"What are some adverse consequences that arise from requiring the axiom of choice?"

[Khong Dong]

Fyi., https://qr.ae/pv47Kc

[Jeff Barnett]

On 7/5/2022 10:22 PM, Khong Dong wrote:
> Fyi., https://qr.ae/pv47Kc

Interesting citation. However, there are several stumbles when stating
one of the simple equivalents of the Axiom. I think a correct version is

For all sets X whose elements are nonempty pairwise disjoint
sets, there exists a set C = C(X) where C intersection x is a
singleton set for all x in X.

The words "nonempty" and "disjoint" are necessary but are several times
omitted in the citation. This is one of the simplest statements of full
Choice and I like it because it simply asserts the existence of a set
that only depends on one you have in hand (X). Thus Choice is just like
almost all of the ZF axioms - it just tells about the existence of a set
based on the existence of other sets. So the axiom stated this way just
helps prove the existence of sets that must be in your domain of
discourse in much the same way the other axioms do.

I have recommended, several times in this forum, a book that in the end
takes literally hundreds of Choice axioms and related theorems and tells
which ones imply which others as well as identifying a zillion open
problems in re the Choice Axioms. (Note, there is a doubly indexed set
of such axioms - indices are cardinality of the x in X and cardinality
of X; and there are Choice axioms that don't fit this scheme.) The book
is principally a history book that documents the battles Zermelo and
Cantor vs most of the rest of the planet. The battle started in the late
1800s and poked into the first quarter of the twentieth century. The
book continues history for fifty years or so past that.

Gregory H. Moore, "Zermelo's Axiom of Choice It's Origins, Development,
and Influence"

Amazon, USA, has a paperback version for $20.07, Prime and that's a
bargain. It's a well written 400+ page book. It's a mystery story in
some ways where you'll run into some of your favorite characters:
Russell, Hilbert, Zorn, Borel, Godel, Cohen, Skolem, Dedekind, Church,
Tarski, Turin, Sierpinski, and on and on and on.
--
Jeff Barnett

[Ross A. Finlayson]

Excuse me, that was well-written.

[Jeff Barnett]

Damn. I promise not to do it again.
--
Jeff Barnett

[Ross A. Finlayson]

Once there was a giant thread here about choice
and what is lesser or "countable choice", where the
first usual contradiction from choice is "closed,
doesn't exist a choice function", over some domain,
then whether are not functions are total over domain.

Then, that coming right up, then "countable choice"
is still allowed, because to close domains is entire,
total, or giant iteration over function, choice function.

That the entire well-ordered set exists or ORD, is why
for example choice and well-ordering go together.

When you pick Skolem and Cohen out from the
above, with Zermelo (or rather Fraenkel) closing
after Russell and Goedel again closing, and re-opening,
from an "arithmetization", for choice, is for
usual regular deciding results for example when to quit.

It's about same as well-ordering principle.

It's usual that making an arithmetization that
there is the external or "outside the theory, if
then back inside the theory, as it were", has the
apparatus of provision besides existence, of
choice functions, for example as how providing
examples of well-orderings go, given they exist.

(According to ordinals or "all regularity in set theory",
or trichotomy and so on.)

So, well-order the reals.

The usual "a well-ordering exists because of the
uniformity of sets as a class, but it's absolutely
not the reals their normal ordering", is about
how that's so for under arithmetic, that the reals
are in their normal ordering first, that "the order
is not an example of a well-ordering".

Anyways I well-order the reals [0,1] that being
countable make for an arithmetization that is
in the same ordinal order as its choice function,
or a well-ordering.

Anyways there are giant threads here about
well-ordering, what is most usually always part
of "the set theory ZFC the foundations of mathematics".
(Add descriptive set theory after standard field, ....)

Or, there are bigger numbers.

[Ross A. Finlayson]

Seems "Ramsey cardinal" is coming up these days.

https://en.wikipedia.org/wiki/Ramsey_cardinal

Of course large cardinals aren't cardinals, nor sets for that matter.

V = L is a usual determinacy.

[Jeff Barnett]

Cantor was convinced that the well ordering principal was an obvious law
of thought and that it could be demonstrated easily. In the beginning he
wasn't aware that using a version of choice was anything special. When
Choice was identified in his proof and many other places, he thought
there was nothing special about it either; After all, everybody used it!
However once Choice was pulled out explicitly and put to work, a large
fraction of working mathematicians were horrified at some of the
results. It took Zermelo then Frankel to do a decent formalization
within set theory so consequences with and without Choice could be
understood.

