Re: A Guide for Cantor Cranks
RussellE
> Russell, I'd like your permission to use the above in a class on set
> theory. My current thought is to use it as a homework assignment, with
> 20 marks given for identifying and refuting each provably incorrect item.
Please do.
Of course, you can't give credit for answers like
"it works in base 4". And, your students aren't
allowed to use induction. Induction assumes
there is a set of all natural numbers.
As my second argument proves, no such set exists.
> > 3) Claim 1.000... =/= 0.999...
> > This has nothing to do with Cantor, but will
> > always start a flame war. Generate the maximum
> > number of responses by claiming this proves
> > general relativity is wrong.
>
> Didn't you get the memo? -- GR is fine, but this disproves the
> Copenhagen interpretation of quantum mechanics.
What?
Isn't it well known God has a gambling problem?
And hates cats?
> > Cross post to as many groups as possible.
>
> No no no! Only post to relevant ones. I suggest
> alt.psychology.personality .
OK
>
> --
> ---------------------------
> | BBB b \ Barbara at LivingHistory stop co stop uk
> | B B aa rrr b |
> | BBB a a r bbb | Quidquid latine dictum sit,
> | B B a a r b b | altum videtur.
> | BBB aa a r bbb |
> ------------------------------
Russell
- The universe is one dimensional
Androcles
"RussellE" <reas...@gmail.com> wrote in message
news:5916f49f-1dbb-4c95...@q40g2000prh.googlegroups.com...
>
> Russell, I'd like your permission to use the above in a class on set
> theory. My current thought is to use it as a homework assignment, with
> 20 marks given for identifying and refuting each provably incorrect item.
Please do.
Of course, you can't give credit for answers like
"it works in base 4". And, your students aren't
allowed to use induction. Induction assumes
there is a set of all natural numbers.
As my second argument proves, no such set exists.
> > 3) Claim 1.000... =/= 0.999...
> > This has nothing to do with Cantor, but will
> > always start a flame war. Generate the maximum
> > number of responses by claiming this proves
> > general relativity is wrong.
>
> Didn't you get the memo? -- GR is fine, but this disproves the
> Copenhagen interpretation of quantum mechanics.
What?
Isn't it well known God has a gambling problem?
======================================
No, it was Einstein that said "God does not play dice".
God said "Yes I do, Einstein is just a poor loser; he's the
one that won't play."
MoeBlee
> Induction assumes
> there is a set of all natural numbers.
There is induction on finite ordinals even if there is not a set of
all the finite ordinals (natural numbers). See, e.g., Suppes
'Axiomatic Set Theory'.
MoeBlee
Marshall
Whatever system or logic we are using, we always have
available the technique of case analysis.
In a universe where the only values that exist are those
that derive from constructors (as is the case with algebraic
data types,) we can use exhaustive case analysis on
constructors as a proof technique, with each case
parameterized by the arguments of the particular
constructor; this is called "structural induction." I haven't
seen it described as a special case of case analysis,
but it so qualifies.
The usual two constructors for the natural numbers are "zero"
and "successor to some natural number". If we take
structural induction on these two constructors and partially
apply them to structural induction, we get (ordinary)
induction.
Thus, induction is just a special case of structural induction,
which is itself just a form of case analysis. These things
all work whether or not there is a set of all natural numbers,
or whether there is any distinction between "potential" and
"actual", or what-have-you.
Marshall
PS. It's possible that I've used terms from sufficiently many
disciplines as to render this post unreadable.
Richard Harter
<marshal...@gmail.com> wrote:
>On Mar 11, 8:32=A0am, MoeBlee <jazzm...@hotmail.com> wrote:
>> On Mar 10, 9:53=A0pm, RussellE <reaste...@gmail.com> wrote:
>>
>> > Induction assumes
>> > there is a set of all natural numbers.
>>
>> There is induction on finite ordinals even if there is not a set of
>> all the finite ordinals (natural numbers). See, e.g., Suppes
>> 'Axiomatic Set Theory'.
>
>Whatever system or logic we are using, we always have
>available the technique of case analysis.
>
>In a universe where the only values that exist are those
>that derive from constructors (as is the case with algebraic
>data types,) ...
Well, that is the sticking point, isn't it. How do you propose to
establish "the only values that exist are those..." within axiomatic
set theory?
Marshall
> On Fri, 11 Mar 2011 09:56:18 -0800 (PST), Marshall
> <marshall.spi...@gmail.com> wrote:
>
> >> > Induction assumes there is a set of all natural numbers.
>
> >> There is induction on finite ordinals even if there is not a set of
> >> all the finite ordinals (natural numbers). See, e.g., Suppes
> >> 'Axiomatic Set Theory'.
>
> >Whatever system or logic we are using, we always have
> >available the technique of case analysis.
>
> >In a universe where the only values that exist are those
> >that derive from constructors (as is the case with algebraic
> >data types,) ...
>
> Well, that is the sticking point, isn't it.
It is indeed, which is why I specifically called it out.
> How do you propose to
> establish "the only values that exist are those..." within axiomatic
> set theory?
I do not so propose, especially since, if I understand correctly,
such is impossible. Do you have any ideas?
On the other hand, this difficulty seems to me to be just
another of the difficulties that arise out of thinking of
theories as coming before models, when it seems to me
it is better to think of them as "above" or abstracted from
models.
