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Cantor's diagonal arguments are invalid
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Julio Di Egidio
Jun 29, 2015, 12:17:44 PM6/29/15
to
Thesis: Cantor's diagonal arguments for the uncountability of the real numbers are all invalid.
Proof: The diagonalisation of a standard list is a standard sequence; a real number is the limit of a standard sequence; hence there is no sense in asking whether the anti-diagonal of a standard list of real numbers belongs to that list. QED
Hint: Formalise a Cantor's diagonal argument and you will get a type error.
Julio
Proof: The diagonalisation of a standard list is a standard sequence; a real number is the limit of a standard sequence; hence there is no sense in asking whether the anti-diagonal of a standard list of real numbers belongs to that list. QED
Hint: Formalise a Cantor's diagonal argument and you will get a type error.
Julio
George Greene
Jun 29, 2015, 1:37:42 PM6/29/15
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On Monday, June 29, 2015 at 12:17:44 PM UTC-4, Julio Di Egidio wrote:
> Hint: Formalise a Cantor's diagonal argument and you will get a type error.
Idiot: THERE ARE NO types in ZFC. Or, equivalently, in Z, EVERYTHING is ONE
> Hint: Formalise a Cantor's diagonal argument and you will get a type error.
and the SAME type: SET.
It is comical to hear you say "formalize Cantor's diagonal argument". It was
BORN formal. Nobody beyond Zermelo in 1903 NEEDS to formalIZE it.
WE KNOW how TO PROVE this theorem formally, and "a type error" is something
that CANNOT EVEN OCCUR in the theory in which we are proving it.
> The diagonalisation of a standard list is a standard sequence;
Sets very famously DO NOT impose any particular ORDER on their elements!
Sets are NOT lists, although some lists CAN be encoded as sets.
There are many different methods of encoding and they are all mere conventions.
It is a fact about sets in general that they just DON'T DO order.
In the set {red,blue,green}, NONE of the 3 elements is FIRST! You can make
6 different lists using all&only the members of that set, but the set ITSELF
is NOT a list and the diagonalization proof does NOT depend IN ANY way on
the concept of LIST! It is a proof ABOUT ANY&EVERY set, that it cannot be
bijected onto its powerset. SET, NOT list.
Dumbass.
Ross A. Finlayson
Jun 29, 2015, 1:41:34 PM6/29/15
to
You are wanting the
completing the diamond
in the ontology.
Is it so defining a 1.0:
real analysis?
Digital sets are uncounting
of their powersets (infinite
ones as is so usual). Yes,
this is all usually in terms
of the infinite terms of these
species, and finite.
So, the direct statement of
the result, is what is valid
(in ZF set theory with
cardinals of regular standard
infinities, in their types).
completing the diamond
in the ontology.
Is it so defining a 1.0:
real analysis?
Digital sets are uncounting
of their powersets (infinite
ones as is so usual). Yes,
this is all usually in terms
of the infinite terms of these
species, and finite.
So, the direct statement of
the result, is what is valid
(in ZF set theory with
cardinals of regular standard
infinities, in their types).
Julio Di Egidio
Jun 29, 2015, 2:04:27 PM6/29/15
to
On Monday, June 29, 2015 at 6:37:42 PM UTC+1, George Greene wrote:
> On Monday, June 29, 2015 at 12:17:44 PM UTC-4, Julio Di Egidio wrote:
> > Hint: Formalise a Cantor's diagonal argument and you will get a type error.
>
> Idiot: THERE ARE NO types in ZFC. Or, equivalently, in Z, EVERYTHING is ONE
> and the SAME type: SET.
You are an idiot, George, just missing the macroscopic point: Use any formalisation you like, the argument is still flawed, logically already. Not to mention that Cantor did not use ZFC. Not to mention that a real number *is* the limit of a sequence, not just a sequence, even in ZFC: i.e. modulo structural equivalence.
> On Monday, June 29, 2015 at 12:17:44 PM UTC-4, Julio Di Egidio wrote:
> > Hint: Formalise a Cantor's diagonal argument and you will get a type error.
>
> Idiot: THERE ARE NO types in ZFC. Or, equivalently, in Z, EVERYTHING is ONE
> and the SAME type: SET.
That said (and I can only scratch the surface, of your nonsense):
> It is comical to hear you say "formalize Cantor's diagonal argument".
> It was BORN formal.
<snip>
Then you do not even know what formal means.
> SET, NOT list.
> Dumbass.
Dumb ass, it just fails in any theory: dumb ass.
Go back fighting the cranks George, as that seems your only inspiration.
Julio
Virgil
Jun 30, 2015, 2:20:49 AM6/30/15
to
In article <7cdade36-3dd3-4b73...@googlegroups.com>,
of the uncountability of the set of real numbers. Perhqps Julio can give
is specifics.
Cantor's anti-diagonal proof shows that any set of all functions from |N
to any set having more than one element is uncountable!
> Not to mention that a real number
> *is* the limit of a sequence, not just a sequence, even in ZFC: i.e. modulo
> structural equivalence.
But there is no more than one decimal digit sequence for any irrational
number and no more than two for any rational number, so if the
countability of the set of decimal digit sequences following a decimal
point is uncountable, as it is, the set of reals is too!
--
Virgil
"Mit der Dummheit kampfen Gotter selbst vergebens." (Schiller)
Julio Di Egidio <ju...@diegidio.name> wrote:
> On Monday, June 29, 2015 at 6:37:42 PM UTC+1, George Greene wrote:
> > On Monday, June 29, 2015 at 12:17:44 PM UTC-4, Julio Di Egidio wrote:
> > > Hint: Formalise a Cantor's diagonal argument and you will get a type
> > > error.
> >
> > Idiot: THERE ARE NO types in ZFC. Or, equivalently, in Z, EVERYTHING is
> > ONE
> > and the SAME type: SET.
>
> You are an idiot, George, just missing the macroscopic point: Use any
> formalisation you like, the argument is still flawed, logically already.
I see no such logical flaw in either of Cantor's two independent proofs
> On Monday, June 29, 2015 at 6:37:42 PM UTC+1, George Greene wrote:
> > On Monday, June 29, 2015 at 12:17:44 PM UTC-4, Julio Di Egidio wrote:
> > > Hint: Formalise a Cantor's diagonal argument and you will get a type
> > > error.
> >
> > Idiot: THERE ARE NO types in ZFC. Or, equivalently, in Z, EVERYTHING is
> > ONE
> > and the SAME type: SET.
>
> You are an idiot, George, just missing the macroscopic point: Use any
> formalisation you like, the argument is still flawed, logically already.
of the uncountability of the set of real numbers. Perhqps Julio can give
is specifics.
Cantor's anti-diagonal proof shows that any set of all functions from |N
to any set having more than one element is uncountable!
> Not to mention that a real number
> *is* the limit of a sequence, not just a sequence, even in ZFC: i.e. modulo
> structural equivalence.
number and no more than two for any rational number, so if the
countability of the set of decimal digit sequences following a decimal
point is uncountable, as it is, the set of reals is too!
--
Virgil
"Mit der Dummheit kampfen Gotter selbst vergebens." (Schiller)
Julio Di Egidio
Jun 30, 2015, 3:42:27 PM6/30/15
to
On Monday, June 29, 2015 at 6:41:34 PM UTC+1, Ross A. Finlayson wrote:
>
> You are wanting the
> completing the diamond
> in the ontology.
>
> Is it so defining a 1.0:
> real analysis?
>
> Digital sets are uncounting
> of their powersets (infinite
> ones as is so usual). Yes,
> this is all usually in terms
> of the infinite terms of these
> species, and finite.
Finite, so nope: within strict finitism, N is not even a set, it is a class; and the powerset P(N) of N is itself a class, collecting all *finite* subsets of N. There P(N) is obviously equipotent with N.
>
> You are wanting the
> completing the diamond
> in the ontology.
>
> Is it so defining a 1.0:
> real analysis?
>
> Digital sets are uncounting
> of their powersets (infinite
> ones as is so usual). Yes,
> this is all usually in terms
> of the infinite terms of these
> species, and finite.
> So, the direct statement of
> the result, is what is valid
> (in ZF set theory with
> cardinals of regular standard
> infinities, in their types).
Julio
Julio Di Egidio
Jun 30, 2015, 7:35:05 PM6/30/15
to
On Tuesday, June 30, 2015 at 7:20:49 AM UTC+1, Virgil wrote:
> In article <7cdade36-3dd3-4b73...@googlegroups.com>,
> Julio Di Egidio <j***@diegidio.name> wrote:
> In article <7cdade36-3dd3-4b73...@googlegroups.com>,
<snip>
> > Use any formalisation you like,
> > the argument is still flawed, logically already.
>
> I see no such logical flaw in either of Cantor's two independent proofs
> of the uncountability of the set of real numbers. Perhqps Julio can give
> is specifics.
Will try:
> > the argument is still flawed, logically already.
>
> I see no such logical flaw in either of Cantor's two independent proofs
> of the uncountability of the set of real numbers. Perhqps Julio can give
> is specifics.
Cantor's First Proof is not the concern here (although, IMO, it fails for the same essential reason), I am focusing on the class of so called "diagonal argument"s here. And the issue is already logical (the very problem statement is broken! so to speak), hence it does not matter the formalisation. In particular, it does not matter whether we take a constructive approach as in Cantor's Diagonal Argument vs. an axiomatic approach as in Cantor's Theorem vs. any approach really, as long as it is a "diagonal argument": a real number *is* what it is regardless.
> Cantor's anti-diagonal proof shows that any set of all functions from |N
> to any set having more than one element is uncountable!
As for what diagonalisation actually is, the mathematical details are up to the mathematicians, but few things seem logical to me (few informal thoughts): i) When we diagonalise a complete list of rational numbers, we certainly do *not* get a real number (as explained). ii) The sequence we get rather is a (kind of) paradoxical construction, of which we can prove, but just per definition (!), that it differs from every entry in the list, yet we cannot (!) prove that it is not in the list by any inductive method (equivalently, the search cannot halt). iii) That specifically means: the formal system cannot prove it, just the logic requires it. Then, whichever the (logical) resolution there (of course there is a resolution), the mathematics either does not halt, or does not compile already, otherwise we need an "extended" mathematics (actual infinities, mathematically).
> > a real number
> > *is* the limit of a sequence, not just
> > a sequence, even in ZFC: i.e. modulo
> > structural equivalence.
>
> But there is no more than one decimal digit sequence for any irrational
> number and no more than two for any rational number,
Julio
Virgil
Jun 30, 2015, 9:24:49 PM6/30/15
to
In article <50b4624a-fd59-42b4...@googlegroups.com>,
Every real number between 0 and 1 is representable by a function from |N
to {1,2,3,4,5,6,7,8,9}
> That is the
> key point, that your proposition is meaningless.
If it were meaningless, nits like you and WM would not even bother to
fight so futilely against it!
>
> As for what diagonalisation actually is, the mathematical details are up to
> the mathematicians, but few things seem logical to me (few informal
> thoughts): i) When we diagonalise a complete list of rational numbers, we
> certainly do *not* get a real number (as explained).
The finite decimal subsequences of ANY non-terminating sequence of
decimal digits following a decimal point form a monotone increasing
bounded sequence of rationals so must have a real number as their limit!
ii) The sequence we get
> rather is a (kind of) paradoxical construction, of which we can prove, but
> just per definition (!), that it differs from every entry in the list, yet we
> cannot (!) prove that it is not in the list by any inductive method
> (equivalently, the search cannot halt).
In any logically coherent system, anything that provably differs from
every member of a list is not listed in that list!
> iii) That specifically means: the
> formal system cannot prove it, just the logic requires it.
The I, for one, put my trust in the logic. But
or a set of all rationals, or anything like infinite sequences to take
limits of.
have a set of all naturals, much less a set of all rationals or a set of
all reals.
> On Tuesday, June 30, 2015 at 7:20:49 AM UTC+1, Virgil wrote:
> > In article <7cdade36-3dd3-4b73...@googlegroups.com>,
> > Julio Di Egidio <j***@diegidio.name> wrote:
> <snip>
> > > Use any formalisation you like,
> > > the argument is still flawed, logically already.
> >
> > I see no such logical flaw in either of Cantor's two independent proofs
> > of the uncountability of the set of real numbers. Perhqps Julio can give
> > is specifics.
>
> Will try:
>
> Cantor's First Proof is not the concern here (although, IMO, it fails for the
> same essential reason), I am focusing on the class of so called "diagonal
> argument"s here. And the issue is already logical (the very problem
> statement is broken! so to speak), hence it does not matter the
> formalisation. In particular, it does not matter whether we take a
> constructive approach as in Cantor's Diagonal Argument vs. an axiomatic
> approach as in Cantor's Theorem vs. any approach really, as long as it is a
> "diagonal argument": a real number *is* what it is regardless.
>
> > Cantor's anti-diagonal proof shows that any set of all functions from |N
> > to any set having more than one element is uncountable!
>
> A real number is just not a function from N to an alphabet. \
> > In article <7cdade36-3dd3-4b73...@googlegroups.com>,
> > Julio Di Egidio <j***@diegidio.name> wrote:
> <snip>
> > > Use any formalisation you like,
> > > the argument is still flawed, logically already.
> >
> > I see no such logical flaw in either of Cantor's two independent proofs
> > of the uncountability of the set of real numbers. Perhqps Julio can give
> > is specifics.
>
> Will try:
>
> Cantor's First Proof is not the concern here (although, IMO, it fails for the
> same essential reason), I am focusing on the class of so called "diagonal
> argument"s here. And the issue is already logical (the very problem
> statement is broken! so to speak), hence it does not matter the
> formalisation. In particular, it does not matter whether we take a
> constructive approach as in Cantor's Diagonal Argument vs. an axiomatic
> approach as in Cantor's Theorem vs. any approach really, as long as it is a
> "diagonal argument": a real number *is* what it is regardless.
>
> > Cantor's anti-diagonal proof shows that any set of all functions from |N
> > to any set having more than one element is uncountable!
>
Every real number between 0 and 1 is representable by a function from |N
to {1,2,3,4,5,6,7,8,9}
> That is the
> key point, that your proposition is meaningless.
fight so futilely against it!
>
> As for what diagonalisation actually is, the mathematical details are up to
> the mathematicians, but few things seem logical to me (few informal
> thoughts): i) When we diagonalise a complete list of rational numbers, we
> certainly do *not* get a real number (as explained).
decimal digits following a decimal point form a monotone increasing
bounded sequence of rationals so must have a real number as their limit!
ii) The sequence we get
> rather is a (kind of) paradoxical construction, of which we can prove, but
> just per definition (!), that it differs from every entry in the list, yet we
> cannot (!) prove that it is not in the list by any inductive method
> (equivalently, the search cannot halt).
every member of a list is not listed in that list!
> iii) That specifically means: the
> formal system cannot prove it, just the logic requires it.
> Then, whichever
> the (logical) resolution there (of course there is a resolution), the
> mathematics either does not halt, or does not compile already, otherwise we
> need an "extended" mathematics (actual infinities, mathematically).
Without actual infinities, one does not even have a set of all naturals
> the (logical) resolution there (of course there is a resolution), the
> mathematics either does not halt, or does not compile already, otherwise we
> need an "extended" mathematics (actual infinities, mathematically).
or a set of all rationals, or anything like infinite sequences to take
limits of.
>
> > > a real number
> > > *is* the limit of a sequence, not just
> > > a sequence, even in ZFC: i.e. modulo
> > > structural equivalence.
> >
> > But there is no more than one decimal digit sequence for any irrational
> > number and no more than two for any rational number,
>
> Hardly a sequitur, anyway essentially wrong: a rational fractional expansion
> consists of two finite strings, anti-period and period, each with arbitrary
> length. Now, in the finite
The whole point is that in your pseudo-finite world one can not even
> > > a real number
> > > *is* the limit of a sequence, not just
> > > a sequence, even in ZFC: i.e. modulo
> > > structural equivalence.
