# Cardinality continuum is aleph_1

Discussion in 'Recreational Math' started by William Elliot, Apr 30, 2006.

1. ### William ElliotGuest

By Cohen's work CH is independent of ZFC.
Thus we're free to chose CH or not.

Thus by Occam's razor CH, the
cardinality of the continuum is aleph_1

Is GCH also independent of ZFC ?
Then yah, GCH by Occam's razor.
Thus we can finish up some theorems, prove others directly and
go on to other notions lacking that immense bother.

Why clutter your abacus with myraids of uselessly colored
beads on an hallucinatorial rod between the 1's and the 10's?

--
Occam also allows the convenient disappearance of
inaccessible cardinals and gods.

IIRC

The denial of inaccessible cardinals is consistence with ZFC. There cannot
be a proof that the existence of inaccessible cardinals is consistence
with set theory. Thus I and Occam, on the firm ground of ZFC consistence,
affirm down with inaccessibles!

William Elliot, Apr 30, 2006

2. ### Arturo MagidinGuest

[.snip sophistry.]
Yes. But C is not independent of ZF+GCH.

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin

Arturo Magidin, Apr 30, 2006

3. ### William ElliotGuest

No problem, I'm a prochoice mathematician.

William Elliot, Apr 30, 2006
4. ### Gene Ward SmithGuest

Only if you exclude aleph_0, which seems prejudicial. Since aleph_0 is
strongly inaccessible unless you deliberately exclude it, and since the
class of all sets behaves like a strongly inaccessible cardinal,
excluding them seems dubious. This is particularly true since we want
there to be strongly inaccessible cardinals, as we want to do things
like work inside of Grothendieck universes for purposes such as doing
algebraic geometry, providing models of set theory, or making the
surreal numbers into a field.

Moreover, unless there is something to prevent large cardinals from
existing, they ought to exist. What stops them? Why should there not be
cardinals which behave like proper classes for many purposes?

Gene Ward Smith, Apr 30, 2006
5. ### William ElliotGuest

Oh yes, just like we want fairy tales, are why we crave elves, gnomes,
batmen, etc. Oh come on, congressional and UN investigation of White
House war crimes, are much more preferable tales to be told the world.

BTW, have you noticed how big those gaps in the surreal numbers are?
Because, as explained above, they come with no warranty and just like gods
and goddesses, are too big for their britches. Just imagine what a world
this would be without big inaccessible gods to argue and fight about.

William Elliot, Apr 30, 2006
6. ### Stephen Montgomery-SmithGuest

I have always seen Occam's razor as a helpful guide rather than a hard
and fast principle.

Stephen Montgomery-Smith, Apr 30, 2006
7. ### Gene Ward SmithGuest

I think applying it to what is postulated by mathematics is a
misapplication of the idea. Mathematical structures are possibilities,
and you can't wave a magical ontological wand and make possibilities
impossible.

Gene Ward Smith, Apr 30, 2006
8. ### William ElliotGuest

Enuf is enough, it's time for a shave, for an end to those incessantly
itchy long hairs. Speak ye of faerie tails? I'll tell you a tale or two.
Once upon a when within a neighborhood of then, there was a mathematician
who had grown long hair ideas longer than the long line, Conway IIRC,
who's very long long line was thicker and denser than any ever before.
Yet upon examination by Dr. Occam IIRC, was found tiny tiny gaps
everywhere which, upon the stresses of flights into fancy, would proof too
fragile and brittle for safe trips. Instead, Rev. Occam IIRC, recommended
Conway's long long line with tiny fine gaps everywhere be used in the
kitchen as a strainer or sieve, to filter out infinitesimals for
non-standard analysis preparations. Anyway, anyway, just a nanosec...
Occam! Occam! Will you sweep up those hairs you shaved off?

William Elliot, May 1, 2006
9. ### RupertGuest

Why does CH follow from Occam's razor rather than ~CH?

Rupert, May 1, 2006
10. ### dkfjdkljGuest

But why do you suppose "they" are possible? Is it some kind of magical
ontological possibility wand? Merely proving an assertion is
consistent with other axioms does not mean that it is possible (in the
apparently absolute sense you are using it).

dkfjdklj, May 1, 2006
11. ### William ElliotGuest

Because CH mops up a mess of cardinals between aleph_0 and 2^aleph_0, rubs
them out in fact so they'll go away and not bother us. Bother us, for
example, requiring longer proofs to show cardinality of a set is >= c when
it's quit apparent the set is uncountable.

GCH is even nicer. aleph_xi = beth_xi settling the nether lands with nice
orderly guys instead of scads of unrulely sorts arguing who's bigger or
smaller and what's there in between that nobody can show yet believe there
are, can or will be some.

Same with CH, quit talking about life on Mars and show us
your unconstructive wonder between aleph_0 and 2^aleph_0.

William Elliot, May 1, 2006
12. ### RupertGuest

~CH requires postulating a cardinal between aleph_0 and 2^aleph_0. But
CH requires postulating a bijection between 2^aleph_0 and aleph_1. It's
not clear to me why one violates Occam's razor more than the other.

Rupert, May 1, 2006
13. ### Dave SeamanGuest

Because CH mops up a mess of cardinals between aleph_0 and 2^aleph_0, rubs
them out in fact so they'll go away and not bother us. Bother us, for
example, requiring longer proofs to show cardinality of a set is >= c when
it's quit apparent the set is uncountable.[/QUOTE]
According to Occam's razor, it is supposed to be assumptions, not
consequences, that we are to avoid multiplying without necessity.

Dave Seaman, May 1, 2006
14. ### William ElliotGuest

They're possible, all you need is axiom and lo and behold, there they are.
They're there if you want them and for those who do, they're fun. Like
pack rats for whom lots of stuff is fun. Others with a less lavish sense
of intellectual fun, prefer thoughts Amish style.

So I put it to ya'll, aleph_omega0 practically has no practical use except
as jump off point to greater fanstasies aleph_omega1, aleph_epsilon_0
for ever and ever, aleph_omega_epsilon_omega_epsilon_omega_epsilon_...

William Elliot, May 1, 2006
15. ### William ElliotGuest

~CH puts more members on the board of directors.
CH revokes the applications of additional board members.

Already most board members have nothing to do but
to look big and aleph_(epsilon_omega) impressive.

William Elliot, May 1, 2006
16. ### William ElliotGuest

CH is an example where the statistical direct proportion of axioms to
results is locally an inverse proportion. It's important not to multiply
and fill the mind with cluttering consequences as for example, we could cut
down on number of assumptions to just one simple axiom that proves almost
everything we want and even perhaps more, more than we want.

William Elliot, May 1, 2006
17. ### NathanGuest

CH has lots of interesting mathematical consequences. But so do many
axioms that are inconsistent with CH. All of these are still
mathematics. Math is a big discipline; there's room for all sort of
interests.

Nathan, May 1, 2006
18. ### Gene Ward SmithGuest

It's supposed to be entities according to Occam, but that gets extended
to assumptions of a particular kind, which have nothing to do with
mathematics. Trying to apply Occam's razor to mathematics is a category
mistake, in that mathematics is not an aspect of the natural world we
are attempting to explain via some model.

Gene Ward Smith, May 2, 2006