Marshall
Graham Cooper
Can anyone explain why there are a half dozen (mainstream) branches of
mathematics, none of which have produced a single practical useful
formula?
These are some of the foundations of various branches of mathematics.
A program that examines itself, if it halts then continues?
A number that is different to all other listed numbers, only by an
explicit infinite clause that splices the opposing digit of all
listed infinite numbers?
A statement that asserts "you can't prove me"?
A *finite* sized algorithm that gives the maximum output length of
*any* sized algorithm?
A set that doesn't contain itself, yet contains all sets that don't
contain themselves?
----------
Maybe it's Jim's Lemma that's at fault?
For all x in X, y not= x. Therefore, y not in X.
It's certainly trivial to find a counterexample working only in N with
X=N.
Richard Harter
<marshal...@gmail.com> wrote:
>On Mar 11, 1:54=A0pm, c...@tiac.net (Richard Harter) wrote:
>> On Fri, 11 Mar 2011 09:56:18 -0800 (PST), Marshall
>> <marshall.spi...@gmail.com> wrote:
>>
>> >> > Induction assumes there is a set of all natural numbers.
>>
>> >> There is induction on finite ordinals even if there is not a set of
>> >> all the finite ordinals (natural numbers). See, e.g., Suppes
>> >> 'Axiomatic Set Theory'.
>>
>> >Whatever system or logic we are using, we always have
>> >available the technique of case analysis.
>>
>> >In a universe where the only values that exist are those
>> >that derive from constructors (as is the case with algebraic
>> >data types,) ...
>>
>> Well, that is the sticking point, isn't it.
>
>It is indeed, which is why I specifically called it out.
>
>
>> How do you propose to
>> establish "the only values that exist are those..." within axiomatic
>> set theory?
>
>I do not so propose, especially since, if I understand correctly,
>such is impossible. Do you have any ideas?
You are right, it is impossible.
>
>On the other hand, this difficulty seems to me to be just
>another of the difficulties that arise out of thinking of
>theories as coming before models, when it seems to me
>it is better to think of them as "above" or abstracted from
>models.
I'm not sure that either of us understands what you are saying in that
paragraph. Be that as it may, the concept of the set of all subsets
of the set of integers has problems. The notion is seductive - after
all, we feel that there is no difficulty with the notion of the power
set of a finite set - there is a simple procedure for constructing
said power set. The continuum, however, is a different matter. That
"all" is irretrievably fuzzy.
Graham Cooper
> paragraph. Be that as it may, the concept of the set of all subsets
> of the set of integers has problems. The notion is seductive - after
> all, we feel that there is no difficulty with the notion of the power
> set of a finite set - there is a simple procedure for constructing
> said power set. The continuum, however, is a different matter. That
> "all" is irretrievably fuzzy
What pattern can the set of all computer programs miss?
The Omega-Halt sequence?
Define a computer program that examines it's own halt value
and keeps looping if it says halt and vice versa?
That's the ONLY obstacle to computing infinite powersets.
n e P(m) IFF UTM(m,n)=1
UTM(m,n) is the mth computer program with input tape n.
fishfry
c...@tiac.net (Richard Harter) wrote:
This has nothing to do with "procedures." The power set of N exists by
the power set axiom. Constructability in the sense of algorithms and
computer science is not required by set theory.
In fact the term "constructable" in set theory means something else, as
in Godel's constructible universe. Given a set such as N, its power set
exists by the power set axiom; and therefore all the members of the
power set are constructible by definition. Note that this is a totally
different meaning of "constructible" than people are thinking about when
they talk about being able to define a set via an algorithm or procedure.
fishfry
fishfry <BLOCKSPA...@your-mailbox.com> wrote:
I just realized that I may be confused about the constructible universe.
Evidently given a set, the power set in the constructible universe
consists of only those subsets definable by a formula. So that doesn't
include the entire conceptual power set.
I wonder if someone can straighten me out about this.
Richard Harter
<BLOCKSPA...@your-mailbox.com> wrote:
That's the essence of the matter. Godel's L is a minimal universe
satisfying ZF. L isn't the only countable model for ZF; there are
many. The trouble with the "entire conceptual power set" is that you
can't guarantee it with any finite set of axioms.
Aatu Koskensilta
> That's the essence of the matter. Godel's L is a minimal universe
> satisfying ZF. L isn't the only countable model for ZF; there are
> many.
As usually understood L, G�del's constructible universe, is not
countable -- it is a proper class. L is minimal in the sense that it
is contained in any inner model of set theory. (There is also a
countable ordinal alpha such that L_alpha is the minimal standard
model of ZF.)
> The trouble with the "entire conceptual power set" is that you can't
> guarantee it with any finite set of axioms.
What sort of guarantees do you have in mind?
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Richard Harter
<aatu.kos...@uta.fi> wrote:
>c...@tiac.net (Richard Harter) writes:
>
>> That's the essence of the matter. Godel's L is a minimal universe
>> satisfying ZF. L isn't the only countable model for ZF; there are
>> many.
>
> As usually understood L, Gödel's constructible universe, is not
>countable -- it is a proper class. L is minimal in the sense that it
>is contained in any inner model of set theory. (There is also a
>countable ordinal alpha such that L_alpha is the minimal standard
>model of ZF.)
Right. Thanks for being precise.