> >
> > But there is no more than one decimal digit sequence for any irrational
> > number and no more than two for any rational number,
>
> Hardly a sequitur, anyway essentially wrong: a rational fractional expansion
> consists of two finite strings, anti-period and period, each with arbitrary
> length. Now, in the finite
have a set of all naturals, much less a set of all rationals or a set of
all reals.
George Greene
Jul 1, 2015, 1:41:00 AM7/1/15
to
On Monday, June 29, 2015 at 2:04:27 PM UTC-4, Julio Di Egidio wrote:
> You are an idiot, George,
No, you are.
> You are an idiot, George,
> just missing the macroscopic point:
> Use any formalisation you like, the argument is still flawed,
logically flawed: you have to show some step in the proof that is a mis-
application of an inference rule. THAT IS ALL.
All the other "flaws" you WRONGLY THINK you know are just proofs that
YOU'RE STUPID.
> logically already.
You should be banned from using the word.
> Not to mention that Cantor did not use ZFC.
has exactly zero effect on the existence or non-existence of a valid
proof of a theorem from some axioms.
> Not to mention that a real number *is* the limit of a sequence,
George Greene
Jul 1, 2015, 1:41:41 AM7/1/15
to
On Monday, June 29, 2015 at 2:04:27 PM UTC-4, Julio Di Egidio wrote:
> Dumb ass, it just fails in any theory: dumb ass.
Dumbass, you can't JUST SAY that a PROOF
> Dumb ass, it just fails in any theory: dumb ass.
fails! You HAVE to PROVE that!
WM
Jul 1, 2015, 5:37:01 AM7/1/15
to
Regards, WM
Alan Smaill
Jul 1, 2015, 5:45:04 AM7/1/15
to
Virgil <VIR...@VIRGIL.com> writes:
> In article <50b4624a-fd59-42b4...@googlegroups.com>,
> Julio Di Egidio <ju...@diegidio.name> wrote:
>
>> On Tuesday, June 30, 2015 at 7:20:49 AM UTC+1, Virgil wrote:
>> >
>> > Cantor's anti-diagonal proof shows that any set of all functions from |N
>> > to any set having more than one element is uncountable!
>>
>> A real number is just not a function from N to an alphabet. \
>
> Every real number between 0 and 1 is representable by a function from |N
> to {1,2,3,4,5,6,7,8,9}
But then there are then multiple representations of the same real --
> In article <50b4624a-fd59-42b4...@googlegroups.com>,
> Julio Di Egidio <ju...@diegidio.name> wrote:
>
>> On Tuesday, June 30, 2015 at 7:20:49 AM UTC+1, Virgil wrote:
>> >
>> > Cantor's anti-diagonal proof shows that any set of all functions from |N
>> > to any set having more than one element is uncountable!
>>
>> A real number is just not a function from N to an alphabet. \
>
> Every real number between 0 and 1 is representable by a function from |N
> to {1,2,3,4,5,6,7,8,9}
so this is not what reals "really" are.
This seems to be the objection -- it is easily enough dealt with,
but it is a fair point to make. After all, the claim is that the reals
are uncountable, not the representations of reals.
--
Alan Smaill
WM
Jul 1, 2015, 6:37:32 AM7/1/15
to
Am Mittwoch, 1. Juli 2015 11:45:04 UTC+2 schrieb Alan Smaill:
> Virgil <VIR...@VIRGIL.com> writes:
>
> > In article <50b4624a-fd59-42b4...@googlegroups.com>,
> > Julio Di Egidio <ju...@diegidio.name> wrote:
> >
> >> On Tuesday, June 30, 2015 at 7:20:49 AM UTC+1, Virgil wrote:
> >> >
> >> > Cantor's anti-diagonal proof shows that any set of all functions from |N
> >> > to any set having more than one element is uncountable!
> >>
> >> A real number is just not a function from N to an alphabet. \
> >
> > Every real number between 0 and 1 is representable by a function from |N
> > to {1,2,3,4,5,6,7,8,9}
>
> But then there are then multiple representations of the same real --
Not by digit sequences.
> Virgil <VIR...@VIRGIL.com> writes:
>
> > In article <50b4624a-fd59-42b4...@googlegroups.com>,
> > Julio Di Egidio <ju...@diegidio.name> wrote:
> >
> >> On Tuesday, June 30, 2015 at 7:20:49 AM UTC+1, Virgil wrote:
> >> >
> >> > Cantor's anti-diagonal proof shows that any set of all functions from |N
> >> > to any set having more than one element is uncountable!
> >>
> >> A real number is just not a function from N to an alphabet. \
> >
> > Every real number between 0 and 1 is representable by a function from |N
> > to {1,2,3,4,5,6,7,8,9}
>
> But then there are then multiple representations of the same real --
> so this is not what reals "really" are.
>
> This seems to be the objection -- it is easily enough dealt with,
> but it is a fair point to make. After all, the claim is that the reals
> are uncountable, not the representations of reals.
Every real has a finite definition. Every real is the meaning of a finite word. There are only countably many finite words. There are only as many meanings as there are languages. Languages have to be constructed. There is a finite number of languages at every time. Thzere can never be more than countably many reals. QED.
Regards, WM
George Greene
Jul 1, 2015, 10:28:19 AM7/1/15
to
On Tuesday, June 30, 2015 at 2:20:49 AM UTC-4, Virgil wrote:
> I see no such logical flaw in either of Cantor's two independent proofs
> of the uncountability of the set of real numbers. Perhqps Julio can give
> us specifics.
> I see no such logical flaw in either of Cantor's two independent proofs
> of the uncountability of the set of real numbers. Perhqps Julio can give
No, he can't.
The best he can do is something like what Moritz Pasch did to Euclid, is
point out some minor matters of hygiene ("a type error"????) that basically
EVERYbody was making UNTIL Frege/Russell/Zermelo FINALLY DISCOVERED
formalization-as-we-know-it. In the course OF ANY formalization WORTHY
of the name, formalizing Cantor's argument would automatically correct/clean-
up those errors, so that is just all a bunch of irrelevant B.S.
There may be some value somewhere in finding out just how the argument
that Cantor originally presented was less than perfectly logical,
but THAT is missing the point.
THE POINT IS that by NOW, we HAVE some set theories, and
IN ALL of them, there is, PROVABLY, NO bijection between a set and its powerset.
THAT is THE point. EVERYBODY *ELSE*, not us, is missing THE point.
George Greene
Jul 1, 2015, 10:30:23 AM7/1/15
to
On Tuesday, June 30, 2015 at 3:42:27 PM UTC-4, Julio Di Egidio wrote:
> You cannot get from potential to actual infinities
Just STFU.
> You cannot get from potential to actual infinities
Nobody gives a shit about your "potential infinity".
You are assuming that somewhere, somehow, somebody started with a potential
infnity. This is simply not the case. There WERE ALWAYS an acutally infinite
number of natural numbers. In Z, there as an actually infinite set
AT THE BEGINNING, IN THE AXIOM of infinity.
It is consistent with the other axioms.
The fact that you think "potential infinity" even IS A THING,
with respect to Cantor's Theorem, makes YOU a crank.
George Greene
Jul 1, 2015, 10:37:07 AM7/1/15
to
On Tuesday, June 30, 2015 at 7:35:05 PM UTC-4, Julio Di Egidio wrote:
> I am focusing on the class of so called "diagonal argument"s here.
This makes you a crank.
> I am focusing on the class of so called "diagonal argument"s here.
Non-cranks do not have any trouble understanding diagonal arguments.
This is the pons asinorum of this forum and has been, approximately forever.
I've been here almost 30 years now.
> And the issue is already logical
You DO NOT KNOW what "logical" *means*!!
AROUND HERE, "logical" means "starting with a countable signature
for a first-order langauge, and SOME AXIOMS written in that language,
and implying/inferring MORE sentences in that language -- theorems -- using
standard rules of inference"!! THAT is logical! What you are doing is
anything but!
> (the very problem statement is broken! so to speak),
broken. Just because YOUR wings are broken doesn't mean OTHER birds can't fly.
> hence it does not matter the formalisation.
exist. The proofs are valid. Since you cannot attack the validity of the proofs, the best you could do as a lame substitute would be to insist that
the proofs don't match the original argument. But WE DON'T *CARE*.
The proofs, the formalizations, are what WE are STICKING to. They will
resemble Cantor's original presentation in some ways and depart from it in
others. We will continue to view THAT glass as half-FULL and concentrate
on the similarities. The very simple bridge between subsets of N and real
numbers is that every real number has one (or two, in a very few cases
of rationals with even denominators) representation AS A BIT-STRING, and
the 1s and 0s in its binary representation become analogous to the presence
or absence of certain natnums in a subset.
George Greene
Jul 1, 2015, 10:38:05 AM7/1/15
to
On Tuesday, June 30, 2015 at 7:35:05 PM UTC-4, Julio Di Egidio wrote:
> A real number is just not a function from N to an alphabet. QED.
You're a fool, QED.
> A real number is just not a function from N to an alphabet. QED.
Alan Smaill
Jul 1, 2015, 11:00:03 AM7/1/15
to
WM <wolfgang.m...@hs-augsburg.de> writes:
> Am Mittwoch, 1. Juli 2015 11:45:04 UTC+2 schrieb Alan Smaill:
>> Virgil <VIR...@VIRGIL.com> writes:
>>
>> > In article <50b4624a-fd59-42b4...@googlegroups.com>,
>> > Julio Di Egidio <ju...@diegidio.name> wrote:
>> >
>> >> On Tuesday, June 30, 2015 at 7:20:49 AM UTC+1, Virgil wrote:
>> >> >
>> >> > Cantor's anti-diagonal proof shows that any set of all functions
>> >> > from |N to any set having more than one element is uncountable!
>> >>
>> >> A real number is just not a function from N to an alphabet. \
>> >
>> > Every real number between 0 and 1 is representable by a function
>> > from |N to {1,2,3,4,5,6,7,8,9}
>>
>> But then there are then multiple representations of the same real --
>
> Not by digit sequences.
By finite definitions that specify potentially infinite digit sequences.
> Am Mittwoch, 1. Juli 2015 11:45:04 UTC+2 schrieb Alan Smaill:
>> Virgil <VIR...@VIRGIL.com> writes:
>>
>> > In article <50b4624a-fd59-42b4...@googlegroups.com>,
>> > Julio Di Egidio <ju...@diegidio.name> wrote:
>> >
>> >> On Tuesday, June 30, 2015 at 7:20:49 AM UTC+1, Virgil wrote:
>> >> >
>> >> > Cantor's anti-diagonal proof shows that any set of all functions
>> >> > from |N to any set having more than one element is uncountable!
>> >>
>> >> A real number is just not a function from N to an alphabet. \
>> >
>> > Every real number between 0 and 1 is representable by a function
>> > from |N to {1,2,3,4,5,6,7,8,9}
>>
>> But then there are then multiple representations of the same real --
>
> Not by digit sequences.
>> so this is not what reals "really" are.
>>
>> This seems to be the objection -- it is easily enough dealt with,
>> but it is a fair point to make. After all, the claim is that the reals
>> are uncountable, not the representations of reals.
>
> But this point is not supported by the diagonal argument.
Please read what I write.
> Further it is easily contradicted.
> Every real has a finite definition. Every real is the meaning of a
> finite word. There are only countably many finite words. There are
> only as many meanings as there are languages. Languages have to be
> constructed. There is a finite number of languages at every
> time.
This of course fails to contradict the fact that two finite definitions
may refer to the same real. There is a difference between the
potentially infinite set of finite definitions, and the
potentially infinite set of reals; and you have failed to
recognise this distinction.
> Regards, WM
--
Alan Smaill
Virgil
Jul 1, 2015, 11:26:00 AM7/1/15
to
In article <72bc7195-9ab3-4bda...@googlegroups.com>,
In order for an infinite digit sequence to represent a rational number,
it must be "eventually periodic", that is, from some digit position
onward it only repeats a finite sequence of digits. Any such infinite
sequence of digits which does not ever become "eventually periodic"
represents an irrational.
So in order for WM to prove that a Cantor decimal antidiagonal must be
the decimal sequence for a rational number, WM must first prove that it
must be eventually periodic, which WM has not done and cannot do because
it is not true!
At least it is not true anywhere outside of WM's worthless world of
WMytheology.
it must be "eventually periodic", that is, from some digit position
onward it only repeats a finite sequence of digits. Any such infinite
sequence of digits which does not ever become "eventually periodic"
represents an irrational.
So in order for WM to prove that a Cantor decimal antidiagonal must be
the decimal sequence for a rational number, WM must first prove that it
must be eventually periodic, which WM has not done and cannot do because
it is not true!
At least it is not true anywhere outside of WM's worthless world of
WMytheology.
Virgil
Jul 1, 2015, 11:53:18 AM7/1/15
to
In article <beccd093-0a53-40e0...@googlegroups.com>,
Every rational whose denominator is a factor of a power of ten has two
decimal representations, one ending with infinitely many zeros and the
other with infinitely many nines!
expressible as fractions whose denominators are factors of some power of
ten, if the set of decimal representations are uncountable, so is the
set of reals represented! At least everywhere outside of WM's worthless
world of WMytheology
>
> Further it is easily contradicted.
By what?
provably the case!
> Every real is the meaning of a finite word.
Where is that claimed?
As far as I am aware, the field of real numbers is finitely defined, but
in that definition there is no explicit requirement that each real shall
have its own separate finite definition.
WM <wolfgang.m...@hs-augsburg.de> wrote:
> Am Mittwoch, 1. Juli 2015 11:45:04 UTC+2 schrieb Alan Smaill:
> > Virgil <VIR...@VIRGIL.com> writes:
> >
> > > In article <50b4624a-fd59-42b4...@googlegroups.com>,
> > > Julio Di Egidio <ju...@diegidio.name> wrote:
> > >
> > >> On Tuesday, June 30, 2015 at 7:20:49 AM UTC+1, Virgil wrote:
> > >> >
> > >> > Cantor's anti-diagonal proof shows that any set of all functions from
> > >> > |N
> > >> > to any set having more than one element is uncountable!
> > >>
> > >> A real number is just not a function from N to an alphabet. \
> > >
> > > Every real number between 0 and 1 is representable by a function from |N
> > > to {1,2,3,4,5,6,7,8,9}
> >
> > But then there are then multiple representations of the same real --
>
> Not by digit sequences.
WRONG! AGAIN!! AS USUAL!!!
> Am Mittwoch, 1. Juli 2015 11:45:04 UTC+2 schrieb Alan Smaill:
> > Virgil <VIR...@VIRGIL.com> writes:
> >
> > > In article <50b4624a-fd59-42b4...@googlegroups.com>,
> > > Julio Di Egidio <ju...@diegidio.name> wrote:
> > >
> > >> On Tuesday, June 30, 2015 at 7:20:49 AM UTC+1, Virgil wrote:
> > >> >
> > >> > Cantor's anti-diagonal proof shows that any set of all functions from
> > >> > |N
> > >> > to any set having more than one element is uncountable!
> > >>
> > >> A real number is just not a function from N to an alphabet. \
> > >
> > > Every real number between 0 and 1 is representable by a function from |N
> > > to {1,2,3,4,5,6,7,8,9}
> >
> > But then there are then multiple representations of the same real --
>
> Not by digit sequences.
Every rational whose denominator is a factor of a power of ten has two
decimal representations, one ending with infinitely many zeros and the
other with infinitely many nines!
>
> > so this is not what reals "really" are.
> >
> > This seems to be the objection -- it is easily enough dealt with,
> > but it is a fair point to make. After all, the claim is that the reals
> > are uncountable, not the representations of reals.
Since only countably many reals have dual decimal representations, those
> > so this is not what reals "really" are.
> >
> > This seems to be the objection -- it is easily enough dealt with,
> > but it is a fair point to make. After all, the claim is that the reals
> > are uncountable, not the representations of reals.
expressible as fractions whose denominators are factors of some power of
ten, if the set of decimal representations are uncountable, so is the
set of reals represented! At least everywhere outside of WM's worthless
world of WMytheology
>
> But this point is not supported by the diagonal argument.
Since it is otherwise supported, it is still true!
