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	MATHEGENESIS, THE BOOK OF MATHEMATICAL FOUNDATIONS

In the beginning there was Cantor, and he gave us Ordinals and Cardinals
and numerable and nondenumerable sets.  And the multitude looked on in
wonder and marveled, for the vistas opened up were uncountable.  But
Kronecker cried out, "This is sin and wickedness, God has made the integers,
all else is the work of the competition."  And some heeded him, but the
multitude was beguiled by the infinite pleasures of Cantor's new theory,
and they mocked Kronecker, and drove him into the wilderness.

And it came to pass that in those days there was a man named Frege, a
most wonderously subtle and wise man.  And he said, behold, I have seen a
vision.  It has been revealed unto me that all mathematics and all reason
are but a species of logic.  And he built a mighty palace that reached
up into the heavens without limit.  And within that palace there were
no end of wondrous things, including contradictions.

And the Lord said, and I will cast down this palace, for it is not mete
that a mortal should thus storm the heavens.  And the Lord called unto
a man named Russell, and said unto him, "Behold, this man Frege hath
displeased me, for he hath made much out of nothing.  Thou art my
instrument to cast him down.  I shall arm you with a dox, nay with a 
pair of dox, and set you forth".  And Russell set forth with his paradox
and Frege was cast down.

The multitude saw this and lamented, for they loth to leave garden of
paradise of absolute infinities, yet the garden was infested with paradox.

And Russell saw this and felt shamed for he had destroyed much and had
built nothing.  So he set about to restore the palace of Frege, free
of paradox, wherein the Garden of Cantor might be placed.  And he labored
mightily, and brought forth a new palace.  And he guarded against 
paradox by all manner of walls and barriers, types and orders.

Now there came unto Russell a Spanish Barber, saying, "Thy work is
wondrous, though somewhat tedious, but where is this marvelous garden
that Cantor gave us."  And Russell replied, "Behold, I have added a path
within the palace from which all can be reached, for it is of infinite
length."  And a great cry went up, for this path was a bit hard to swallow.

Whilst Russell was raising the palace that he had razed, a new prophet
arose, saying, "Cantor gave unto a thing of mathematics, and Frege hath
made of it a thing of logic.  Let us hark unto the vision of Cantor."
And he said, "The false prophet, Frege, gave us law.  And he said it
was good, for law is surely good.  But the dox, both of them, hast shown
that Frege's law was false law.  Now I say that Cantor was a man of
truth, and his vision was true.  But truth needs law, as law needs
truth.  So let us lay down law for Cantor's truth."  And he did.
And his name was Zermelo.

Now Zermelo was a man of great order, yea beyond all measure.  But
some saw that he was arbitrary in the ways he chose to attain that
order, and they rejected his order.

And it came to pass that the spirit of Kronecker whispered unto two
men and prevailed on them to take on the quest of restoring the spirit
of Mathematics.  And each did so, each in his own way, taking separate
paths.  And their names were Hilbert and Brouwer, and the names of their
paths were Formalism and Intuitionism.  And they labored mightily, and
they traveled far.  But in the end, nothing much came of it.

The law of Zermelo took hold in the land, and the people praised it,
Theologians formulated it and elaborated it and tested it, and
created all manner of axiomatic set theories.  And their names were
Frankel, and Bernays, and Skolem.

But Skolem was a holy man, and there came unto him a vision.  And he
saw that the great garden of Cantor was an illusion, a great shadow
cast by countable models.  And he preached that it was so.  And the
multitude said, yes it is so, and we will it in our proof theory.
But their agreement was with their lips, but not in their hearts.
And they said, "None-the-less the continuum is not countable", and
they hearkened unto Platonism.

Now the Lord looked down, and he saw that many believed that Man is the
measure of all things, and that all problems could be solved.  And the
Lord grew wroth, for this sort of thing was lese majeste, and He called
unto Him a man named Goedel, and He armed him with theorems, and He set
Goedel forth to preach.

And Goedel preached unto the multitude, saying "Whatever thou knowest,
if it be truly worth knowing, thou canst never be sure of".  And he
said unto them, "For every truth that thou canst prove, there is another
that thou cannot prove."  And the multitude was abashed and sore perplexed.

In the wilderness of Harvard there came a man named Quine.  And he 
surveyed the ruins of the palace that Russell had built on the ruins
of the palace that Frege had built.  And he said, "I can repair this
mess."  And he did.  And he called his labors, New Foundations, and
he cast them unto the multitude.  And the multitude said, "Can we do
Mathematics in the ways of our Fathers?"   And Quine said, "Not
exactly."  And the multitude asked, "Can we follow your law and the
law of Zermelo at the same time?".  And Quine replied, "Not completely."
And the multitude rejected Quine, and Quine returned to the wilderness
of Harvard and wrote books.

The spirit of Kronecker was wroth, for the multitude followed the law
of Zermelo and froliced in the garden of Cantor.  So he appeared unto
a man named Turing, and a man named Church, saying unto them, "The
multitude hath donned all manner of fantastic garments, but they can
do no more than their forefathers did, for the integers are all there
really is."  And Turing and Church hearkened unto the spirit of Kronecker
and they devised theories of computatability.  And Church laid down
a great Thesis, saying that all theories of computability are equivalent,
and that what ye may do is forever limited.  And he laid down a universal
law prescribing what Man may or may not do.  And he shewed that the great
law could never be proven.

And thus it came to pass that inconsistency was removed from set theory
in many divers and inconsistent ways.  And the Lord looked upon this
and saw that it was good.  And the multitude wrote papers and taught
courses and garnered grants, and saw that that was good too.

-- 
Richard Harter: SMDS Inc.  Net address: rh@smds.com Phone: 508-369-7398 
US Mail: SMDS Inc., PO Box 555, Concord MA 01742.    Fax: 508-369-8272
In the fields of Hell where the grass grows high
Are the graves of dreams allowed to die.