One of the things I found really interesting was that a great deal of
the early research in the area was pushed by those working in real
analysis. Real analysis typically treats the reals like urelements
(atoms or simply non sets) and the rest of the world was trying to
establish a bases for mathematics through set theory without urelements.
Typically real analysis (aka point-set topology) asked questions about
what you could conclude given separation axioms and the cardinality of
those non-set elements.

Since you can embed reals and sets of reals in a set only theory, nobody
worried too much about this. However it turns out that different things
are true when urelements are allowed and when they are not. Some of
these differences involved how Choice could be woven into set theory.

The history of all this is fascinating and took so many twists and turns
that only a scholar in this area could come close to remembering even a
fraction of the events and confrontations. The comical aspects where
provided by mathematicians like Borel and Lebesgue who fought tooth and
nail against the axiom while their own work was total nonsense without
it, i.e., it was used in the their proofs the first day of class than
used even more as the days went by.
--
Jeff Barnett

[Ross A. Finlayson]

Ha. Comprehension in the naive is really quite usual and very well-explored.

The finite combinatorics of course, make for that while lots of naive
theorems immediately allow usual deduction for example about bounds,
actually constructively enumerating the bounds, is itself a task in bounds.

Those worked all the way out make closures and arithmetics,
models of arithmetic and arithmetizations. Then mostly those
are as simply defined as orders, in class comprehensions, even
in the same orders, as the usual naive deductive apparatus about it.

Cantor's paradise might be his domain
but Knuth's paradise is a system of counting arguments.

It nice to have both paradises.

Basically, in set theory, "types" are in set theory, and "classes" are out.
It's "group nouns", for what sets model like ordinals, that basically
"ordering theory is actually primitive in set theory".

In "Borel vs. Combinatorics" is for examples of conjectures not
necessarily just "decided" but for example "independent", the
existence any "standard, regular, complete model of integers",
that for example itself is independent the theory, besides axioms.

For example "Knuth's 2-normal and absolutely normal or 2-distributed
and *-distributed, sequences, occupy half of a binary Cantor space,
according to a partitioning, in sets". That's along the lines of a conjecture
independent ordinary set theory, and about what exists in extra-ordinary set theory.

I found an identity like Stirling's in very usual concrete terms in this,
about number theory, function theory, and those being objects for
models of arithmetic in set theory, and even primary, primitive, ur-elements.

I.e. after "half the sequences in Cantor space are half 0's and half 1's",
and some differential parameterization, I wrote out some equations
to compute the factorial, which isn't even decided by ordinary set theory.

About "why is choice/well-ordering an axiom", it's primary for ordering theory,
which "decides all the ordinary".

It's like "why sets instead of parts", "Brentano has a theory of parts".

The "existence results according to axioms" and "structure results
an example after enumeration", are very different in their strengths
as independent, proving an example exists and providing an example.

The only relation in set theory is "elt", but: sets are defined by their membership.

So, being able to prove "well-ordering the reals exists" but not "here is
an example, but according to comprehension a subset exists their
normal order, contradiction of each to next containing rationals",
it is a usual balk, that is for point-set topologists to establish in
the real-valued, where continuum theory is primary. (Elementary.)


Anyways the old "Factorial/Exponential Identity, Infinity" thread involves
both the undecideable and independent, "Borel vs. Combinatorics".

Besides for ordering then number theory for function theory
that "infinitesimals are integrable and sum to 1, even though
standardly modeled sum to 1/2", makes for "re-Vitali-izing measure
theory", both demonstrates "well-ordering the reals" and builds
out from set theory its usual objects the ordinals, numbers, functions,
and so on, all ordinary and decided.

Then "point-sets live in a space".

That's a set theory, ....

The unbounded finite combinatorics doesn't need an axiom, it's
as defined all together, already complete its constructions.
But, unrestricted comprehensions also makes for one. Then,
is for that unbounded finite combinatorics, is for a Cantor's domain,
Russell's types, Goedel's arithmetic, still independent that there
is one, thus according to comprehension, what exists.


Anyways decideability and independence are two different things.