> Further it is easily contradicted.
>
>
> Every real has a finite definition.
Not if there are more reals than there are finite definitions, which is
>
> Every real has a finite definition.
provably the case!
> Every real is the meaning of a finite word.
As far as I am aware, the field of real numbers is finitely defined, but
in that definition there is no explicit requirement that each real shall
have its own separate finite definition.
Virgil
Jul 1, 2015, 12:19:11 PM7/1/15
to
having more things than individual names for them
> Every real is the meaning of a
> finite word.
Certainly no other!
George Greene
Jul 1, 2015, 12:52:28 PM7/1/15
to
On Wednesday, July 1, 2015 at 5:37:01 AM UTC-4, WM wrote:
> Am Mittwoch, 1. Juli 2015 07:41:41 UTC+2 schrieb George Greene:
> > On Monday, June 29, 2015 at 2:04:27 PM UTC-4, Julio Di Egidio wrote:
> > > Dumb ass, it just fails in any theory: dumb ass.
> >
> > Dumbass, you can't JUST SAY that a PROOF
> > fails! You HAVE to PROVE that!
>
> The antidiagonal differs from every irrational number.
No, it doesn't. It may itself be rational or irrational;
> Am Mittwoch, 1. Juli 2015 07:41:41 UTC+2 schrieb George Greene:
> > On Monday, June 29, 2015 at 2:04:27 PM UTC-4, Julio Di Egidio wrote:
> > > Dumb ass, it just fails in any theory: dumb ass.
> >
> > Dumbass, you can't JUST SAY that a PROOF
> > fails! You HAVE to PROVE that!
>
> The antidiagonal differs from every irrational number.
That depends on the construction of the original underlying sequence.
There are no a-priori constraints on the values in the original underlying sequence. ANY sequence of the appropriate size and shape must have an anti-diagonal. Probabilistically,
its anti-diagonal is probably irrational. The anti-diagonal in any case
is NOT ON the list, since it differs from every element that IS on the list.
George Greene
Jul 1, 2015, 1:08:46 PM7/1/15
to
On Wednesday, July 1, 2015 at 5:45:04 AM UTC-4, Alan Smaill wrote:
> But then there are then multiple representations of the same real --
> so this is not what reals "really" are.
That's our problem as architects of the representors, not any actual problem with the representands.
> But then there are then multiple representations of the same real --
> so this is not what reals "really" are.
> This seems to be the objection -- it is easily enough dealt with,
> but it is a fair point to make.
> After all, the claim is that the reals
> are uncountable, not the representations of reals.
IS NOT REMOTELY legitimate! This does NOT become "a fair point to make"
until and unless there is some reason to suspect that the NUMBER OR CARDINALITY
of the representations MIGHT SOMEHOW DIFFER from that of the "actual" reals!
AND THERE IS NOTHING REMOTELY RESEMBLING A SUSPICION of that in the discussion!
SO WHAT if some reals have two representations under this scheme?!?!?
SO *FUCKING* WHAT???!???
In the first place, EVEN IF THEY *ALL* had a finite number of differing
representations, that STILL WOULD NOT INCREASE the cardinality OF ANY
set that WAS INFINITE to begin with ( , *moron* -- NOT you Alan, but anyone
who would be FOOL, and I DO MEAN *F*O*O*L*, enough to raise THAT objection)!
In the second place, THE ONLY reals that wind up with 2 representations
under the relevant encoding are rationals with denominators whose factors are
factors of the base! The rationals
are a VERY SMALL countable slice of the reals, so EVEN doubling THEM will STILL
not increase the cardinality OF ANY infinite set! If doubling ALL of them
doesn't matter, then certainly doubling this thin slice of the rationals with
the right denominator-factors is NOT going to change SQUAT!!
George Greene
Jul 1, 2015, 1:12:50 PM7/1/15
to
On Wednesday, July 1, 2015 at 5:45:04 AM UTC-4, Alan Smaill wrote:
> Virgil <VIR...@VIRGIL.com> writes:
>
> > In article <50b4624a-fd59-42b4...@googlegroups.com>,
> > Julio Di Egidio <ju...@diegidio.name> wrote:
> >
> >> On Tuesday, June 30, 2015 at 7:20:49 AM UTC+1, Virgil wrote:
> >> >
> >> > Cantor's anti-diagonal proof shows that any set of all functions from |N
> >> > to any set having more than one element is uncountable!
Of course, and the right base to use for this IS *2*, because the {0,1}
>
> > In article <50b4624a-fd59-42b4...@googlegroups.com>,
> > Julio Di Egidio <ju...@diegidio.name> wrote:
> >
> >> On Tuesday, June 30, 2015 at 7:20:49 AM UTC+1, Virgil wrote:
> >> >
> >> > Cantor's anti-diagonal proof shows that any set of all functions from |N
> >> > to any set having more than one element is uncountable!
alphabet will help you map to not-being-in, vs. being-in, the relevant SET,
which will allow you to encode reals AS SUBSETS OF N, which IS THE BRIDGE BETWEEN the ZFC version of the general theorem ("no set is as big as its powerset") and the list-of-reals via list-of-digits version, which people
seem to be taking here as the original, even though it isn't "the original", either, really.
> >>
> >> A real number is just not a function from N to an alphabet. \
> >
> > Every real number between 0 and 1 is representable by a function from |N
> > to {1,2,3,4,5,6,7,8,9}
to make this CLEAR, is {0,1}.
George Greene
Jul 1, 2015, 1:19:31 PM7/1/15
to
On Monday, June 29, 2015 at 12:17:44 PM UTC-4, Julio Di Egidio wrote:
>
> Proof:
This is not a proof.
> The diagonalisation of a standard list is a standard sequence;
Of course.
> a real number is the limit of a standard sequence;
Of course.
> hence
You are not using a rule of inference here. Illegal use of the hence.
> there is no sense in asking whether the anti-diagonal
> of a standard list of real numbers belongs to that list.
YOU are the one making the type error.
The things ON the list are standard sequences (of digits).
The anti-diagonal itself is a standard sequence (of digits).
You DO NOT NEED to take ANY LIMITS to pass from the digit-sequence
to the actual real here. You are also making an illegitimate use of
"it makes no sense". It makes no sense to ask whether the anti-diagonal
of any square list is on the list, because OBVIOUSLY IT ISN'T, SO OF COURSE
YOU DID NOT NEED TO ASK, but that does not mean that the assertion is
nonsensical. If you assert that the anti-diagonal of the list is not
on the list, that is banally, trivially, true, and it is shocking that
there exist people cranky enough to dispute it. Similarly, if you suggest
that the anti-diagonal of the list might somehow still be on the list, that
is trivially obviously false and brands you a fool (or at least mistaken)
for suggesting it.
The "list of real numbers" in question here ISN'T a list "of real numbers" -- IT IS a list of their representations according to some scheme or another, but you can't deny that the representations exist or that they represent the reals they represent; everybody is free to choose whatever numerals THEY want -- YOU DON'T GET to object!
>
> Proof:
This is not a proof.
> The diagonalisation of a standard list is a standard sequence;
> a real number is the limit of a standard sequence;
> hence
You are not using a rule of inference here. Illegal use of the hence.
> there is no sense in asking whether the anti-diagonal
> of a standard list of real numbers belongs to that list.
The things ON the list are standard sequences (of digits).
The anti-diagonal itself is a standard sequence (of digits).
You DO NOT NEED to take ANY LIMITS to pass from the digit-sequence
to the actual real here. You are also making an illegitimate use of
"it makes no sense". It makes no sense to ask whether the anti-diagonal
of any square list is on the list, because OBVIOUSLY IT ISN'T, SO OF COURSE
YOU DID NOT NEED TO ASK, but that does not mean that the assertion is
nonsensical. If you assert that the anti-diagonal of the list is not
on the list, that is banally, trivially, true, and it is shocking that
there exist people cranky enough to dispute it. Similarly, if you suggest
that the anti-diagonal of the list might somehow still be on the list, that
is trivially obviously false and brands you a fool (or at least mistaken)
for suggesting it.
The "list of real numbers" in question here ISN'T a list "of real numbers" -- IT IS a list of their representations according to some scheme or another, but you can't deny that the representations exist or that they represent the reals they represent; everybody is free to choose whatever numerals THEY want -- YOU DON'T GET to object!
George Greene
Jul 1, 2015, 1:32:22 PM7/1/15
to
On Tuesday, June 30, 2015 at 3:42:27 PM UTC-4, Julio Di Egidio wrote:
> Finite, so nope: within strict finitism, N is not even a set, it is a class;
Well, DUH, you can certainly define all sets as finite and all infinite
> Finite, so nope: within strict finitism, N is not even a set, it is a class;
classes as proper if you WANT to. That is consistent. It's just not that
EASY -- it's not clear that finitude is even first-order-definable AT ALL.
Even something as strong as PA (if you encode sets as numbers), which makes
a great finite set theory, still cannot actually TELL finite from infinite (the non-standard models have some numbers that are bigger than all naturals).
> and the powerset P(N) of N is itself a class,
> collecting all *finite* subsets of N.
N *itself* IS ALSO a proper class and you can't even APPLY P(.) to it.
In pure set theories like ZFC, the proper classes are not even elements
of the domain.
> There P(N) is obviously equipotent with N.
but that is a very loose use of "obviously equipotent". The kinds of bijections
that prove equipollence will be similarly infinite if you are trying to
prove them between infinite classes, and therefore similarly suspect in their
own right. You have to migrate away from set theory to some sort of class
theory to make THAT work.
George Greene
Jul 1, 2015, 1:34:57 PM7/1/15
to
On Tuesday, June 30, 2015 at 3:42:27 PM UTC-4, Julio Di Egidio wrote:
You have a square list of digits and you produce a row that it is not on it.
After that, YOU CAN go from there to reals "for free", if you have a brain.
You don't seem to know what reals even ARE.
Maybe this will help you. These are some axioms.
"The reals" are "the domain of" ANY -- ANY -- model, WHATSOEVER -- of
THESE axioms. OF COURSE there are "multiple representations".
That NEVER matters.
(addition is associative) (∀a,b,c) a+(b+c)=(a+b)+c
(additive identity) (∃0)(∀a) a+0=a
(additive inverse) (∀a)(∃b) a+b=0
(addition is commutative) (∀a,b) a+b=b+a
(multiplication is associative) (∀a,b,c) a(bc)=(ab)c
(multiplicative identity) (∃1)(∀a) a⋅1=a
(multiplicative inverse) (∀a≠0)(∃b) ab=1
(multiplication is commutative) (∀a,b) ab=ba
(distributive law) (∀a,b,c) a(b+c)=ab+ac
(trichotomy) (∀x)[x<0∨x=0∨x>0]
(positives closed under addition) (∀a>0,b>0)[a+b>0]
(positives closed under multiplication) (∀a>0,b>0)[ab>0]
(least upper bound) (∀S⊆R)[if S has an upper bound, then S has a least upper bound]
Alan Smaill
Jul 1, 2015, 1:55:03 PM7/1/15
to
George Greene <gre...@email.unc.edu> writes:
> On Wednesday, July 1, 2015 at 5:45:04 AM UTC-4, Alan Smaill wrote:
>> But then there are then multiple representations of the same real --
>> so this is not what reals "really" are.
>
> That's our problem as architects of the representors, not any actual
> problem with the representands.
Fine.
> On Wednesday, July 1, 2015 at 5:45:04 AM UTC-4, Alan Smaill wrote:
>> But then there are then multiple representations of the same real --
>> so this is not what reals "really" are.
>
> That's our problem as architects of the representors, not any actual
> problem with the representands.
>> This seems to be the objection -- it is easily enough dealt with,
>
> Not easily enough for julio, apparently.
>
>> but it is a fair point to make.
>
> NO, IT ISN'T!!
>
>> After all, the claim is that the reals
>> are uncountable, not the representations of reals.
>
> IF we are talking about NUMERATION AND CARDINALITY then this objection
> IS NOT REMOTELY legitimate! This does NOT become "a fair point to
> make" until and unless there is some reason to suspect that the NUMBER
> OR CARDINALITY of the representations MIGHT SOMEHOW DIFFER from that
> of the "actual" reals!
Otherwise all we know is that the number of representations
of reals >= the number of reals, and the number
of representations of reals is > than the number of naturals.
Not hard to fix, but necessary for a proof.
--
Alan Smaill
WM
Jul 2, 2015, 8:32:43 AM7/2/15
to
Am Mittwoch, 1. Juli 2015 17:00:03 UTC+2 schrieb Alan Smaill:
> >> But then there are then multiple representations of the same real --
> >
> > Not by digit sequences.
>
> By finite definitions that specify potentially infinite digit sequences.
That is absolutely correct.
> >> But then there are then multiple representations of the same real --
> >
> > Not by digit sequences.
>
> By finite definitions that specify potentially infinite digit sequences.
>
> >> After all, the claim is that the reals
> >> are uncountable, not the representations of reals.
> >
> > But this point is not supported by the diagonal argument.
>
> Did I say it was?
Other supporters will be hard to find.
> >> After all, the claim is that the reals
> >> are uncountable, not the representations of reals.
> >
> > But this point is not supported by the diagonal argument.
>
> Did I say it was?
> > Further it is easily contradicted.
>
> Not so.
>
> > Every real has a finite definition. Every real is the meaning of a
> > finite word. There are only countably many finite words. There are
> > only as many meanings as there are languages. Languages have to be
> > constructed. There is a finite number of languages at every
> > time.
> > There can never be more than countably many reals. QED.
>
> This of course fails to contradict the fact that two finite definitions
> may refer to the same real.
The set of definitions is countable. The set of real numbers is countable too since many definitions do not concern real numbers and others concern one and the same real number.
> There is a difference between the
> potentially infinite set of finite definitions, and the
> potentially infinite set of reals; and you have failed to
> recognise this distinction.
Regards, WM
WM
Jul 2, 2015, 8:35:29 AM7/2/15
to
Am Mittwoch, 1. Juli 2015 16:30:23 UTC+2 schrieb George Greene:
> The fact that you think "potential infinity" even IS A THING,
> with respect to Cantor's Theorem, makes YOU a crank.
Everybody unable to understand potential infinity is at least very unprofessional. Even you should be able to grasp that every downwards sequence in the ordinal numbers is termninating, i.e., finite. But for every sequence there is a longer one. That means the set of downwards sequences is potentially infinite. The set of upwards sequences is actually infinite, i.e., longer than every downwards sequence.
> The fact that you think "potential infinity" even IS A THING,
> with respect to Cantor's Theorem, makes YOU a crank.
Regards, WM
WM
Jul 2, 2015, 8:40:29 AM7/2/15
to
Am Mittwoch, 1. Juli 2015 17:53:18 UTC+2 schrieb Virgil:
> > Every real has a finite definition.
>
> Not if there are more reals than there are finite definitions, which is
> provably the case!
It is not "proven". It is "believed" but only by matheologians.
> > Every real has a finite definition.
>
> Not if there are more reals than there are finite definitions, which is
> provably the case!
>
>
>
> > Every real is the meaning of a finite word.
>
> Where is that claimed?
In mathematics where every real number can be used.
>
>
> > Every real is the meaning of a finite word.
>
> Where is that claimed?
>
> As far as I am aware, the field of real numbers is finitely defined, but
> in that definition there is no explicit requirement that each real shall
> have its own separate finite definition.
Then you should try to increase your awareness. The set or field of real numbers is not what is used for calculating. The real numbers are used.
> As far as I am aware, the field of real numbers is finitely defined, but
> in that definition there is no explicit requirement that each real shall
> have its own separate finite definition.
Regards, WM
WM
Jul 2, 2015, 8:40:47 AM7/2/15
to
Am Mittwoch, 1. Juli 2015 18:19:11 UTC+2 schrieb Virgil:
> > Every real has a finite definition.
>
> Assumed by WM but not by anyone else! There is no logical objection to
> having more things than individual names for them
Not in reality. But if the "things" are merely ideas, then they cannot exist without the possibility to be thought of. Ideas that nobody can have are not ideas.
