Re: Set Theory: Should You Believe


Kevin Karn wrote:
This is a question from the sociological angle, but I'm curious to know
what you think. Why, in your opinion, is the orthodoxy in set theory
etc. so entrenched?

Because it's adequate to encode most things.
It's a comprehensive neutral framework.
It makes a good model-construction language.
But it is not what "practicing mathematicians"
actually use, in any case, so from that standpoint, it is
NOT entrenched. It is entrenched as a foundation and
AS A TOOL FOR INVESTIGATING MATHEMATICAL *LOGIC*.
It is NOT entrenched in other ways.

Why is the idea of questioning/rejecting infinity so threatening?

It's not threatening. It's just stupid. It's so stupid that EVEN
after
people (like NW and others) prove that we could actually, in an
Occam's Razor sense, get along without it, people keep it ANYway.
Because pretending that infinity did not exist WOULD JUST BE STUPID.
Infinity OBVIOUSLY exists. Infinity OBSERVABLY exists, once you
think you know what a natural number is, once you notice that EVERY
natural number has the property that you can add 1 to it and get
something
differing from ALL SMALLER natural numbers. Suppose you TRIED
to confine yourself to considering ONLY FINITE things. Well, guess
what:
there are AN INFINITE NUMBER OF finite natural numbers. So you have
ALREADY COMMITTED yourself to an infinity of things. So pretending
that you haven't, is, I repeat, JUST OBVIOUSLY STUPID. So that's
why people who (futilely) try to do it get dismissed as idiots.

Mathemaitcs is about producing proofs
and proofs do require axioms as starting points. To say that
mathematics does not require axioms is the same is saying that
mathematics does not require formal systems or even logic itself. But
that will quickly lead us into contradictions and just plain confusion
as well.

When NW said that "You don't need axioms", I understood him to be
saying something more nuanced -- i.e. that "You don't need set theory,
or the axioms of set theory, to do mathematics."

Well, you're even stupider than he is.
He really WAS saying "you don't need axioms"
and he really was proving about himself, thereby, that
he is too stupid to realize that HE needs axioms and
is in fact relying on them constantly.


As he said:
" Whenever discussions about the foundations of mathematics arise, we
pay lip service to the `Axioms' of Zermelo-Fraenkel, but do we every
use them? Hardly ever. With the notable exception of the `Axiom of
Choice', I bet that fewer than 5% of mathematicians have ever employed
even one of these `Axioms' explicitly in their published work. The
average mathematician probably can't even remember the `Axioms'. I
think I am typical---in two weeks time I'll have retired them to their
usual spot in some distant ballpark of my memory, mostly beyond recall.
"

This is just silly.
Seriously, despite the fact that all of this is true, he marks himself
as a crank by having dared to say it. OF COURSE he doesn't use
the axioms OF SET theory because HE is NOT a SET THEORIST!
HE uses the axioms of WHATEVER FIELD HE SPECIALIZES IN!

The value and relevance of set theory is that the axioms of THAT
field AND of ALL THE OTHERS can ALL be SIMULTANEOUSLY
ENCODED in this ONE dialect (ZFC) -- that is why IT gets to BE
the FOUNDATION. SOME of these concepts (like category theory)
are also equally comprehensive and could serve as alternative
foundations, and are in fact invited to, sometimes, especially when
their take on the universe gets hard to translate into set theory
(which in the case of category theory, it is).

It's very clear that you don't need set theory or the axioms of set
theory to do mathematics.

Well, of course, any old axioms will do.
It goes withOUT saying that once you pick ANY consistent
axiom-set and investigate what follows from it, YOU ARE "DOING
mathematics". That is not the point. Set theory is special because
it is a comprehensive neutral framework, because it is one size that
can fit all. What is equally clear (and what you are overlooking,
because
you're just STUPID like that) is that you DO need something AS RICH
or AS POWERFUL as set theory in order to be able to do ALL of
mathematics
WITHIN THIS ONE framework.

After all, virtually the entire body of
pre-20th century mathematics was developed without set theory,
or axioms of set theory.

As I said, any old simpler body of axioms will get you
a little something. The mathematical endeavor that motivated
set theory was describing the sets of points where infinite series
failed to converge. So yes, a whole bunch of calculus and geometry
all got developed first before anybody ever knew they NEEDED to
care about infinite set theory. But the point is, to the extent that
it got developed RIGHT, it got developed axiomatically.

Even today, mathematicians like NW do productive
work without even knowing set theory, or the axioms of set theory, and
this clearly shows that the whole field is an unnecessary appendix.

Dip***: you can't tell the difference between an appendix
and a foundation. What you have said makes about as much
sense as saying: "Game developers write massively complicated
pieces of graphics software without knowing anything about integrated
circuits or registers and accumulators; that just proves that the
whole field of microcode is an unnecessary appendix."
It proves nothing of the kind. It proves that as long as other
people are maintaining the integrity of the lower levels, YOU
are free to focus on the HIGHER ones. The fact that you write
your computer code in HTML instead of machine language does
not mean that machine language is irrelevant. It is in fact
what your code gets translated into and executed as.

.

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