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From: Lee Davidson on 10 Jun 2010 23:20
It’s likely that some of you have had thoughts like mine before, so I
am directing this to those of you who haven’t taken to time to think
of things from the perspective I will develop here.

I’ve seen some threads in sci.logic and sci.math initiated by posters
that were confident that they have found flaws in Cantor’s diagonal
argument etc. I have to admit that I couldn’t make that much sense out
of their postings, but perhaps I didn’t study these postings closely
enough. I have just so much time.

At any rate, all proofs of the uncountability of the reals that I know
of make use of some machinery of set theory (e.g. ZFC) and/or assume
the reals form a complete metric space.

If we define the reals using ZFC as, e.g., Dedekind cuts, we can of
course prove that the reals form a complete metric space and all
proofs of uncountability carry through.

However, suppose we begin with what I would like to call the idealized
empirical continuum. This is the sort of thing you draw pictures of in
10th grade plane geometry, you know, points, lines etc. in two
dimensions.

Consider an idealized one dimensional continuum, that is, an infinite
straight line.

When you say that this forms a complete metric space, intuitively,
this means that the line is solid, that is, has no gaps.

Now, empirically, in order to investigate whether this line is really
solid, equip yourself with microscopes, 2x, 4x, 8x, 16x, etc., through
all the powers of 2.

Now suppose, for example, that you have reason to believe that there
are points on this line corresponding to all the rationals. (Don’t ask
me how you know this.)

Now try to use your microscopes to determine whether there is a point
on the line corresponding to the square root of 2. Suppose you can
make an infinite number of observations using your microscopes, in an
attempt to home in on the square root of 2. What do you see in your
microscopic observations? This line always looks solid.

But does this tell you there is a point there, corresponding to the
square root of 2? No. All this tells you is at most that, between any
two points on the line, the line looks (and maybe is) solid. That is,
that you have a dense ordering.

In other words, empirically – if you can perform infinitely many
microscopic observations – all you can verify is that the line is a
dense ordering.

But that it is a dense ordering does not imply, logically, that it is
a complete metric space.

But the situation is even worse than I have depicted. You can’t even
prove, empirically, that points on the line corresponding to all
rationals exist. Though you might be able to base this on some
considerations that seem self-evident from your spatial intuition –
e.g., translation invariance, isotropy, reflection invariance –
whatever. I won’t go into any details here; I’m trying to think about
what’s reasonable here.

The general point here: sure, you can show me a Cauchy sequence. But
how to you know that the non-existence of a point to which that
sequence converges constitutes a “gap”?

Anyway, I suspect that you need to bring in the machinery of set
theory in a significant way here. But those who disbelieve in the
ontology of set theory and want to base real number theory on our
spatial intuition alone have some reason for skepticism.

Me, I don’t really have any trouble with the ontology of set theory.
But, from a foundational point of view, I see a problem here.

By the way, you can certainly get the square root of 2, and lots else
besides, if you can build up an empirical theory, in my sense, of
ruler and compass construction. But, of course, this still leaves you
with, at most, a proof of a countable two dimensional continuum.