> > Every real has a finite definition.
>
> Assumed by WM but not by anyone else! There is no logical objection to
> having more things than individual names for them
>
>
> > Every real is the meaning of a
> > finite word.
>
> What axiom set includes that as an axiom or theorem?
The axiom of trichotomy says: For every pair of different real numbers we can determine which one is less.
>
> > Every real is the meaning of a
> > finite word.
>
> What axiom set includes that as an axiom or theorem?
Regards, WM
WM
Jul 2, 2015, 8:45:52 AM7/2/15
to
Am Mittwoch, 1. Juli 2015 18:52:28 UTC+2 schrieb George Greene:
> > The antidiagonal differs from every irrational number.
>
> No, it doesn't.
Irrational means not rational, not given by digits, not even by infinitely many. There are infinitely many rational approximations.
> > The antidiagonal differs from every irrational number.
>
> No, it doesn't.
> It may itself be rational or irrational;
> That depends on the construction of the original underlying sequence.
> The anti-diagonal in any case
> is NOT ON the list, since it differs from every element that IS on the list.
Regards, WM
WM
Jul 2, 2015, 9:00:40 AM7/2/15
to
Am Mittwoch, 1. Juli 2015 19:55:03 UTC+2 schrieb Alan Smaill:
> Otherwise all we know is that the number of representations
> of reals >= the number of reals,
Correct. Cp Newton's and Euler's representation of e.
> Otherwise all we know is that the number of representations
> of reals >= the number of reals,
> and the number
> of representations of reals is > than the number of naturals.
> Not hard to fix,
only if stupidness triumphs over logic.
Regards, WM
WM
Jul 2, 2015, 9:00:52 AM7/2/15
to
Am Mittwoch, 1. Juli 2015 19:34:57 UTC+2 schrieb George Greene:
>
> You DON'T NEED to go the limits, AT ALL.
> You have a square list of digits and you produce a row that it is not on it.
Ridiculous! There is no "square". And if there was a square, then double each line. Then you fail to have a square.
>
> You DON'T NEED to go the limits, AT ALL.
> You have a square list of digits and you produce a row that it is not on it.
Your sillieness becomes obvious by the fact thzat the antidiagonal has to have precisely as many digits as the list has lines and columns. But this is impossible, if the length is arbirarily changed.
The fact that two sets like naturals and primes can have the same cardinality is completely irrelevant in the diagoal argument. There must be a precise correspondence between lines of the list and digits of the antidiagonal and the columns of the list. And there is no "open end", because a digit sequence with open end cannot determine an irrational number.
It is unbelievable to see what nonsensical beliefs and confusion prevail in matheology.
Regards, WM
Martin Shobe
Jul 2, 2015, 10:02:41 AM7/2/15
to
On 7/2/2015 7:35 AM, WM wrote:
> Am Mittwoch, 1. Juli 2015 16:30:23 UTC+2 schrieb George Greene:
>> The fact that you think "potential infinity" even IS A THING,
>> with respect to Cantor's Theorem, makes YOU a crank.
> Am Mittwoch, 1. Juli 2015 16:30:23 UTC+2 schrieb George Greene:
>> The fact that you think "potential infinity" even IS A THING,
>> with respect to Cantor's Theorem, makes YOU a crank.
> Even you should be able to grasp that every downwards sequence in the ordinal numbers is termninating, i.e., finite. But for every sequence there is a longer one. That means the set of downwards sequences is potentially infinite.
It also means that the set of downward sequences is actually infinite.
> The set of upwards sequences is actually infinite, i.e., longer than every downwards sequence.
Martin Shobe
WM
Jul 2, 2015, 10:19:34 AM7/2/15
to
Am Donnerstag, 2. Juli 2015 16:02:41 UTC+2 schrieb Martin Shobe:
> On 7/2/2015 7:35 AM, WM wrote:
> > Am Mittwoch, 1. Juli 2015 16:30:23 UTC+2 schrieb George Greene:
> >> The fact that you think "potential infinity" even IS A THING,
> >> with respect to Cantor's Theorem, makes YOU a crank.
>
> > Even you should be able to grasp that every downwards sequence in the ordinal numbers is termninating, i.e., finite. But for every sequence there is a longer one. That means the set of downwards sequences is potentially infinite.
>
> It also means that the set of downward sequences is actually infinite.
Agreed. Like the set of finite approximations 0.1, 0.11, 0.111, ... to 1/9 is actually infinite without containing the atually infinite sequence 0.111...
> On 7/2/2015 7:35 AM, WM wrote:
> > Am Mittwoch, 1. Juli 2015 16:30:23 UTC+2 schrieb George Greene:
> >> The fact that you think "potential infinity" even IS A THING,
> >> with respect to Cantor's Theorem, makes YOU a crank.
>
> > Even you should be able to grasp that every downwards sequence in the ordinal numbers is termninating, i.e., finite. But for every sequence there is a longer one. That means the set of downwards sequences is potentially infinite.
>
> It also means that the set of downward sequences is actually infinite.
But here we get 0.111... when we write all finite digit sequences in one and the same line. What about writing all downwards sequences in one single line? Do we get an infinite downward sequence then?
>
> > The set of upwards sequences is actually infinite, i.e., longer than every downwards sequence.
>
> What's the length of a set?
My mistake. The upwards *sequence* (there is only one complete upwards sequence) is actually infinite.
> > The set of upwards sequences is actually infinite, i.e., longer than every downwards sequence.
>
> What's the length of a set?
Regards, WM
Martin Shobe
Jul 2, 2015, 11:19:40 AM7/2/15
to
On 7/2/2015 9:19 AM, WM wrote:
> Am Donnerstag, 2. Juli 2015 16:02:41 UTC+2 schrieb Martin Shobe:
>> On 7/2/2015 7:35 AM, WM wrote:
>>> Am Mittwoch, 1. Juli 2015 16:30:23 UTC+2 schrieb George Greene:
>>>> The fact that you think "potential infinity" even IS A THING,
>>>> with respect to Cantor's Theorem, makes YOU a crank.
>>
>>> Even you should be able to grasp that every downwards sequence in the ordinal numbers is termninating, i.e., finite. But for every sequence there is a longer one. That means the set of downwards sequences is potentially infinite.
>>
>> It also means that the set of downward sequences is actually infinite.
>
> Agreed. Like the set of finite approximations 0.1, 0.11, 0.111, ... to 1/9 is actually infinite without containing the atually infinite sequence 0.111...
>
> But here we get 0.111... when we write all finite digit sequences in one and the same line. What about writing all downwards sequences in one single line? Do we get an infinite downward sequence then?
First of all, your question is ambiguous. However, if you were to
> Am Donnerstag, 2. Juli 2015 16:02:41 UTC+2 schrieb Martin Shobe:
>> On 7/2/2015 7:35 AM, WM wrote:
>>> Am Mittwoch, 1. Juli 2015 16:30:23 UTC+2 schrieb George Greene:
>>>> The fact that you think "potential infinity" even IS A THING,
>>>> with respect to Cantor's Theorem, makes YOU a crank.
>>
>>> Even you should be able to grasp that every downwards sequence in the ordinal numbers is termninating, i.e., finite. But for every sequence there is a longer one. That means the set of downwards sequences is potentially infinite.
>>
>> It also means that the set of downward sequences is actually infinite.
>
> Agreed. Like the set of finite approximations 0.1, 0.11, 0.111, ... to 1/9 is actually infinite without containing the atually infinite sequence 0.111...
>
> But here we get 0.111... when we write all finite digit sequences in one and the same line. What about writing all downwards sequences in one single line? Do we get an infinite downward sequence then?
superimpose every downward sequence so that the same elements are always
written on top of each other, the result would be the reversal of the
original well-order. It would be a sequence if and only if the original
well-order was finite.
Martin Shobe
Ross A. Finlayson
Jul 2, 2015, 12:36:37 PM7/2/15
to
It is the sweep function
that is uniquely being a
counterexample among the
number-theoretic uncount-
ability results.
For the set theoretic results
there is a theory of ubiquitous
ordinals where powerset is simply
enough order type and successor
(i.e, "bigger").
There is also "A function surjects
the rational mumbers onto the
irrational numbers", another number-
theoretic result due their density,
but that is of the ordered field,
not the integers.
that is uniquely being a
counterexample among the
number-theoretic uncount-
ability results.
For the set theoretic results
there is a theory of ubiquitous
ordinals where powerset is simply
enough order type and successor
(i.e, "bigger").
There is also "A function surjects
the rational mumbers onto the
irrational numbers", another number-
theoretic result due their density,
but that is of the ordered field,
not the integers.
Ross A. Finlayson
Jul 2, 2015, 12:41:42 PM7/2/15
to
Cantor's arguments
prove the line is drawn!
Your interpretation
of Cantor's results
is otherwise not
complete.
So, draw a line.
prove the line is drawn!
Your interpretation
of Cantor's results
is otherwise not
complete.
So, draw a line.
George Greene
Jul 2, 2015, 1:38:36 PM7/2/15
to
On Thursday, July 2, 2015 at 9:00:52 AM UTC-4, WM wrote:
> Ridiculous! There is no "square".
You're just a damn fool,plain and simple. OF FUCKING COURSE
> Ridiculous! There is no "square".
there is a square because the number of nubmers (whatEVER it is) ACROSS
and the number of numbers DOWN are THE SAME number.
Its a square because EVERY ROW has an index and EVERY COLUMN has an index
and the indices ARE THE SAME. BOth across and down, they are ALL the
natural numbers, IN THE SAME order.
> And if there was a square, then double each line.
> Then you fail to have a square.
There are THE SAME NUMBER OF natural numbers divisible by 3 as not.
There ARE NOT twice as many divisible by 3 as not. If you double each line
THEN YOU STILL have a square because because that's just HOW omega WORKS.
If you have a hotel with a room for each natural number, and you have one
person in each room, and you have the same number of people as rooms, and
you then double and put TWO people in each room, THEN YOU STILL have the
same number of people as rooms.
But you're *A*DAMN*FOOL*, so you didn't know that.
Virgil
Jul 2, 2015, 1:44:15 PM7/2/15
to
In article <73ef6f6e-7d70-4f32...@googlegroups.com>,
inaginings.
> But if the "things" are merely ideas, then they cannot exist
> without the possibility to be thought of. Ideas that nobody can have are not
> ideas.
Actually finite sets are an IDEA that anyone can have. And mathematics
is a world of ideas
word"? Reals are ideas, not physical objects and
Mathematics is a world of pure ideas, not of physical realities!
>
> Regards, WM
WM <wolfgang.m...@hs-augsburg.de> wrote:
> Am Mittwoch, 1. Juli 2015 18:19:11 UTC+2 schrieb Virgil:
>
>
> > > Every real has a finite definition.
> >
> > Assumed by WM but not by anyone else! There is no logical objection to
> > having more things than individual names for them
>
> Not in reality.
But mathemtics is not a world of "reality" but a world of ideas and
> Am Mittwoch, 1. Juli 2015 18:19:11 UTC+2 schrieb Virgil:
>
>
> > > Every real has a finite definition.
> >
> > Assumed by WM but not by anyone else! There is no logical objection to
> > having more things than individual names for them
>
> Not in reality.
inaginings.
> But if the "things" are merely ideas, then they cannot exist
> without the possibility to be thought of. Ideas that nobody can have are not
> ideas.
is a world of ideas
> >
> >
> > > Every real is the meaning of a
> > > finite word.
> >
> > What axiom set includes that as an axiom or theorem?
>
> The axiom of trichotomy says: For every pair of different real numbers we can
> determine which one is less.
How does trichotomy require every real to be "the meaning of a finite
> >
> > > Every real is the meaning of a
> > > finite word.
> >
> > What axiom set includes that as an axiom or theorem?
>
> The axiom of trichotomy says: For every pair of different real numbers we can
> determine which one is less.
word"? Reals are ideas, not physical objects and
Mathematics is a world of pure ideas, not of physical realities!
>
> Regards, WM
Julio Di Egidio
Jul 2, 2015, 1:48:23 PM7/2/15
to
On Wednesday, July 1, 2015 at 6:55:03 PM UTC+1, Alan Smaill wrote:
> >> This seems to be the objection -- it is easily enough dealt with,
<snip>
That said, whatever you have up your sleeve, it is maybe time to get it out: no more cards on the table that I can see. -- I could have a fight with George on many details, maybe I will, but that is marginal to the present topic.
Julio
> George Greene <g***@email.unc.edu> writes:
> > On Wednesday, July 1, 2015 at 5:45:04 AM UTC-4, Alan Smaill wrote:
>
> >> But then there are then multiple representations of the same real --
> >> so this is not what reals "really" are.
> >
> > That's our problem as architects of the representors, not any actual
> > problem with the representands.
>
> Fine.
No, it is the opposite of fine. Firstly, the argument rather goes from a conclusion on the "representors" to a conclusion on the "representands", not the other way round (!); then the objection just remains, that those "representors" do not "represent" real numbers. Indeed, the claim is that they cannot: we just cannot get from potential to actual "for free".
> > On Wednesday, July 1, 2015 at 5:45:04 AM UTC-4, Alan Smaill wrote:
>
> >> But then there are then multiple representations of the same real --
> >> so this is not what reals "really" are.
> >
> > That's our problem as architects of the representors, not any actual
> > problem with the representands.
>
> Fine.
> >> This seems to be the objection -- it is easily enough dealt with,
> It does need to be ruled out, though.
> Otherwise all we know is that the number of representations
> of reals >= the number of reals, and the number
> of representations of reals is > than the number of naturals.
No, you do not know any of that! Again, that and only that is the objection: that the argument tells *nothing* about *real numbers*.
> Otherwise all we know is that the number of representations
> of reals >= the number of reals, and the number
> of representations of reals is > than the number of naturals.
That said, whatever you have up your sleeve, it is maybe time to get it out: no more cards on the table that I can see. -- I could have a fight with George on many details, maybe I will, but that is marginal to the present topic.
Julio
Virgil
Jul 2, 2015, 1:54:42 PM7/2/15
to
In article <77480409-c03d-4274...@googlegroups.com>,
complete Archimedean ordered field as defining the set of reals!
Archimedean ordered field, where and when does mathematics ever say that
every real is the meaning of a finite word?
the definition of any complete Archimedean ordered field of real numbers
requires every real number to have a numeral.
WM <wolfgang.m...@hs-augsburg.de> wrote:
> Am Mittwoch, 1. Juli 2015 17:53:18 UTC+2 schrieb Virgil:
>
>
> > > Every real has a finite definition.
> >
> > Not if there are more reals than there are finite definitions, which is
> > provably the case!
>
> It is not "proven". It is "believed" but only by matheologians.
It is provable by anyone who starts with a proper definition of a
> Am Mittwoch, 1. Juli 2015 17:53:18 UTC+2 schrieb Virgil:
>
>
> > > Every real has a finite definition.
> >
> > Not if there are more reals than there are finite definitions, which is
> > provably the case!
>
> It is not "proven". It is "believed" but only by matheologians.
complete Archimedean ordered field as defining the set of reals!
> >
> >
> >
> > > Every real is the meaning of a finite word.
> >
> > Where is that claimed?
>
> In mathematics where every real number can be used.
While mathematics does claim that every real is member of a complete
> >
> >
> > > Every real is the meaning of a finite word.
> >
> > Where is that claimed?
>
> In mathematics where every real number can be used.
Archimedean ordered field, where and when does mathematics ever say that
every real is the meaning of a finite word?
> >
> > As far as I am aware, the field of real numbers is finitely defined, but
> > in that definition there is no explicit requirement that each real shall
> > have its own separate finite definition.
>
> Then you should try to increase your awareness. The set or field of real
> numbers is not what is used for calculating.
Numerals, not numbers, are what are used for calculating, but nothing in
> > As far as I am aware, the field of real numbers is finitely defined, but
> > in that definition there is no explicit requirement that each real shall
> > have its own separate finite definition.
>
> Then you should try to increase your awareness. The set or field of real
> numbers is not what is used for calculating.
the definition of any complete Archimedean ordered field of real numbers
requires every real number to have a numeral.
