics, Faith, and Life by Dick Armey
The Complete Book of Zingers by Croft M. Pentz
The Zurich Axioms by Max Gunther
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Ten-Tronck's Celebrity Directory 2004-2005 (Celebrity Directory, 2004-2005) by Axiom Information Resources
Axiom: The Scientific Computation System by Richard D. Jenks
Venture Capitalists & Angel Investors Directory: The Ultimate Guide to Locate V by Axiom Information Resources
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Millionaire Directory: The Ultimate Guide to Identify and Contact Thousands of Millionaires from All the Fields of Endeavor (Millionaire Directory, 2) by Axiom Information Resources
Bond Market Rules: 50 Investing Axioms To Master Bonds for Income or Trading by Michael D. Sheimo
Culturegrams 2004: World Edition (Culturegrams: World Edition, 2004) by Axiom Press
The axioms of religion by Herschel H. Hobbs
The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal (De Gruyter Series in Logic and Its Application, 1) by W. Hugh Woodin
Freyds Models for the Independence of the Axiom of Choice (Memoirs of the American Mathematical Society, 404) by Andreas Blass
Reflections for the Effective Nonprofit Volunteer : Quotes, axioms and observations to help you serve our important institutions by Jim Norvell
Axiom
The word axiom comes from the Greek word
αξιωμα (axioma), which means that which is deemed worthy or fit or that which is considered self-evident. The word comes from αξιοειν (axioein), meaning to deem worthy, which in turn comes from αξιος (axios), meaning worthy. Among the philosophers of the ancient Greeks an axiom was a claim which could be seen to be true without any need for proof.
In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. To say the least, not all epistemologists agree that axioms, understood in that sense, exist.
As the word axiom is understood in modern mathematics, an axiom is not a proposition that is self-evident. Rather, it simply means a starting point in a logical system. For example, in some
rings, the operation of multiplication is commutative, and in some it is not; those rings in which it is are said to satisfy the "axiom of commutativity of multiplication." Another name for an axiom is postulate. An axiom is an elementary basis for a formal logic system that together with the rules of inference define a logic.
For instance, (misquoting Peano) simple arithmetic including addition can be defined and many theorems proven by assuming
a number called 0 exists
every number X has a successor called inc(X)
X+0 = X
inc(X) + Y = X + inc(Y)
Using these axioms, and defining the customary short names 1, 2, 3, and so on for inc(0), inc(inc(0)), inc(inc(inc(0))) respectively, we can show that:
inc(X) = X + 1
and
1 + 2 = 1 + inc(1) Expansion of abbreviation (2 = inc(1))
1 + 2 = inc(1) + 1 Axiom 4
1 + 2 = 2 + 1 Abbreviation (2 = inc(1))
1 + 2 = 2 + inc(0) Expansion of abbreviation (1 = inc(0))
1 + 2 = inc(2) + 0 Axiom 4
1 + 2 = 3 Axiom 3 and Use of abbreviation (inc(2) = 3)
Any fact that we can derive from the axioms is not needed as an axiom. Anything that we cannot derive from the axioms and for which we also cannot derive the negation might reasonably be added as an axiom.
Probably the most famous very early set of axiom are the 4+1 postulates of Euclid. These turn out to be fairly incomplete, actually, and many more postulates are necessary to completely characterize his geometry (Hilbert used 23).
I say 4+1 since the 5th postulate (through a point outside a line there is exactly one parallel) was suspected to be derivable from the first 4 for nearly two millennia. Ultimately, the fifth postulate was found to be independent of the first four. Indeed, one can assume that no parallels through a point outside a line exist, that exactly one exists, or that infinitely many exist. These choices give us alternative forms of geometry in which the interior angles of a triangle add up to less than, exactly or more than a straight line respectively and are known as elliptic, Euclidean and hyperbolic geometries. The general theory of relativity is essentially a claim that mass gives space hyperbolic geometry.
The fact that alternative forms of geometry might exist was very troubling to mathematicians of the 19th century and in similar developments, say Boolean algebra, there were generally elaborate efforts taken to derive the system from normal arithmetic systems. Galois showed just before his untimely death that these efforts were largely wasted but that the grand parallels between axiomatic systems could be put to good use as he algebraicly solved many classical geometrical problems. Ultimately, the abstract parallels between algebraic systems were seen to be more important than the details and modern algebra was born.
In the twentieth century, Gödel's incompleteness theorem showed that no explicit (i.e. recursive) set of axioms sufficiently large for ordinary mathematics could be both (1) complete (i.e. every statement can be either proved or disproved) and (2) consistent (i.e. no statement can be both proved and disproved).
See also:
External links
The above article is adapted from from Wikipedia All Wikipedia article text is available under the terms of the GNU Free Documentation License
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