But where I’m stuck here is: how do you know that non-parallel lines
intersect in a common point? If you can’t know this, you can’t even
have ruler and compass construction.
From: Rupert on 11 Jun 2010 00:38
On Jun 11, 1:20 pm, Lee Davidson <l...(a)meta5.com> wrote:
> It’s likely that some of you have had thoughts like mine before, so I
> am directing this to those of you who haven’t taken to time to think
> of things from the perspective I will develop here.
>
> I’ve seen some threads in sci.logic and sci.math initiated by posters
> that were confident that they have found flaws in Cantor’s diagonal
> argument etc. I have to admit that I couldn’t make that much sense out
> of their postings, but perhaps I didn’t study these postings closely
> enough. I have just so much time.
>
> At any rate, all proofs of the uncountability of the reals that I know
> of make use of some machinery of set theory (e.g. ZFC) and/or assume
> the reals form a complete metric space.
>
> If we define the reals using ZFC as, e.g., Dedekind cuts, we can of
> course prove that the reals form a complete metric space and all
> proofs of uncountability carry through.
>
> However, suppose we begin with what I would like to call the idealized
> empirical continuum. This is the sort of thing you draw pictures of in
> 10th grade plane geometry, you know, points, lines etc. in two
> dimensions.
>
> Consider an idealized one dimensional continuum, that is, an infinite
> straight line.
>
> When you say that this forms a complete metric space, intuitively,
> this means that the line is solid, that is, has no gaps.
>
> Now, empirically, in order to investigate whether this line is really
> solid, equip yourself with microscopes, 2x, 4x, 8x, 16x, etc., through
> all the powers of 2.
>
> Now suppose, for example, that you have reason to believe that there
> are points on this line corresponding to all the rationals. (Don’t ask
> me how you know this.)
>
> Now try to use your microscopes to determine whether there is a point
> on the line corresponding to the square root of 2. Suppose you can
> make an infinite number of observations using your microscopes, in an
> attempt to home in on the square root of 2. What do you see in your
> microscopic observations? This line always looks solid.
>
> But does this tell you there is a point there, corresponding to the
> square root of 2? No. All this tells you is at most that, between any
> two points on the line, the line looks (and maybe is) solid. That is,
> that you have a dense ordering.
>
> In other words, empirically – if you can perform infinitely many
> microscopic observations – all you can verify is that the line is a
> dense ordering.
>
> But that it is a dense ordering does not imply, logically, that it is
> a complete metric space.
>
> But the situation is even worse than I have depicted. You can’t even
> prove, empirically, that points on the line corresponding to all
> rationals exist. Though you might be able to base this on some
> considerations that seem self-evident from your spatial intuition –
> e.g., translation invariance, isotropy, reflection invariance –
> whatever. I won’t go into any details here; I’m trying to think about
> what’s reasonable here.
>
> The general point here: sure, you can show me a Cauchy sequence. But
> how to you know that the non-existence of a point to which that
> sequence converges constitutes a “gap”?
>
> Anyway, I suspect that you need to bring in the machinery of set
> theory in a significant way here. But those who disbelieve in the
> ontology of set theory and want to base real number theory on our
> spatial intuition alone have some reason for skepticism.
>
> Me, I don’t really have any trouble with the ontology of set theory.
> But, from a foundational point of view, I see a problem here.
>
> By the way, you can certainly get the square root of 2, and lots else
> besides, if you can build up an empirical theory, in my sense, of
> ruler and compass construction. But, of course, this still leaves you
> with, at most, a proof of a countable two dimensional continuum.
>
> But where I’m stuck here is: how do you know that non-parallel lines
> intersect in a common point? If you can’t know this, you can’t even
> have ruler and compass construction.

I don't think I understand what question you're asking.

Are you asking a question about physical reality?
From: MoeBlee on 11 Jun 2010 14:39
On Jun 10, 10:20 pm, Lee Davidson <l...(a)meta5.com> wrote:

> At any rate, all proofs of the uncountability of the reals that I know
> of make use of some machinery of set theory (e.g. ZFC) and/or assume
> the reals form a complete metric space.

Usually, I find that the reals are characterized as the carrier set of
a complete ordered field (and all complete ordered fields are
isomorphic).

And we don't need full ZFC to prove that any such carrier set is
uncountable. Just a portion of Z set theory and an intuitionistically
acceptable portion of the logic suffice.

As to tjhe line you mentioned. If I'm not mistaken, it is usually an
axiom of ordinary geometry (the ruler postulate) that every line is
1-1 with the set of real numbers. I don't know that there is a way to
physically "examine" to confirm such a thing.