Virgil
Jul 2, 2015, 2:07:03 PM7/2/15
to
In article <6c7746a9-95b6-4752...@googlegroups.com>,
infinitely many columns, one for each of each number's infinitely many
digits.
whole point of the diagonal argument!
There must be a precise
> correspondence between lines of the list and digits of the antidiagonal
And there is!
> It is unbelievable to see what nonsensical beliefs and confusion prevail
Since WM cannot see out anywhere beyond the boundaries of the worthless
world of WMytheology in which he has imprisoned himself, it is only
inside that world that the nonsensical beliefs and confusion that he
sees prevails.
WM <wolfgang.m...@hs-augsburg.de> wrote:
> Am Mittwoch, 1. Juli 2015 19:34:57 UTC+2 schrieb George Greene:
>
>
> >
> > You DON'T NEED to go the limits, AT ALL.
> > You have a square list of digits and you produce a row that it is not on
> > it.
>
> Ridiculous! There is no "square". And if there was a square, then double each
> line. Then you fail to have a square.
>
> Your sillieness becomes obvious by the fact thzat the antidiagonal has to
> have precisely as many digits as the list has lines and columns.
False! Every finite list of reals, whether rational or not, stkk has
> Am Mittwoch, 1. Juli 2015 19:34:57 UTC+2 schrieb George Greene:
>
>
> >
> > You DON'T NEED to go the limits, AT ALL.
> > You have a square list of digits and you produce a row that it is not on
> > it.
>
> Ridiculous! There is no "square". And if there was a square, then double each
> line. Then you fail to have a square.
>
> Your sillieness becomes obvious by the fact thzat the antidiagonal has to
> have precisely as many digits as the list has lines and columns.
infinitely many columns, one for each of each number's infinitely many
digits.
>
> The fact that two sets like naturals and primes can have the same cardinality
> is completely irrelevant in the diagoal argument.
The fact that two infinite sets can have different cardinalities is the
> The fact that two sets like naturals and primes can have the same cardinality
> is completely irrelevant in the diagoal argument.
whole point of the diagonal argument!
There must be a precise
> correspondence between lines of the list and digits of the antidiagonal
> It is unbelievable to see what nonsensical beliefs and confusion prevail
world of WMytheology in which he has imprisoned himself, it is only
inside that world that the nonsensical beliefs and confusion that he
sees prevails.
Virgil
Jul 2, 2015, 2:13:36 PM7/2/15
to
In article <16aa4c96-50af-469e...@googlegroups.com>,
them have none at all, since there are a lot more reals than explicit
representations possible for them.
And until WM can produce AND PROVE a surjection from |N to |R (or an
injection from |R to |N), which he can't, his claim remains totally
false!
WM <wolfgang.m...@hs-augsburg.de> wrote:
> Am Mittwoch, 1. Juli 2015 19:55:03 UTC+2 schrieb Alan Smaill:
>
>
> > Otherwise all we know is that the number of representations
> > of reals >= the number of reals,
>
> Correct. Cp Newton's and Euler's representation of e.
WHile one real number may have more than one representation, most of
> Am Mittwoch, 1. Juli 2015 19:55:03 UTC+2 schrieb Alan Smaill:
>
>
> > Otherwise all we know is that the number of representations
> > of reals >= the number of reals,
>
> Correct. Cp Newton's and Euler's representation of e.
them have none at all, since there are a lot more reals than explicit
representations possible for them.
And until WM can produce AND PROVE a surjection from |N to |R (or an
injection from |R to |N), which he can't, his claim remains totally
false!
Virgil
Jul 2, 2015, 2:23:57 PM7/2/15
to
In article <66769f09-5b00-4b50...@googlegroups.com>,
That does not require any number to be both!
list of rational, the set of rationals , being unlistable, is
uncountable! At least in WM's worthless world of WMytheology.
WM <wolfgang.m...@hs-augsburg.de> wrote:
> Am Mittwoch, 1. Juli 2015 18:52:28 UTC+2 schrieb George Greene:
>
>
> > > The antidiagonal differs from every irrational number.
> >
> > No, it doesn't.
>
> Irrational means not rational, not given by digits, not even by infinitely
> many. There are infinitely many rational approximations.
Irrelevant!
> Am Mittwoch, 1. Juli 2015 18:52:28 UTC+2 schrieb George Greene:
>
>
> > > The antidiagonal differs from every irrational number.
> >
> > No, it doesn't.
>
> Irrational means not rational, not given by digits, not even by infinitely
> many. There are infinitely many rational approximations.
>
> > It may itself be rational or irrational;
>
> Learn a bit of mathematic
> > It may itself be rational or irrational;
>
> Learn a bit of mathematic
> Irrational rationals do not exist - not even in
> finished infinity.
LEARN TO READ! Every real number is itself rational OR irrational!
> finished infinity.
That does not require any number to be both!
>
> > That depends on the construction of the original underlying sequence.
>
> No. The limit may be irrational, but not any digit sequence.
The limit is named by that infinite digit sequence!
> > That depends on the construction of the original underlying sequence.
>
> No. The limit may be irrational, but not any digit sequence.
>
> > The anti-diagonal in any case
> > is NOT ON the list, since it differs from every element that IS on the
> > list.
>
> This shows very nicely that an actually infinite list is a self-contradicting
> object.
Then, since in WM; world there cannot be any actually infinite complete
> > The anti-diagonal in any case
> > is NOT ON the list, since it differs from every element that IS on the
> > list.
>
> This shows very nicely that an actually infinite list is a self-contradicting
> object.
list of rational, the set of rationals , being unlistable, is
uncountable! At least in WM's worthless world of WMytheology.
Virgil
Jul 2, 2015, 2:32:24 PM7/2/15
to
In article <c7f96320-be16-44f3...@googlegroups.com>,
there is an ACTUAL infinity of starting points, the NUMBER of such
downward sequences is still ACTUALLY infinite!
> The set of upwards sequences is actually infinite
The set of downward sequences is at least as large as the set of
starting points!
WM <wolfgang.m...@hs-augsburg.de> wrote:
> Am Mittwoch, 1. Juli 2015 16:30:23 UTC+2 schrieb George Greene:
>
>
> > The fact that you think "potential infinity" even IS A THING,
> > with respect to Cantor's Theorem, makes YOU a crank.
>
> Everybody unable to understand potential infinity is at least very
> unprofessional. Even you should be able to grasp that every downwards
> sequence in the ordinal numbers is termninating, i.e., finite. But for every
> sequence there is a longer one. That means the set of downwards sequences is
> potentially infinite.
WM goofs again! While each such downard sequence may be finite, since
> Am Mittwoch, 1. Juli 2015 16:30:23 UTC+2 schrieb George Greene:
>
>
> > The fact that you think "potential infinity" even IS A THING,
> > with respect to Cantor's Theorem, makes YOU a crank.
>
> Everybody unable to understand potential infinity is at least very
> unprofessional. Even you should be able to grasp that every downwards
> sequence in the ordinal numbers is termninating, i.e., finite. But for every
> sequence there is a longer one. That means the set of downwards sequences is
> potentially infinite.
there is an ACTUAL infinity of starting points, the NUMBER of such
downward sequences is still ACTUALLY infinite!
> The set of upwards sequences is actually infinite
starting points!
Virgil
Jul 2, 2015, 2:36:06 PM7/2/15
to
In article <5d4dd439-0e73-49da...@googlegroups.com>,
one gets 0.1, 0.11, 0.111, ...
As WM himself did above!
> My mistake.
As always!
WM <wolfgang.m...@hs-augsburg.de> wrote:
> Am Donnerstag, 2. Juli 2015 16:02:41 UTC+2 schrieb Martin Shobe:
> > On 7/2/2015 7:35 AM, WM wrote:
> > > Am Mittwoch, 1. Juli 2015 16:30:23 UTC+2 schrieb George Greene:
> > >> The fact that you think "potential infinity" even IS A THING,
> > >> with respect to Cantor's Theorem, makes YOU a crank.
> >
> > > Even you should be able to grasp that every downwards sequence in the
> > > ordinal numbers is termninating, i.e., finite. But for every sequence
> > > there is a longer one. That means the set of downwards sequences is
> > > potentially infinite.
> >
> > It also means that the set of downward sequences is actually infinite.
>
> Agreed. Like the set of finite approximations 0.1, 0.11, 0.111, ... to 1/9 is
> actually infinite without containing the atually infinite sequence 0.111...
>
> But here we get 0.111... when we write all finite digit sequences in one and
> the same line.
OUTSIDE of WM's worthless world of WMytheology
> Am Donnerstag, 2. Juli 2015 16:02:41 UTC+2 schrieb Martin Shobe:
> > On 7/2/2015 7:35 AM, WM wrote:
> > > Am Mittwoch, 1. Juli 2015 16:30:23 UTC+2 schrieb George Greene:
> > >> The fact that you think "potential infinity" even IS A THING,
> > >> with respect to Cantor's Theorem, makes YOU a crank.
> >
> > > Even you should be able to grasp that every downwards sequence in the
> > > ordinal numbers is termninating, i.e., finite. But for every sequence
> > > there is a longer one. That means the set of downwards sequences is
> > > potentially infinite.
> >
> > It also means that the set of downward sequences is actually infinite.
>
> Agreed. Like the set of finite approximations 0.1, 0.11, 0.111, ... to 1/9 is
> actually infinite without containing the atually infinite sequence 0.111...
>
> But here we get 0.111... when we write all finite digit sequences in one and
> the same line.
one gets 0.1, 0.11, 0.111, ...
As WM himself did above!
> My mistake.
As always!
Virgil
Jul 2, 2015, 2:47:31 PM7/2/15
to
In article <a1fdd64c-4325-409b...@googlegroups.com>,
The set of real numbers is not defined in vacuo, but as a consequence of
the definition of the reals as a complete Archimedean ordered field,
from which definition the uncountability of the the SET of real is
derivable. And remains valid until WM or someone else has enumerated |R
the definition of the reals as a complete Archimedean ordered field,
from which definition the uncountability of the the SET of real is
derivable. And remains valid until WM or someone else has enumerated |R
>
> > There is a difference between the
> > potentially infinite set of finite definitions, and the
> > potentially infinite set of reals; and you have failed to
> > recognise this distinction.
>
> Your logic is broken!
WM's "logic" is non-existent!
> > There is a difference between the
> > potentially infinite set of finite definitions, and the
> > potentially infinite set of reals; and you have failed to
> > recognise this distinction.
>
> Your logic is broken!
Julio Di Egidio
Jul 2, 2015, 4:17:22 PM7/2/15
to
On Wednesday, July 1, 2015 at 3:37:07 PM UTC+1, George Greene wrote:
> On Tuesday, June 30, 2015 at 7:35:05 PM UTC-4, Julio Di Egidio wrote:
>
> > I am focusing on the class of so called "diagonal argument"s here.
>
> This makes you a crank.
> Non-cranks do not have any trouble understanding diagonal arguments.
> This is the pons asinorum of this forum and has been, approximately forever.
> I've been here almost 30 years now.
Learning how to shout out fallacies without even noticing. Yes, it is a pons asinorum, no doubt on that.
> > And the issue is already logical
>
> Oh, SHUT UP.
> You DO NOT KNOW what "logical" *means*!!
> AROUND HERE, "logical" means "starting with a countable signature
> for a first-order langauge, and SOME AXIOMS written in that language,
> and implying/inferring MORE sentences in that language -- theorems -- using
> standard rules of inference"!! THAT is logical! What you are doing is
> anything but!
What you call logic is mathematical logic: mathematics. It is you who don't know what logic means, around here or anywhere: nor can you know what formal means. And you will not be able to really understand logical arguments either, their very nature escapes you.
> > hence it does not matter the formalisation.
>
> A more ANTI-logical statement could hardly be devised. The formalizations
> exist.
Indeed, you do not know what logic is or means, nor its relationship to mathematics and formality. That again said:
> The proofs are valid.
So far, you have not even managed to address my objection. Here it is again:
> The very simple bridge between subsets of N and real
> numbers is that every real number has one (or two, in a very few cases
> of rationals with even denominators) representation AS A BIT-STRING,
Nonsense: a real number is THE LIMIT of a bit string, the anti-diagonal is a bit string, hence the anti-diagonal is not a real number: IOW, you have not constructed a real number and you have proved nothing about real numbers. QED.
> and
> the 1s and 0s in its binary representation become
> analogous to the presence
> or absence of certain natnums in a subset.
Circular reasoning is invalid. Rather, put up (show the putative missing link in my objection) or shut up.
Julio
> On Tuesday, June 30, 2015 at 7:35:05 PM UTC-4, Julio Di Egidio wrote:
>
> > I am focusing on the class of so called "diagonal argument"s here.
>
> This makes you a crank.
> Non-cranks do not have any trouble understanding diagonal arguments.
> This is the pons asinorum of this forum and has been, approximately forever.
> I've been here almost 30 years now.
Learning how to shout out fallacies without even noticing. Yes, it is a pons asinorum, no doubt on that.
> > And the issue is already logical
>
> Oh, SHUT UP.
> You DO NOT KNOW what "logical" *means*!!
> AROUND HERE, "logical" means "starting with a countable signature
> for a first-order langauge, and SOME AXIOMS written in that language,
> and implying/inferring MORE sentences in that language -- theorems -- using
> standard rules of inference"!! THAT is logical! What you are doing is
> anything but!
What you call logic is mathematical logic: mathematics. It is you who don't know what logic means, around here or anywhere: nor can you know what formal means. And you will not be able to really understand logical arguments either, their very nature escapes you.
> > hence it does not matter the formalisation.
>
> A more ANTI-logical statement could hardly be devised. The formalizations
> exist.
Indeed, you do not know what logic is or means, nor its relationship to mathematics and formality. That again said:
> The proofs are valid.
So far, you have not even managed to address my objection. Here it is again:
> The very simple bridge between subsets of N and real
> numbers is that every real number has one (or two, in a very few cases
> of rationals with even denominators) representation AS A BIT-STRING,
Nonsense: a real number is THE LIMIT of a bit string, the anti-diagonal is a bit string, hence the anti-diagonal is not a real number: IOW, you have not constructed a real number and you have proved nothing about real numbers. QED.
> and
> the 1s and 0s in its binary representation become
> analogous to the presence
> or absence of certain natnums in a subset.
Circular reasoning is invalid. Rather, put up (show the putative missing link in my objection) or shut up.
Julio
Virgil
Jul 2, 2015, 5:03:55 PM7/2/15
to
In article <62902785-6909-47ab...@googlegroups.com>,
https://en.wikipedia.org/wiki/Pons_asinorum
In geometry, the statement that the angles opposite the equal sides of
an isosceles triangle are themselves equal is known as the pons
asinorum, Latin for "bridge of donkeys". This statement is Proposition 5
of Book 1 in Euclid's Elements, and is also known as the isosceles
triangle theorem. Its converse is also true: if two angles of a triangle
are equal, then the sides opposite them are also equal.
The name of this statement is also used metaphorically for a problem or
challenge which will separate the sure of mind from the simple, the
fleet thinker from the slow, the determined from the dallier; to
represent a critical test of ability or understanding.
And Julio Di Egidio , for some reason, fails to cross it!
> It is you who don't know what logic means.
The evidence does not support your claim!
bit string itself?
Or, more precisely, how is the real number represented by an infinite
bit string different from the real number represented by the real
represented by the limit of the reals represented by its finite initia
subsequences?
In really real mathematics there is no difference!
> the anti-diagonal is a
> bit string, hence the anti-diagonal is not a real number
All the sequences in the list from which it is formed are also bit
strings so they are not real numbers either, but there are more such bit
strings than can be listed and for each list, as many bit strings as yet
listed! There is not and cannot be a list of all bitstrings but there
can be and must be set of a bitstrings.
: IOW, you have not
> constructed a real number and you have proved nothing about real numbers.
Nonsense yourself! A real number may be defined as the limit of a
suitably nested SEQUENCE of infinitely many finite bit strings, but for
each one of uncountably many bitstrings there is one and only one
corresponding real number, though there are occasionally two different
bitstrings for the same real number.
