It is impossible to axiomatize the arithmetic. If you create any finite set of axioms, you can always find a new arithmetic property that can't be proved starting from the axioms.  ''This is not what Goedel proved.''

That assertion ''was'' proved by Goedel. See: http://kilby.stanford.edu/~rvg/154/handouts/incompleteness.html.

Mathematical description has been merged into GoedelsIncompletenessTheorem

That leaves the following, which I'm not sure what to do with...

-- AndrewMcGuinness

An Experiential excursion into Goedel.

Pseudo-axiomatic definitions

	* ''Pseudo-philosophers'' The set of self-appointed philosophers who abuse famous theorems to prove hobby horses are real.

	* ''actual philosophers'' The set of self-appointed philosophers who use famous theorems to prove hobby horses are real.

	* ''God'' The set of entities who can reliably classify all entities into one of the previous two sets.

The ''Goedelic aspect of these axioms'' is that it has since been proved that God is intrinsically incapable classifying himself as either an actual or a pseudo philosopher.

(This attempt at a goedellic is lame(or wrong).) -- Author

(Anyone got DouglasHofstadter's email address to ask for a better example for this context?) -- desperate author

(Anyone here got Goedel's email address?) -- confused Author

(Any "actual" mathematicians here) - but then we will need God to identify the pseudo mathematicians' ones from the actual mathematicians'. Damnation, this logic stuff is hard. -- Desperately confused Author.

Note; For the humor impaired :)

''How about: actual philosopher = someone with a degree in philosophy and recognized by his peers as a philosopher?''

This would be fine just so long as we are prepared to conclude that Galileo was not an astronomer. I suspect other problematic examples exist.

At various times in science, half of a person's peers recognize them and the other half do not.

Note the same kinds of problems exist with defining living, dead, plant, animal,
intelligence, instinctive, nature, nurture, tall people, short people, species boundaries, Justice, Truth, Beauty, Charm, and Strangeness, ...

... and yet ''somehow'' humans know the meanings of these words and are able to communicate usefully.


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Explanation:

A theory is a domain and a set of theorems that apply to that domain.

A theorem is an statement that:

1. Is always true (for all objects in the domain).
2. Can be proven true through something called a demonstration.

A demonstration is a sorted linear linked list of statements that, based on that the previous statement is true, and a set of statements that allow for transformation of the previous statement:

3. Prove the next statement is true.
4. The proven theorem is true and therefore can be used as a transformation for proving new thoerems.

An statement can be true or false given a set of objects in the domain.

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Given that a demonstration is a list of statements, it turns out that given a possible theorem, the only way to prove would be:

a) to find an infinite number of statements, for which the first we would never have and therefore this is not a demonstration.
b) to find circular statements, so that there is no beginning and no end, and therefore this would be circular reasoning, also not a demonstration.
c) to have a defined first statement and walk its way into the result into a finite number of steps. This is the only possible logical answer.

Given that that is true, some statements must be assumed to be true from the beginning, otherwise we would never have a first statement. Those first statements are called axioms of the theory and they are assumed to be true. If you take different axioms you end up with a different theory, but in some cases different axioms lead exactly to the same theory, because the set {a,b,c} of axioms allow to demonstrate {d} and the set {a,b,d} allow to demonstrate {c}, etc.