MoeBlee






>
> If we define the reals using ZFC as, e.g., Dedekind cuts, we can of
> course prove that the reals form a complete metric space and all
> proofs of uncountability carry through.
>
> However, suppose we begin with what I would like to call the idealized
> empirical continuum. This is the sort of thing you draw pictures of in
> 10th grade plane geometry, you know, points, lines etc. in two
> dimensions.
>
> Consider an idealized one dimensional continuum, that is, an infinite
> straight line.
>
> When you say that this forms a complete metric space, intuitively,
> this means that the line is solid, that is, has no gaps.
>
> Now, empirically, in order to investigate whether this line is really
> solid, equip yourself with microscopes, 2x, 4x, 8x, 16x, etc., through
> all the powers of 2.
>
> Now suppose, for example, that you have reason to believe that there
> are points on this line corresponding to all the rationals. (Don’t ask
> me how you know this.)
>
> Now try to use your microscopes to determine whether there is a point
> on the line corresponding to the square root of 2. Suppose you can
> make an infinite number of observations using your microscopes, in an
> attempt to home in on the square root of 2. What do you see in your
> microscopic observations? This line always looks solid.
>
> But does this tell you there is a point there, corresponding to the
> square root of 2? No. All this tells you is at most that, between any
> two points on the line, the line looks (and maybe is) solid. That is,
> that you have a dense ordering.
>
> In other words, empirically – if you can perform infinitely many
> microscopic observations – all you can verify is that the line is a
> dense ordering.
>
> But that it is a dense ordering does not imply, logically, that it is
> a complete metric space.
>
> But the situation is even worse than I have depicted. You can’t even
> prove, empirically, that points on the line corresponding to all
> rationals exist. Though you might be able to base this on some
> considerations that seem self-evident from your spatial intuition –
> e.g., translation invariance, isotropy, reflection invariance –
> whatever. I won’t go into any details here; I’m trying to think about
> what’s reasonable here.
>
> The general point here: sure, you can show me a Cauchy sequence. But
> how to you know that the non-existence of a point to which that
> sequence converges constitutes a “gap”?
>
> Anyway, I suspect that you need to bring in the machinery of set
> theory in a significant way here. But those who disbelieve in the
> ontology of set theory and want to base real number theory on our
> spatial intuition alone have some reason for skepticism.
>
> Me, I don’t really have any trouble with the ontology of set theory.
> But, from a foundational point of view, I see a problem here.
>
> By the way, you can certainly get the square root of 2, and lots else
> besides, if you can build up an empirical theory, in my sense, of
> ruler and compass construction. But, of course, this still leaves you
> with, at most, a proof of a countable two dimensional continuum.
>
> But where I’m stuck here is: how do you know that non-parallel lines
> intersect in a common point? If you can’t know this, you can’t even
> have ruler and compass construction.

From: George Greene on 11 Jun 2010 17:40
On Jun 10, 11:20 pm, Lee Davidson <l...(a)meta5.com> wrote:
> At any rate, all proofs of the uncountability of the reals that I know
> of make use of some machinery of set theory (e.g. ZFC)

OK.

> and/or assume the reals form a complete metric space.

Please. You can't MIX the two of these!

> If we define the reals using ZFC as, e.g., Dedekind cuts, we can of
> course prove that the reals form a complete metric space and all
> proofs of uncountability carry through.