Julio Di Egidio <ju...@diegidio.name> wrote:
> On Wednesday, July 1, 2015 at 3:37:07 PM UTC+1, George Greene wrote:
> > On Tuesday, June 30, 2015 at 7:35:05 PM UTC-4, Julio Di Egidio wrote:
> >
> > > I am focusing on the class of so called "diagonal argument"s here.
> >
> > This makes you a crank.
> > Non-cranks do not have any trouble understanding diagonal arguments.
> > This is the pons asinorum of this forum and has been, approximately
> > forever.
> > I've been here almost 30 years now.
>
> Learning how to shout out fallacies without even noticing. Yes, it is a pons
> asinorum, no doubt on that.
With Julio the one not able to cross it.
> On Wednesday, July 1, 2015 at 3:37:07 PM UTC+1, George Greene wrote:
> > On Tuesday, June 30, 2015 at 7:35:05 PM UTC-4, Julio Di Egidio wrote:
> >
> > > I am focusing on the class of so called "diagonal argument"s here.
> >
> > This makes you a crank.
> > Non-cranks do not have any trouble understanding diagonal arguments.
> > This is the pons asinorum of this forum and has been, approximately
> > forever.
> > I've been here almost 30 years now.
>
> Learning how to shout out fallacies without even noticing. Yes, it is a pons
> asinorum, no doubt on that.
https://en.wikipedia.org/wiki/Pons_asinorum
In geometry, the statement that the angles opposite the equal sides of
an isosceles triangle are themselves equal is known as the pons
asinorum, Latin for "bridge of donkeys". This statement is Proposition 5
of Book 1 in Euclid's Elements, and is also known as the isosceles
triangle theorem. Its converse is also true: if two angles of a triangle
are equal, then the sides opposite them are also equal.
The name of this statement is also used metaphorically for a problem or
challenge which will separate the sure of mind from the simple, the
fleet thinker from the slow, the determined from the dallier; to
represent a critical test of ability or understanding.
And Julio Di Egidio , for some reason, fails to cross it!
>
> > > And the issue is already logical
> >
> > Oh, SHUT UP.
> > You DO NOT KNOW what "logical" *means*!!
> > AROUND HERE, "logical" means "starting with a countable signature
> > for a first-order langauge, and SOME AXIOMS written in that language,
> > and implying/inferring MORE sentences in that language -- theorems -- using
> > standard rules of inference"!! THAT is logical! What you are doing is
> > anything but!
>
> What you call logic is mathematical logic: mathematics.
Mathematics uses the same logic as formal logics use.
> > > And the issue is already logical
> >
> > Oh, SHUT UP.
> > You DO NOT KNOW what "logical" *means*!!
> > AROUND HERE, "logical" means "starting with a countable signature
> > for a first-order langauge, and SOME AXIOMS written in that language,
> > and implying/inferring MORE sentences in that language -- theorems -- using
> > standard rules of inference"!! THAT is logical! What you are doing is
> > anything but!
>
> What you call logic is mathematical logic: mathematics.
> It is you who don't know what logic means.
The evidence does not support your claim!
>
> > > hence it does not matter the formalisation.
> >
> > A more ANTI-logical statement could hardly be devised. The formalizations
> > exist.
>
> Indeed, you do not know what logic is or means, nor its relationship to
> mathematics and formality.
>
> > > hence it does not matter the formalisation.
> >
> > A more ANTI-logical statement could hardly be devised. The formalizations
> > exist.
>
> Indeed, you do not know what logic is or means, nor its relationship to
> mathematics and formality.
>
> > The proofs are valid.
>
> So far, you have not even managed to address my objection.
Your objections so far do not address the Cantor proofs.
>
> So far, you have not even managed to address my objection.
>
> > The very simple bridge between subsets of N and real
> > numbers is that every real number has one (or two, in a very few cases
> > of rationals with even denominators) representation AS A BIT-STRING,
>
> Nonsense: a real number is THE LIMIT of a bit string
How is the limit of an infinite bit string different from the infinite
> > The very simple bridge between subsets of N and real
> > numbers is that every real number has one (or two, in a very few cases
> > of rationals with even denominators) representation AS A BIT-STRING,
>
> Nonsense: a real number is THE LIMIT of a bit string
bit string itself?
Or, more precisely, how is the real number represented by an infinite
bit string different from the real number represented by the real
represented by the limit of the reals represented by its finite initia
subsequences?
In really real mathematics there is no difference!
> the anti-diagonal is a
> bit string, hence the anti-diagonal is not a real number
strings so they are not real numbers either, but there are more such bit
strings than can be listed and for each list, as many bit strings as yet
listed! There is not and cannot be a list of all bitstrings but there
can be and must be set of a bitstrings.
: IOW, you have not
> constructed a real number and you have proved nothing about real numbers.
suitably nested SEQUENCE of infinitely many finite bit strings, but for
each one of uncountably many bitstrings there is one and only one
corresponding real number, though there are occasionally two different
bitstrings for the same real number.
>
> > and
> > the 1s and 0s in its binary representation become
> > analogous to the presence
> > or absence of certain natnums in a subset.
>
> Circular reasoning is invalid.
Then stop doing it!
> > and
> > the 1s and 0s in its binary representation become
> > analogous to the presence
> > or absence of certain natnums in a subset.
>
> Circular reasoning is invalid.
WM
Jul 3, 2015, 7:07:55 AM7/3/15
to
Am Donnerstag, 2. Juli 2015 20:32:24 UTC+2 schrieb Virgil:
> The set of downward sequences is at least as large as the set of
> starting points!
This set is finite. Otherwise you could extend the first downward sequence 1, 0 step by step:
> The set of downward sequences is at least as large as the set of
> starting points!
2, 1, 0
3, 2, 1, 0
...
and you would get as many sequences as there are starting points. In addition, the lengths of the sequences would become as large as the number of starting points. Both magnitudes are in bijection.
Regards, WM
WM
Jul 3, 2015, 7:10:14 AM7/3/15
to
Am Donnerstag, 2. Juli 2015 20:23:57 UTC+2 schrieb Virgil:
> > Irrational rationals do not exist - not even in
> > finished infinity.
>
> LEARN TO READ! Every real number is itself rational OR irrational!
> That does not require any number to be both!
Every digit sequence like every sum of fractions is rational.
> > Irrational rationals do not exist - not even in
> > finished infinity.
>
> LEARN TO READ! Every real number is itself rational OR irrational!
> That does not require any number to be both!
> >
> > > That depends on the construction of the original underlying sequence.
> >
> > No. The limit may be irrational, but not any digit sequence.
>
> The limit is named by that infinite digit sequence!
An infinite sequence is not a name.
> > > That depends on the construction of the original underlying sequence.
> >
> > No. The limit may be irrational, but not any digit sequence.
>
> The limit is named by that infinite digit sequence!
Regards, WM
WM
Jul 3, 2015, 7:17:00 AM7/3/15
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Am Donnerstag, 2. Juli 2015 19:38:36 UTC+2 schrieb George Greene:
> there is a square because the number of nubmers (whatEVER it is) ACROSS
> and the number of numbers DOWN are THE SAME number.
No. They have same cardinality. But they are not the same number.
> there is a square because the number of nubmers (whatEVER it is) ACROSS
> and the number of numbers DOWN are THE SAME number.
There are obviously more naturals than primes. Even Cantor understood this.
In the diagonal argument, however, it is not enough that the antidiagonal has the same cardinality of digits as the list has lines. In the diagonal argument there must be a precise equality.
> Its a square because EVERY ROW has an index and EVERY COLUMN has an index
> and the indices ARE THE SAME. BOth across and down, they are ALL the
> natural numbers, IN THE SAME order.
>
> > And if there was a square, then double each line.
> > Then you fail to have a square.
>
> Again, HORSESHIT.
> There are THE SAME NUMBER OF natural numbers divisible by 3 as not.
Wrong. There is the same cardinality. The numbers are different. Even if you wish to ignore this fact, it remains a fact.
> > And if there was a square, then double each line.
> > Then you fail to have a square.
>
> Again, HORSESHIT.
> There are THE SAME NUMBER OF natural numbers divisible by 3 as not.
Regards, WM
Regards, WM
WM
Jul 3, 2015, 7:51:25 AM7/3/15
to
1,0
2,1,0
...
If this set is actually infinite, then either it contains an actually infinite sequence or there are more sequences than any sequence has terms. The latter case however has been contradicted by the nested arithmogeometrical triangle:
1
1
22
3
31
322
3
31
322
4444
5
53
531
5322
54444
...
which cannot have a larger extension, neither in height nor in width.
Regards, WM
Martin Shobe
Jul 3, 2015, 8:18:21 AM7/3/15
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On 7/3/2015 6:51 AM, WM wrote:
> Am Donnerstag, 2. Juli 2015 17:19:40 UTC+2 schrieb Martin Shobe:
>> On 7/2/2015 9:19 AM, WM wrote:
>>>
>> However, if you were to
>> superimpose every downward sequence so that the same elements are always
>> written on top of each other, the result would be the reversal of the
>> original well-order.
>
> 0
> 1,0
> 2,1,0
> ...
>
> If this set is actually infinite, then either it contains an actually infinite sequence or there are more sequences than any sequence has terms.
What your this is referring to isn't a set.
> Am Donnerstag, 2. Juli 2015 17:19:40 UTC+2 schrieb Martin Shobe:
>> On 7/2/2015 9:19 AM, WM wrote:
>>>
>> However, if you were to
>> superimpose every downward sequence so that the same elements are always
>> written on top of each other, the result would be the reversal of the
>> original well-order.
>
> 0
> 1,0
> 2,1,0
> ...
>
> If this set is actually infinite, then either it contains an actually infinite sequence or there are more sequences than any sequence has terms.
[snip inanities]
Martin Shobe
WM
Jul 3, 2015, 8:40:58 AM7/3/15
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Regards, WM
Alan Smaill
Jul 3, 2015, 9:10:03 AM7/3/15
to
Julio Di Egidio <ju...@diegidio.name> writes:
> On Wednesday, July 1, 2015 at 6:55:03 PM UTC+1, Alan Smaill wrote:
>> George Greene <g***@email.unc.edu> writes:
>> > On Wednesday, July 1, 2015 at 5:45:04 AM UTC-4, Alan Smaill wrote:
>>
>> >> But then there are then multiple representations of the same real --
>> >> so this is not what reals "really" are.
>> >
>> > That's our problem as architects of the representors, not any actual
>> > problem with the representands.
>>
>> Fine.
>
> No, it is the opposite of fine.
I was agreeing that there is a problem for those who come up
> On Wednesday, July 1, 2015 at 6:55:03 PM UTC+1, Alan Smaill wrote:
>> George Greene <g***@email.unc.edu> writes:
>> > On Wednesday, July 1, 2015 at 5:45:04 AM UTC-4, Alan Smaill wrote:
>>
>> >> But then there are then multiple representations of the same real --
>> >> so this is not what reals "really" are.
>> >
>> > That's our problem as architects of the representors, not any actual
>> > problem with the representands.
>>
>> Fine.
>
> No, it is the opposite of fine.
with representations for real numbers, nothing else.
> Firstly, the argument rather goes
> from a conclusion on the "representors" to a conclusion on the
> "representands", not the other way round (!); then the objection just
> remains, that those "representors" do not "represent" real numbers.
(the latter if your notion of "represent" entails one-to-one correspondence
between representors and representands).
> Indeed, the claim is that they cannot: we just cannot get from
> potential to actual "for free".
>> >> This seems to be the objection -- it is easily enough dealt with,
> <snip>
>> It does need to be ruled out, though.
>> Otherwise all we know is that the number of representations
>> of reals >= the number of reals, and the number
>> of representations of reals is > than the number of naturals.
>
> No, you do not know any of that!
If you object to that, then you will not accept these
claims.
> Again, that and only that is the objection: that the argument tells
> *nothing* about *real numbers*.
> That said, whatever you have up your sleeve, it is maybe time to get
> it out: no more cards on the table that I can see. -- I could have a
> fight with George on many details, maybe I will, but that is marginal
> to the present topic.
>
> Julio
--
Alan Smaill
Alan Smaill
Jul 3, 2015, 9:40:03 AM7/3/15
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WM <wolfgang.m...@hs-augsburg.de> writes:
> Am Mittwoch, 1. Juli 2015 17:00:03 UTC+2 schrieb Alan Smaill:
>
> Am Mittwoch, 1. Juli 2015 17:00:03 UTC+2 schrieb Alan Smaill:
>
>> >> But then there are then multiple representations of the same real --
>> >
>> >
>> > Not by digit sequences.
>>
>> By finite definitions that specify potentially infinite digit sequences.
>
> That is absolutely correct.
>>
>> >> After all, the claim is that the reals
>> >> are uncountable, not the representations of reals.
>> >
>> > But this point is not supported by the diagonal argument.
>>
>> Did I say it was?
>
> Other supporters will be hard to find.
Why would the diagonal argument as given by Cantor support the claim
>>
>> By finite definitions that specify potentially infinite digit sequences.
>
> That is absolutely correct.
>>
>> >> After all, the claim is that the reals
>> >> are uncountable, not the representations of reals.
>> >
>> > But this point is not supported by the diagonal argument.
>>
>> Did I say it was?
>
> Other supporters will be hard to find.
that the reals are uncountable, rather than the representations
of reals? It doesn't.
>> > Further it is easily contradicted.
>>
>> Not so.
>>
>> > Every real has a finite definition. Every real is the meaning of a
>> > finite word. There are only countably many finite words. There are
>> > only as many meanings as there are languages. Languages have to be
>> > constructed. There is a finite number of languages at every
>> > time.
>> > There can never be more than countably many reals. QED.
>>
>> This of course fails to contradict the fact that two finite definitions
>> may refer to the same real.
>
> Tht is not contradicted.
finite definitions may refer to the same real.
I am glad to see that you now accept this obvious fact.
> "pi" or "the ratio of circumference to
> diameter" are two finite definitions referring to pi. In n languages
> number.
> That does cause any uncountability.
apply directly to reals.
>> There is a difference between the
>> potentially infinite set of finite definitions, and the
>> potentially infinite set of reals; and you have failed to
>> recognise this distinction.
>
> real number do not increase the set of real numbers and do not make it
> uncountable.
Please address what I wrote.
Multiple definitions for reals invalidate the inference:
set of representations of reals uncountable
=> set of reals uncountable.
(where "set of representations of reals uncountable" is
an *assumption*, not a claim).
You have the direction of inference backwards!!
>
> Regards, WM
--
Alan Smaill
Martin Shobe
Jul 3, 2015, 12:02:22 PM7/3/15
to
claims become provable in set theory, not some inconsistent theory you
dreamed up to mislead the unwary. As things currently stand, you've said
nothing about sets in this.
Martin Shobe
WM
Jul 3, 2015, 12:36:44 PM7/3/15
to
On Friday, 3 July 2015 18:02:22 UTC+2, Martin Shobe wrote:
> > The downward sequences are the elements of a set as well as the numbers 1, 22, 333, 4444, ... . The geometrical arrangement is mathematics, used to analyze the claims of set theory.
>
> Then you need to map those sequences to sets in such a way that your
> claims become provable in set theory,
The proof is done in mathematics. It is enough to show that set theory is incompatible with mathematics.
> > The downward sequences are the elements of a set as well as the numbers 1, 22, 333, 4444, ... . The geometrical arrangement is mathematics, used to analyze the claims of set theory.
>
> Then you need to map those sequences to sets in such a way that your
> claims become provable in set theory,
> not some inconsistent theory
I have not applied any "theory" other than the mathematical truth that there is no infinite natural number. By the way, this truth is even shared by set theory.
> you dreamed up to mislead the unwary.
Regards, WM
WM
Jul 3, 2015, 12:37:55 PM7/3/15
to
On Friday, 3 July 2015 15:40:03 UTC+2, Alan Smaill wrote:
> >> >> After all, the claim is that the reals
> >> >> are uncountable, not the representations of reals.
