Re: how to list all of the real numbers

On Aug 5, 10:16 pm, lwal...@xxxxxxxxx wrote:

The word "crank" is thrown around often on this newsgroup,

I don't see it "thrown around" that much. Usually, I see it applied
appropriately.

and there is much controversy to what the word means.

Where is that controversy posted?

It is
very often used to describe someone who does not fully accept
a standard set theory, such as ZFC.

I've never seen it used that way.

Ironically, the word is
most often used to describe the notorious poster JSH, yet
although JSH has contradicted standard number theory, ring
theory, etc., I have never seen him challenge set theory.

What's ironic about that? One can be a crank about subjects other than
set theory.

Among the nonstandard mathematicians who do contradict
standard set theory, the most commonly challenged axiom is,
of course, the Axiom of Infinity.

Once in a while, a crank may get specific enough to even mention an
axiom, but usually cranks are not so specific. (And, by the way, I do
not hold that one is a crank merely for rejecting the axiom of
infinity.)

Although ZFC with Infinity
replaced by its negation is consistent if ZFC itself is,

Yes, we know that.

By the way, if you replace the axiom of infinity with its negation,
you don't need the 'C' in 'ZFC'.

and
there is a branch of mathematics called "finitism,"

I don't know whether its a branch or better described as a philosophy
or approach or school. And there are different forms of it too.

it is
the general belief of most standard mathematicians that
anyone who denies Infinity in the 21st century merits the
label "crank."

I don't know how you determined that. It's not my impression that
merely choosing to work only with finite sets gets one to be called a
'crank'. Perhaps there are such instances, but I've not seen them or
at least not very many.

Even denying Axiom of Choice, considered controversial a few
short years ago, may possibly make the Axiom's opponent a
borderline "crank."

Where have you seen that? Maybe it's around, but I haven't seen it.

I've seen a few threads challenging
Choice recently, and the standard mathematicians who reply
in such threads defend Choice fully.

So what if they do? That doesn't entail that they claim that a mere
rejection of the axiom of choice makes one a crank.

There are only a few well-known axioms which one may safely
deny without being labeled a "crank." One such axiom is the
Continuum Hypothesis.

I've never seen the continuum hypothesis as an axiom.

Also, again, I've not seen all this 'crank'-calling just from people
choosing not to adopt certain axioms.

An adherent of ZFC+CH is not a "crank,"
nor is an adherent of ZFC+not CH. Also, there are the various
large cardinal axioms, some of which are supported by
different axioms. But try denying AC, or especially Infinity,
and standard mathematicians will not hesitate to use the
"crank" label.

Who are all these mathematicians who do all this non-hesitating
'crank'-calling just on the basis of one choosing to eschew choice and/
or infinity?

MoeBlee


.

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