You have completely jumped the shark here.
The ZFC proof of the uncountability of the reals neither
KNOWS NOR CARES ANYTHING WHATSOEVER
about THE REALS! The ZFC proof of Cantor's theorem is a proof about
ALL SETS!!
It says that EVERY set has the property of not being as big as its
powerset!
EVERY set! Infinite sets in general and the reals in particular ARE
NOT SPECIAL
in this regard!
From: Rupert on 11 Jun 2010 17:41
On Jun 12, 4:39 am, MoeBlee <jazzm...(a)hotmail.com> wrote:
> On Jun 10, 10:20 pm, Lee Davidson <l...(a)meta5.com> wrote:
>
> > At any rate, all proofs of the uncountability of the reals that I know
> > of make use of some machinery of set theory (e.g. ZFC) and/or assume
> > the reals form a complete metric space.
>
> Usually, I find that the reals are characterized as the carrier set of
> a complete ordered field (and all complete ordered fields are
> isomorphic).
>
> And we don't need full ZFC to prove that any such carrier set is
> uncountable. Just a portion of Z set theory and an intuitionistically
> acceptable portion of the logic suffice.
>
> As to tjhe line you mentioned. If I'm not mistaken, it is usually an
> axiom of ordinary geometry (the ruler postulate) that every line is
> 1-1 with the set of real numbers. I don't know that there is a way to
> physically "examine" to confirm such a thing.
>
> MoeBlee
>

There is a weak theory called RCA_0, which is conservative over PRA,
discussed in "Subsystems of Second Order Arithmetic", in which it can
be proved that no countable sequence of real numbers contains all real
numbers.
>
>
>
>
> > If we define the reals using ZFC as, e.g., Dedekind cuts, we can of
> > course prove that the reals form a complete metric space and all
> > proofs of uncountability carry through.
>
> > However, suppose we begin with what I would like to call the idealized
> > empirical continuum. This is the sort of thing you draw pictures of in
> > 10th grade plane geometry, you know, points, lines etc. in two
> > dimensions.
>
> > Consider an idealized one dimensional continuum, that is, an infinite
> > straight line.
>
> > When you say that this forms a complete metric space, intuitively,
> > this means that the line is solid, that is, has no gaps.
>
> > Now, empirically, in order to investigate whether this line is really
> > solid, equip yourself with microscopes, 2x, 4x, 8x, 16x, etc., through
> > all the powers of 2.
>
> > Now suppose, for example, that you have reason to believe that there
> > are points on this line corresponding to all the rationals. (Don’t ask
> > me how you know this.)
>
> > Now try to use your microscopes to determine whether there is a point
> > on the line corresponding to the square root of 2. Suppose you can
> > make an infinite number of observations using your microscopes, in an
> > attempt to home in on the square root of 2. What do you see in your
> > microscopic observations? This line always looks solid.
>
> > But does this tell you there is a point there, corresponding to the
> > square root of 2? No. All this tells you is at most that, between any
> > two points on the line, the line looks (and maybe is) solid. That is,
> > that you have a dense ordering.
>
> > In other words, empirically – if you can perform infinitely many
> > microscopic observations – all you can verify is that the line is a
> > dense ordering.
>
> > But that it is a dense ordering does not imply, logically, that it is
> > a complete metric space.
>
> > But the situation is even worse than I have depicted. You can’t even
> > prove, empirically, that points on the line corresponding to all
> > rationals exist. Though you might be able to base this on some
> > considerations that seem self-evident from your spatial intuition –
> > e.g., translation invariance, isotropy, reflection invariance –
> > whatever. I won’t go into any details here; I’m trying to think about
> > what’s reasonable here.
>
> > The general point here: sure, you can show me a Cauchy sequence. But
> > how to you know that the non-existence of a point to which that
> > sequence converges constitutes a “gap”?
>
> > Anyway, I suspect that you need to bring in the machinery of set
> > theory in a significant way here. But those who disbelieve in the
> > ontology of set theory and want to base real number theory on our
> > spatial intuition alone have some reason for skepticism.
>
> > Me, I don’t really have any trouble with the ontology of set theory.
> > But, from a foundational point of view, I see a problem here.
>
> > By the way, you can certainly get the square root of 2, and lots else
> > besides, if you can build up an empirical theory, in my sense, of
> > ruler and compass construction. But, of course, this still leaves you
> > with, at most, a proof of a countable two dimensional continuum.
>
> > But where I’m stuck here is: how do you know that non-parallel lines
> > intersect in a common point? If you can’t know this, you can’t even
> > have ruler and compass construction.- Hide quoted text -
>
> - Show quoted text -

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