> >> >
> >> > But this point is not supported by the diagonal argument.
> >>
> >> Did I say it was?
> >
> > Other supporters will be hard to find.
>
> Why would the diagonal argument as given by Cantor support the claim
> that the reals are uncountable,
Cantor said so. People believe it.
> >> >> After all, the claim is that the reals
> >> >> are uncountable, not the representations of reals.
> >> >
> >> > But this point is not supported by the diagonal argument.
> >>
> >> Did I say it was?
> >
> > Other supporters will be hard to find.
>
> Why would the diagonal argument as given by Cantor support the claim
> that the reals are uncountable,
> rather than the representations
> of reals? It doesn't.
> Please address what I wrote.
>
> Multiple definitions for reals invalidate the inference:
>
> set of representations of reals uncountable
> => set of reals uncountable.
Regards, WM
George Greene
Jul 3, 2015, 1:14:26 PM7/3/15
to
On Friday, July 3, 2015 at 7:10:14 AM UTC-4, WM wrote:
> Every digit sequence like every sum of fractions is rational.
It IS NOT, you IDIOT! It is obviously possible to have a 3-element list of
> Every digit sequence like every sum of fractions is rational.
actually-infinite decimal-representations-of-a-real on which the first element
is pi and the second element is v/2, and the third element is e, NONE OF
which is rational! It is obviously possible to construct FINITE Turing Machines (with FINITE numbers of states and the same FINITE alphabet) THAT PRINT these sequences (despite the fact that they would need to run actually infinitely many steps to "finish" them, if you want them finished). The fact that no machine has yet physically computed or printed the 9^(9^(9^(9^9)))th digit of pi or e or 2^.5 DOES NOT STOP those FINITE sequences from existing and by the SAME argument does NOT STOP the actually infinite sequences following them from existing either!!
George Greene
Jul 3, 2015, 1:19:41 PM7/3/15
to
On Friday, July 3, 2015 at 7:10:14 AM UTC-4, WM wrote:
> An infinite sequence is not a name.
IT *IS*SO*TOO, DUMBASS.
The (relevant) infinite sequence that starts
1.4142... IS A NAME for the square root of 2.
The (relevant) infinite sequence that starts
3.14159265... IS A NAME for pi.
The (relevant) infinite sequence that starts
2.718281828... IS A NAME for e. Why is it (for that matter) that a one-letter
name LIKE *e*, for e, COUNTS as a name for e, but this actually-infinitely-wide
digit-sequence doesn't??? Why is it that both pi and count as names for
the relevant ratio, and that digit sequence doesn't?
Yes, the things I have typed are NOT those sequences, but the fact that something can't be pragmatically written or spoken does not stop it from naming what it names. Good grief.
Virgil
Jul 3, 2015, 1:25:00 PM7/3/15
to
In article <78aa779a-950a-42c7...@googlegroups.com>,
666666
extends it both in height and width!
So WM is
WRONG !
AGAIN ! !
AS USUAL ! ! !
extends it both in height and width!
So WM is
WRONG !
AGAIN ! !
AS USUAL ! ! !
WM
Jul 3, 2015, 1:28:59 PM7/3/15
to
On Friday, 3 July 2015 19:19:41 UTC+2, George Greene wrote:
> On Friday, July 3, 2015 at 7:10:14 AM UTC-4, WM wrote:
> > An infinite sequence is not a name.
>
> On Friday, July 3, 2015 at 7:10:14 AM UTC-4, WM wrote:
> > An infinite sequence is not a name.
>
> The (relevant) infinite sequence that starts
> 1.4142... IS A NAME for the square root of 2.
Yes, it is a name, but it is not an infinite sequence. "1.4142..." is a finite word with nine symbols, and every sequence which can be used as a name is finite. This very confusion, as shown by you here, is the basis of set theory.
> 1.4142... IS A NAME for the square root of 2.
Regards, WM
WM
Jul 3, 2015, 1:30:48 PM7/3/15
to
But it does not extend larger than any natural number.
Regards, WM
Virgil
Jul 3, 2015, 1:31:27 PM7/3/15
to
In article <9043b8a2-4ce8-4f75...@googlegroups.com>,
Until your alleged set is far more precisely defined you cannot prove
such properties for it as you have claimed for it!
WM <muec...@rz.fh-augsburg.de> wrote:
> Am Freitag, 3. Juli 2015 14:18:21 UTC+2 schrieb Martin Shobe:
> > On 7/3/2015 6:51 AM, WM wrote:
> > > Am Donnerstag, 2. Juli 2015 17:19:40 UTC+2 schrieb Martin Shobe:
> > >> On 7/2/2015 9:19 AM, WM wrote:
> > >>>
> > >> However, if you were to
> > >> superimpose every downward sequence so that the same elements are always
> > >> written on top of each other, the result would be the reversal of the
> > >> original well-order.
> > >
> > > 0
> > > 1,0
> > > 2,1,0
> > > ...
> > >
> > > If this set is actually infinite, then either it contains an actually
> > > infinite sequence or there are more sequences than any sequence has
> > > terms.
> >
> > What your this is referring to isn't a set.
>
> The downward sequences are the elements of a set
What set would that be?
> Am Freitag, 3. Juli 2015 14:18:21 UTC+2 schrieb Martin Shobe:
> > On 7/3/2015 6:51 AM, WM wrote:
> > > Am Donnerstag, 2. Juli 2015 17:19:40 UTC+2 schrieb Martin Shobe:
> > >> On 7/2/2015 9:19 AM, WM wrote:
> > >>>
> > >> However, if you were to
> > >> superimpose every downward sequence so that the same elements are always
> > >> written on top of each other, the result would be the reversal of the
> > >> original well-order.
> > >
> > > 0
> > > 1,0
> > > 2,1,0
> > > ...
> > >
> > > If this set is actually infinite, then either it contains an actually
> > > infinite sequence or there are more sequences than any sequence has
> > > terms.
> >
> > What your this is referring to isn't a set.
>
> The downward sequences are the elements of a set
Until your alleged set is far more precisely defined you cannot prove
such properties for it as you have claimed for it!
Virgil
Jul 3, 2015, 1:46:46 PM7/3/15
to
In article <d7b35914-1ab6-4a95...@googlegroups.com>,
set theory is a part of mathematics, as both are incompatible with WM's
worthless world of WMytheology
>
itself a natural number, IS infinite!
WM claims a set F such that F \ F = |N.
Inside WM's delusional world of WMytheology,
WM claims That a mapping from |N to Q+ in which
the mth power of the nth prime maps to m/n
cannot cover every m/n in Q+.
SO why should anyone ever be deluded by what WM
in his own wild weird wacky worthless world of WMytheology
claims!
WM <muec...@rz.fh-augsburg.de> wrote:
> On Friday, 3 July 2015 18:02:22 UTC+2, Martin Shobe wrote:
>
>
> > > The downward sequences are the elements of a set as well as the numbers
> > > 1, 22, 333, 4444, ... . The geometrical arrangement is mathematics, used
> > > to analyze the claims of set theory.
> >
> > Then you need to map those sequences to sets in such a way that your
> > claims become provable in set theory, not some inconsistent theory
> On Friday, 3 July 2015 18:02:22 UTC+2, Martin Shobe wrote:
>
>
> > > The downward sequences are the elements of a set as well as the numbers
> > > 1, 22, 333, 4444, ... . The geometrical arrangement is mathematics, used
> > > to analyze the claims of set theory.
> >
> > Then you need to map those sequences to sets in such a way that your
>
> The proof is done in mathematics. It is enough to show that set theory is
> incompatible with mathematics.
The only real issue is whether mathematics is a part of set theory or
> The proof is done in mathematics. It is enough to show that set theory is
> incompatible with mathematics.
set theory is a part of mathematics, as both are incompatible with WM's
worthless world of WMytheology
>
> I have not applied any "theory" other than the mathematical truth that there
> is no infinite natural number.
No one claims otherwise, But the number of natural numbers, not being
> is no infinite natural number.
itself a natural number, IS infinite!
>
> > you dreamed up to mislead the unwary.
>
> My argument is obviously unrefutable.
Inside WM's delusional world of WMytheology,
> > you dreamed up to mislead the unwary.
>
> My argument is obviously unrefutable.
WM claims a set F such that F \ F = |N.
Inside WM's delusional world of WMytheology,
WM claims That a mapping from |N to Q+ in which
the mth power of the nth prime maps to m/n
cannot cover every m/n in Q+.
SO why should anyone ever be deluded by what WM
in his own wild weird wacky worthless world of WMytheology
claims!
Virgil
Jul 3, 2015, 1:58:58 PM7/3/15
to
In article <33e21c8b-6ab0-4b9f...@googlegroups.com>,
> There are obviously more naturals than primes.
For every n in |N there is an nth prime in |N,
so where is it that there are more natural than primes?
NOWHERE outside of WM's worthless world of WMytheology!
> In the diagonal argument, however, it is not enough that the antidiagonal has
> the same cardinality of digits as the list has lines. In the diagonal
> argument there must be a precise equality.
If the list has a countably infinity of lines and the antidiagonal has a
countably infinity of digits, as is the case, there IS a precise
equality.
successful, but that does not make the original process any less
successful.
At least not anywhere outside of WM's worthless world of WMytheology.
And what WM dreams of inside his own wild weird wacky worthless world
of WMytheology is mathematically irrelevant
WM <wolfgang.m...@hs-augsburg.de> wrote:
> Am Donnerstag, 2. Juli 2015 19:38:36 UTC+2 schrieb George Greene:
>
>
> > there is a square because the number of nubmers (whatEVER it is) ACROSS
> > and the number of numbers DOWN are THE SAME number.
>
> No. They have same cardinality.
NO but yes?
> Am Donnerstag, 2. Juli 2015 19:38:36 UTC+2 schrieb George Greene:
>
>
> > there is a square because the number of nubmers (whatEVER it is) ACROSS
> > and the number of numbers DOWN are THE SAME number.
>
> No. They have same cardinality.
> There are obviously more naturals than primes.
so where is it that there are more natural than primes?
NOWHERE outside of WM's worthless world of WMytheology!
> In the diagonal argument, however, it is not enough that the antidiagonal has
> the same cardinality of digits as the list has lines. In the diagonal
> argument there must be a precise equality.
countably infinity of digits, as is the case, there IS a precise
equality.
r.
>
> If you double every line without applying the diagonalization, the argument
> fails.
Changing any successful process may make it into something less
>
> If you double every line without applying the diagonalization, the argument
> fails.
successful, but that does not make the original process any less
successful.
At least not anywhere outside of WM's worthless world of WMytheology.
And what WM dreams of inside his own wild weird wacky worthless world
of WMytheology is mathematically irrelevant
Virgil
Jul 3, 2015, 2:09:00 PM7/3/15
to
In article <5d3fb3dc-4577-4938...@googlegroups.com>,
is rational.
At least not outside of WM's worthless world of WMytheology.
value is irrational. Unless WM can represent it as a m/n for natural
umes m and n.
Well can you, punk?
WM <wolfgang.m...@hs-augsburg.de> wrote:
> Am Donnerstag, 2. Juli 2015 20:23:57 UTC+2 schrieb Virgil:
>
>
> > > Irrational rationals do not exist - not even in
> > > finished infinity.
> >
> > LEARN TO READ! Every real number is itself rational OR irrational!
> > That does not require any number to be both!
>
> Every digit sequence like every sum of fractions is rational.
NOT every infinite digits sequence nor sum of infinitely many fractions
> Am Donnerstag, 2. Juli 2015 20:23:57 UTC+2 schrieb Virgil:
>
>
> > > Irrational rationals do not exist - not even in
> > > finished infinity.
> >
> > LEARN TO READ! Every real number is itself rational OR irrational!
> > That does not require any number to be both!
>
> Every digit sequence like every sum of fractions is rational.
is rational.
At least not outside of WM's worthless world of WMytheology.
> > >
> > > > That depends on the construction of the original underlying sequence.
> > >
> > > No. The limit may be irrational, but not any digit sequence.
Sum_(n in |N) 1/10^(n^2) names an infinite decimal digit sequence whose
> > > > That depends on the construction of the original underlying sequence.
> > >
> > > No. The limit may be irrational, but not any digit sequence.
value is irrational. Unless WM can represent it as a m/n for natural
umes m and n.
Well can you, punk?
Virgil
Jul 3, 2015, 2:17:25 PM7/3/15
to
In article <c1fc2312-31e9-4909...@googlegroups.com>,
which outside WM's worthless world of WMytheology is infinite,
and the set of downward sequences is at least as large!
WM <wolfgang.m...@hs-augsburg.de> wrote:
> Am Donnerstag, 2. Juli 2015 20:32:24 UTC+2 schrieb Virgil:
>
>
> > The set of downward sequences is at least as large as the set of
> > starting points!
>
> This set is finite.
The set of starting points for downward sequences in |N is |N,
> Am Donnerstag, 2. Juli 2015 20:32:24 UTC+2 schrieb Virgil:
>
>
> > The set of downward sequences is at least as large as the set of
> > starting points!
>
> This set is finite.
which outside WM's worthless world of WMytheology is infinite,
and the set of downward sequences is at least as large!
Martin Shobe
Jul 3, 2015, 2:32:38 PM7/3/15
to
On 7/3/2015 11:36 AM, WM wrote:
> On Friday, 3 July 2015 18:02:22 UTC+2, Martin Shobe wrote:
>
>
>>> The downward sequences are the elements of a set as well as the numbers 1, 22, 333, 4444, ... . The geometrical arrangement is mathematics, used to analyze the claims of set theory.
>>
>> Then you need to map those sequences to sets in such a way that your
>> claims become provable in set theory,
>
> The proof is done in mathematics. It is enough to show that set theory is incompatible with mathematics.
It doesn't even touch set theory, neither in its set-up or reasoning.
> On Friday, 3 July 2015 18:02:22 UTC+2, Martin Shobe wrote:
>
>
>>> The downward sequences are the elements of a set as well as the numbers 1, 22, 333, 4444, ... . The geometrical arrangement is mathematics, used to analyze the claims of set theory.
>>
>> Then you need to map those sequences to sets in such a way that your
>> claims become provable in set theory,
>
> The proof is done in mathematics. It is enough to show that set theory is incompatible with mathematics.
The contradiction is all you.
>> not some inconsistent theory
> I have not applied any "theory" other than the mathematical truth that there is no infinite natural number. By the way, this truth is even shared by set theory.
>> you dreamed up to mislead the unwary.
>
> My argument is obviously unrefutable. Otherwise you would offer a logical argument.
Martin Shobe
Virgil
Jul 3, 2015, 2:35:04 PM7/3/15
to
In article <d8aeafdb-30cf-4aac...@googlegroups.com>,
only one for any irrational, so that WM's remark is, as usual,
irrelevant.
> So a proof about representations would be
> useless.
Not when limited to only decimal or only binary representations!
> >
> > Multiple definitions for reals invalidate the inference:
> >
> > set of representations of reals uncountable
> > => set of reals uncountable.
NOT when the number of different representations per real
is limited to 2, as it is for decimal representation or binary
representation or any other basal representation.
And the Cantor antidiagonal arguments used either base 2 or base 10,
so WM's objections again and as usual fail to invalidate Cantor!
WM <muec...@rz.fh-augsburg.de> wrote:
> On Friday, 3 July 2015 15:40:03 UTC+2, Alan Smaill wrote:
>
>
> > >> >> After all, the claim is that the reals
> > >> >> are uncountable, not the representations of reals.
> > >> >
> > >> > But this point is not supported by the diagonal argument.
> > >>
> > >> Did I say it was?
> > >
> > > Other supporters will be hard to find.
> >
> > Why would the diagonal argument as given by Cantor support the claim
> > that the reals are uncountable,
>
> Cantor said so. People believe it.
>
> > rather than the representations
> > of reals? It doesn't.
>
> There are more representations than reals. Every real number has infinitely
> many finite representations.
But at most two decimal (or binary) representations for any rational and
> On Friday, 3 July 2015 15:40:03 UTC+2, Alan Smaill wrote:
>
>
> > >> >> After all, the claim is that the reals
> > >> >> are uncountable, not the representations of reals.
