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From: David Libert on 27 Jun 2010 13:21
Transfer Principle (lwalke3(a)lausd.net) writes:
> On Jun 26, 7:51=A0pm, Tim Little <t...(a)little-possums.net> wrote:
>> On 2010-06-26, R. Srinivasan <sradh...(a)in.ibm.com> wrote:
>> > The theory ZF-Inf+~Inf clearly proves ~Inf ("Infinite sets do not
>> > exist").
>> Actually ~Inf does not assert "Infinite sets do not exist". =A0It only
>> asserts "there does not exist a successor-closed set containing the
>> empty set".
>
> This has come up time and time again. I myself have claimed that
> the theory ZF-Infinity+~Infinity proves that every set is finite,
> and someone (usually MoeBlee or Rupert) points out that this
> theory only proves that there's no _successor-inductive_ set
> containing 0, not that there is no infinite set.
>
> And every time this comes up, I want to say _fine_ -- so if
> ZF-Inf+~Inf _doesn't_ prove that every set is finite, then there
> should exist a model M of ZF-Inf+~Inf in which "there is an
> infinite set" is true, even though "there exists a set containing
> 0 that is successor-inductive" is clearly false (assuming, of
> course, that ZF is itself consistent), just as the fact that ZFC
> doesn't prove CH implies that there is a model of ZFC in which
> CH is false (once again, assuming that ZF is itself consistent).
>
> Yet no one seems to accept the existence of this model M.
>
> Either this model M exists, or ZF-Inf+~Inf really does prove that
> every set is finite. There are no other possibilities.
>
> So let's settle this once and for all. Assuming that ZF is
> consistent, I ask:
>
> 1. Is there a proof in ZF-Inf+~Inf that every set is finite?
> 2. Does there exist a model M of ZF-Inf+~Inf in which "there
> is an infinite set" is true?
>
> Notice that exactly one of these questions has a "yes" answer
> and exactly one has a "no" answer. (Actually, come to think
> of it, since the base theory is ZF and not ZFC, it's possible
> that the answer to 1. is "yes" if by "finite" we mean one
> type of finite, say Dedekind finite, and "no" if we mean some
> other type of finite. In this case, I'd like to know which
> types of finite produce a "yes" answer.)
>
> If 1. is "yes," then I hope that I will never again see a post
> claiming that ZF-Inf+~Inf doesn't prove that every set is in
> fact finite. In fact, I'll go as far as to suggest that if 1.
> is "yes," then those who claim that ZF-Inf+~Inf doesn't prove
> that every set is finite deserve to be called five-letter
> insults -- if posters are going to call those who deny the
> proof of Cantor's Theorem by five-letter insults, then those
> who deny the proof of "every set is finite" in ZF-Inf+~Inf
> also ought to be called the same.
>
> If 2. is "yes," then what I'd like to know is how can I take
> _advantage_ of this fact? Suppose I want to consider a theory,
> based on ZF-Inf, which actually proves that an infinite set
> exists, yet also proves that no successor-inductive set
> containing 0 exists.
>
> In the current Tony Orlow thread, there is a discussion about
> whether TO is defining N+ to be a successor-inductive set. It
> is possible that the theory that I mentioned above might be
> useful to discussing TO's ideas.
>
> But of course, we can't proceed until we know, once and for
> all, whether ZF-Inf+~Inf proves every set to be finite or not.


I posted a couple of articles in an old thread related to the
questions above, I will give references below.

One issue in all this is how to define infinite in the posing of
the question.

In usual ZF or ZFC, in the presence of the usual axiom of
infinity, we can prove there exists a smallest set having as
member emptyset and closed under the successor operation
on von Neumann ordinals.

(The axiom of infinity does not directly assert the existence
of such a set. It asserts directly that there is a set having
emptyset as memeber and closed under trhe von Neumann successor
operation. It does not directly assert there is a smallest such
set. But that axiom of infinity together with the separation
axiom proves as a theorem there is a smallest such set.)

We then define this smallest such set to be the set omega.

The usual definition of finiteness in ordinary ZF or ZFC
is having von Neumann cardinaity a member of omega. Infinite
is defined as not finite in that sense.

These definitions rest on the axiom of infinity itself, so
it at least bears some discussion whether to try working with
this definition or some alternatives.

One alternative pair of definitions would be Dedekind finite
and Dedekind infinite. See my references below for more details.

In my references below I also give another definition of finite,
not presupposing the usual axiom of infinity.

With these definitions, we can get various rephrasings of the
question, that could be more suitable for this theory.

My articles especially concentrate on Dedekind finite. But
from the discussion there you can also extract information
about the other cases.

The articles:

[1] David Libert "Axiom of Infinity (AxF) in ZF"
sci.logic Aug 26, 2003
http://groups.google.com/group/sci.logic/msg/8b55a452fdf641d9


[2] David Libert "Axiom of Infinity (AxF) in ZF"
sci.logic Aug 29, 2003
http://groups.google.com/group/sci.logic/msg/9417324657c5b242



--
David Libert ah170(a)FreeNet.Carleton.CA
From: Jesse F. Hughes on 29 Jun 2010 08:30
herbzet <herbzet(a)gmail.com> writes:

> "Jesse F. Hughes" wrote:
>
>> But, Walker, you really have the wrong impression of me. I come to
>> sci.math mostly to read the cranks. I'm not proud of that fact
>
> *I* am proud of you, that you would make this startling announcement.

Coincidentally, I am reading /Idiot America/ by Charlie Pierce. The
book has much to do with cranks, though not of the mathematical sort so
much as the political and other sorts. There's a vaguely relevant
passage.

The value of the crank is in the effort that it takes either to refute
what the crank is saying, or to assimilate it into the mainstream. In
either case, political and cultural imaginations expand. Intellectual
horizons expand.

Now, contrary to Walker's fantasy, none of the cranks here have
anything worth "assimilating" into mathematics, of course, but there is
something to be said with the effort required to refute a bad argument.
Of course, no one here really thinks that any argument will effectively
end a crank's pursuit of nonsense, but that is nonetheless the
intellectually interesting bit.

Of course, I'm really here for lower entertainment. I want posts about
the Hammer, about how surrogate factoring moves the stock market, about
the most influential mathematicians on the planet. But still I pretend
to care about arguments, if only for appearance's sake.

--
Jesse F. Hughes

"Most of my research is irreducibly complex."
-- James S. Harris
From: herbzet on 29 Jun 2010 21:51


"Jesse F. Hughes" wrote:

> Of course, I'm really here for lower entertainment. I want posts about
> the Hammer, about how surrogate factoring moves the stock market, about
> the most influential mathematicians on the planet. But still I pretend
> to care about arguments, if only for appearance's sake.

Well, I didn't come here for the low comedy -- that's simply not an idea
that had occurred to me. But, it being made explicitly an option -- I
suppose it's a valid choice, even one of some value.

If I have any reservation, it would just be that I don't think that
encouraging crankery for its entertainment value is such a great idea,
all things considered. Not that you're suggesting that.

Nevertheless, thanks for enlarging the realm of possibilities.

--
hz
From: herbzet on 29 Jun 2010 21:51


"Jesse F. Hughes" wrote:

> Coincidentally, I am reading /Idiot America/ by Charlie Pierce. The
> book has much to do with cranks, though not of the mathematical sort so
> much as the political and other sorts.

Sounds interesting, but -- please don't get me started.

--
hz
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