> > >> >
> > >> > But this point is not supported by the diagonal argument.
> > >>
> > >> Did I say it was?
> > >
> > > Other supporters will be hard to find.
> >
> > Why would the diagonal argument as given by Cantor support the claim
> > that the reals are uncountable,
>
> Cantor said so. People believe it.
>
> > rather than the representations
> > of reals? It doesn't.
>
> There are more representations than reals. Every real number has infinitely
> many finite representations.
only one for any irrational, so that WM's remark is, as usual,
irrelevant.
> So a proof about representations would be
> useless.
> >
> > Multiple definitions for reals invalidate the inference:
> >
> > set of representations of reals uncountable
> > => set of reals uncountable.
is limited to 2, as it is for decimal representation or binary
representation or any other basal representation.
And the Cantor antidiagonal arguments used either base 2 or base 10,
so WM's objections again and as usual fail to invalidate Cantor!
Ben Bacarisse
Jul 3, 2015, 3:11:08 PM7/3/15
to
WM <muec...@rz.fh-augsburg.de> writes:
> This very confusion, as shown by you here, is the
> basis of set theory.
No. The axioms are the basis of set theory. You've not been able to
> This very confusion, as shown by you here, is the
> basis of set theory.
come with any alternative axioms because the results you want are a
mishmash of standard mathematics (your textbook) and the silly claims
made here (that are not in your textbook).
--
Ben.
Virgil
Jul 3, 2015, 4:00:20 PM7/3/15
to
In article <add591ca-e681-46a9...@googlegroups.com>,
On the contrary, it extends beyond every natural number,
until WM can name one that it does not extend beyond!
But even thus extended, does not prove any of the foolish things WM
claims for it!
until WM can name one that it does not extend beyond!
But even thus extended, does not prove any of the foolish things WM
claims for it!
Virgil
Jul 3, 2015, 4:02:30 PM7/3/15
to
In article <efc09353-de17-4ccb...@googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:
> On Friday, 3 July 2015 19:19:41 UTC+2, George Greene wrote:
> > On Friday, July 3, 2015 at 7:10:14 AM UTC-4, WM wrote:
> > > An infinite sequence is not a name.
> >
>
> > The (relevant) infinite sequence that starts
> > 1.4142... IS A NAME for the square root of 2.
>
> Yes, it is a name, but it is not an infinite sequence.
It names one! And a fairly familiar one at that!
> On Friday, 3 July 2015 19:19:41 UTC+2, George Greene wrote:
> > On Friday, July 3, 2015 at 7:10:14 AM UTC-4, WM wrote:
> > > An infinite sequence is not a name.
> >
>
> > The (relevant) infinite sequence that starts
> > 1.4142... IS A NAME for the square root of 2.
>
> Yes, it is a name, but it is not an infinite sequence.
Julio Di Egidio
Jul 3, 2015, 4:23:33 PM7/3/15
to
On Friday, July 3, 2015 at 2:10:03 PM UTC+1, Alan Smaill wrote:
> Julio Di Egidio <j***@diegidio.name> writes:
<snip>
> >> >> This seems to be the objection -- it is easily enough dealt with,
> > <snip>
> >> It does need to be ruled out, though.
> >> Otherwise all we know is that the number of representations
> >> of reals >= the number of reals, and the number
> >> of representations of reals is > than the number of naturals.
> >
> > No, you do not know any of that!
>
> Indeed this assumes a set-theoretic account of cardinality.
> If you object to that, then you will not accept these
> claims.
Nonsense. Liar.
> > Again, that and only that is the objection: that the argument tells
> > *nothing* about *real numbers*.
>
> So then you need to tell us what real numbers really are.
Another one who does not know what formality or mathematics or logic even mean. No, you should rather explain to us how it is that you claim the argument proves the real numbers uncountable: that is your claim, and that it is unsupported and in fact wrong is my objection. (But you'll still maintain you don't get it.)
> > That said, whatever you have up your sleeve, it is maybe time to get
> > it out: no more cards on the table that I can see. -- I could have a
> > fight with George on many details, maybe I will, but that is marginal
> > to the present topic.
>
> I cannot tell what exactly you are objecting to.
Idiot. But I am not holding my breath for an answer: you never answer questions anyway. Indeed until further news, the discussion is simply over: as the respondents here, including you, not even can manage to engage... as usual.
Julio
> Julio Di Egidio <j***@diegidio.name> writes:
<snip>
> > Indeed, the claim is that they cannot: we just cannot get from
> > potential to actual "for free".
>
> I was not addressing the actual/potential issue.
I know you will not address it, you never address anything at all.
> > potential to actual "for free".
>
> I was not addressing the actual/potential issue.
> >> >> This seems to be the objection -- it is easily enough dealt with,
> > <snip>
> >> It does need to be ruled out, though.
> >> Otherwise all we know is that the number of representations
> >> of reals >= the number of reals, and the number
> >> of representations of reals is > than the number of naturals.
> >
> > No, you do not know any of that!
>
> Indeed this assumes a set-theoretic account of cardinality.
> If you object to that, then you will not accept these
> claims.
> > Again, that and only that is the objection: that the argument tells
> > *nothing* about *real numbers*.
>
> So then you need to tell us what real numbers really are.
> > That said, whatever you have up your sleeve, it is maybe time to get
> > it out: no more cards on the table that I can see. -- I could have a
> > fight with George on many details, maybe I will, but that is marginal
> > to the present topic.
>
> I cannot tell what exactly you are objecting to.
Julio
Virgil
Jul 3, 2015, 5:53:50 PM7/3/15
to
In article <2b5be537-31c9-407f...@googlegroups.com>,
for JDE to comprehend, but his second "diagonal" proof is clearly a pons
asinorum separating Julio from mathematics.
Julio Di Egidio <ju...@diegidio.name> wrote:
> No, you should rather explain to us how it is that you claim the argument
> proves the real numbers uncountable
Cantor explained it twice. His first proof is clearly far too esoteric
> No, you should rather explain to us how it is that you claim the argument
> proves the real numbers uncountable
for JDE to comprehend, but his second "diagonal" proof is clearly a pons
asinorum separating Julio from mathematics.
Zeit Geist
Jul 3, 2015, 6:13:55 PM7/3/15
to
On Friday, July 3, 2015 at 6:40:03 AM UTC-7, Alan Smaill wrote:
> WM <wolfgang.m...@hs-augsburg.de> writes:
>
> > Am Mittwoch, 1. Juli 2015 17:00:03 UTC+2 schrieb Alan Smaill:
> >
> >> >> But then there are then multiple representations of the same real --
> >> >
> >> > Not by digit sequences.
> >>
> >> By finite definitions that specify potentially infinite digit sequences.
> >
> > That is absolutely correct.
> >>
> >> >> After all, the claim is that the reals
> >> >> are uncountable, not the representations of reals.
> >> >
> >> > But this point is not supported by the diagonal argument.
> >>
> >> Did I say it was?
> >
> > Other supporters will be hard to find.
>
> Why would the diagonal argument as given by Cantor support the claim
> that the reals are uncountable, rather than the representations
> of reals? It doesn't.
There is another proof of the uncountability of the real numbers which does NOT appeal to any representation of the real numbers. The proof simply uses the properties of a complete totally ordered field ( which R is ), and sequences ( that is, lists ) of real numbers. No need to worry that our "usual" representation of real numbers does give a injection from the representations to the real numbers themselves ( whatever they are ).
> WM <wolfgang.m...@hs-augsburg.de> writes:
>
> > Am Mittwoch, 1. Juli 2015 17:00:03 UTC+2 schrieb Alan Smaill:
> >
> >> >> But then there are then multiple representations of the same real --
> >> >
> >> > Not by digit sequences.
> >>
> >> By finite definitions that specify potentially infinite digit sequences.
> >
> > That is absolutely correct.
> >>
> >> >> After all, the claim is that the reals
> >> >> are uncountable, not the representations of reals.
> >> >
> >> > But this point is not supported by the diagonal argument.
> >>
> >> Did I say it was?
> >
> > Other supporters will be hard to find.
>
> Why would the diagonal argument as given by Cantor support the claim
> that the reals are uncountable, rather than the representations
> of reals? It doesn't.
> > Regards, WM
>
> --
> Alan Smaill
Zeit Geist
Jul 3, 2015, 6:18:25 PM7/3/15
to
On Friday, July 3, 2015 at 9:36:44 AM UTC-7, WM wrote:
> On Friday, 3 July 2015 18:02:22 UTC+2, Martin Shobe wrote:
>
>
> > > The downward sequences are the elements of a set as well as the numbers 1, 22, 333, 4444, ... . The geometrical arrangement is mathematics, used to analyze the claims of set theory.
> >
> > Then you need to map those sequences to sets in such a way that your
> > claims become provable in set theory,
>
> The proof is done in mathematics. It is enough to show that set theory is incompatible with mathematics.
Euclidean geometry used to be believed, and still by some of your fellow cranks, to be the only mathematical geometry. Hence, according to your premise, non-Euclidean geometry is incompatible with mathematics. Therefore, it is nonsense. Please, get a clue.
> On Friday, 3 July 2015 18:02:22 UTC+2, Martin Shobe wrote:
>
>
> > > The downward sequences are the elements of a set as well as the numbers 1, 22, 333, 4444, ... . The geometrical arrangement is mathematics, used to analyze the claims of set theory.
> >
> > Then you need to map those sequences to sets in such a way that your
> > claims become provable in set theory,
>
> The proof is done in mathematics. It is enough to show that set theory is incompatible with mathematics.
> Regards, WM
ZG
Zeit Geist
Jul 3, 2015, 6:27:23 PM7/3/15
to
> Regards, WM
ZG
Virgil
Jul 3, 2015, 8:29:31 PM7/3/15
to
On Friday, July 3, 2015 at 9:36:44 AM UTC-7, WM wrote:
> On Friday, 3 July 2015 18:02:22 UTC+2, Martin Shobe wrote:
>
>
> > > The downward sequences are the elements of a set as well as the
> > > numbers 1, 22, 333, 4444, ... . The geometrical arrangement is
> > > mathematics, used to analyze the claims of set theory.
> >
> > Then you need to map those sequences to sets in such a way that
> > your claims become provable in set theory,
>
> The proof is done in mathematics.
All the mathematics WM is capable of doing can be done in set theory via
> On Friday, 3 July 2015 18:02:22 UTC+2, Martin Shobe wrote:
>
>
> > > The downward sequences are the elements of a set as well as the
> > > numbers 1, 22, 333, 4444, ... . The geometrical arrangement is
> > > mathematics, used to analyze the claims of set theory.
> >
> > Then you need to map those sequences to sets in such a way that
> > your claims become provable in set theory,
>
> The proof is done in mathematics.
the Peano postulates and what can be derived from them, although not in
the WM corrupt form of set theory WM tries to sell in his WMytheology.
> It is enough to show that set
> theory is incompatible with mathematics.
are also incompatible with any proper form of set theory, so WM is in no
position to judge either!
Virgil
Jul 3, 2015, 8:49:08 PM7/3/15
to
On Friday, July 3, 2015 at 6:40:03 AM UTC-7, Alan Smaill wrote:
> WM <wolfgang.m...@hs-augsburg.de> writes:
>
> > Am Mittwoch, 1. Juli 2015 17:00:03 UTC+2 schrieb Alan Smaill:
> >
> >> >> But then there are then multiple representations of the same real --
> >> >
> >> > Not by digit sequences.
> >>
> >> By finite definitions that specify potentially infinite digit sequences.
> >
> > That is absolutely correct.
> >>
> >> >> After all, the claim is that the reals
> >> >> are uncountable, not the representations of reals.
> >> >
> >> > But this point is not supported by the diagonal argument.
> >>
> >> Did I say it was?
> >
> > Other supporters will be hard to find.
>
> Why would the diagonal argument as given by Cantor support the claim
> that the reals are uncountable, rather than the representations
> of reals?
The particular class of representations under consideration,
> WM <wolfgang.m...@hs-augsburg.de> writes:
>
> > Am Mittwoch, 1. Juli 2015 17:00:03 UTC+2 schrieb Alan Smaill:
> >
> >> >> But then there are then multiple representations of the same real --
> >> >
> >> > Not by digit sequences.
> >>
> >> By finite definitions that specify potentially infinite digit sequences.
> >
> > That is absolutely correct.
> >>
> >> >> After all, the claim is that the reals
> >> >> are uncountable, not the representations of reals.
> >> >
> >> > But this point is not supported by the diagonal argument.
> >>
> >> Did I say it was?
> >
> > Other supporters will be hard to find.
>
> Why would the diagonal argument as given by Cantor support the claim
> that the reals are uncountable, rather than the representations
> of reals?
non-terminating decimal digit sequences is provably uncountable if and
only if the set of reals it represents is uncountable, as any proof of
either being uncountable proves both uncountable!
Or does now WM claim that when no more then two such decimal sequEnces
can represent any one real number that more than "HALF" of a proven to
be UNcountable set of decimal sequences can be UNcountable?
Even as little as 1% of an uncountable set is itself uncountable!
At least everywhere outside of WM's worthless world of WMytheology!
Nam Nguyen
Jul 3, 2015, 9:42:43 PM7/3/15
to
On 03/07/2015 4:18 PM, Zeit Geist wrote:
> On Friday, July 3, 2015 at 9:36:44 AM UTC-7, WM wrote:
>> On Friday, 3 July 2015 18:02:22 UTC+2, Martin Shobe wrote:
>>
>>
>>>> The downward sequences are the elements of a set as well as the numbers 1, 22, 333, 4444, ... . The geometrical arrangement is mathematics, used to analyze the claims of set theory.
>>>
>>> Then you need to map those sequences to sets in such a way that your
>>> claims become provable in set theory,
>>
>> The proof is done in mathematics. It is enough to show that set theory is incompatible with mathematics.
>
> Euclidean geometry used to be believed, and still by some of your fellow cranks, to be the only mathematical geometry.
Yeah. Like some people still believe there's only one kind of the
> On Friday, July 3, 2015 at 9:36:44 AM UTC-7, WM wrote:
>> On Friday, 3 July 2015 18:02:22 UTC+2, Martin Shobe wrote:
>>
>>
>>>> The downward sequences are the elements of a set as well as the numbers 1, 22, 333, 4444, ... . The geometrical arrangement is mathematics, used to analyze the claims of set theory.
>>>
>>> Then you need to map those sequences to sets in such a way that your
>>> claims become provable in set theory,
>>
>> The proof is done in mathematics. It is enough to show that set theory is incompatible with mathematics.
>
> Euclidean geometry used to be believed, and still by some of your fellow cranks, to be the only mathematical geometry.
natural numbers.
It seems in some areas of believing and mathematical reasoning,
the differences between some of the crank and the non-crank have
evaporated.
--
-----------------------------------------------------
There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI
WM
Jul 4, 2015, 8:12:13 AM7/4/15
to
On Saturday, 4 July 2015 00:27:23 UTC+2, Zeit Geist wrote:
> There are names, which are finite strings of symbols, for real numbers; and ALL of them have more than one name. Now, if we take the set of names for real numbers and mod out over names that refer to the same real number, then we find that IN ANY REALITY, there is no one-to-one function from N ONTO those equivalence classes.
Your argumnt is rubbish. We do not need a further proof of countability if the set is provably a subset of a countable set.
> There are names, which are finite strings of symbols, for real numbers; and ALL of them have more than one name. Now, if we take the set of names for real numbers and mod out over names that refer to the same real number, then we find that IN ANY REALITY, there is no one-to-one function from N ONTO those equivalence classes.
Regards, WM
WM
Jul 4, 2015, 8:12:24 AM7/4/15
to
On Saturday, 4 July 2015 00:13:55 UTC+2, Zeit Geist wrote:
> > Why would the diagonal argument as given by Cantor support the claim
> > that the reals are uncountable, rather than the representations
> > of reals? It doesn't.
>
> There is another proof of the uncountability of the real numbers
Don't chnge the topic Here we are concerned with the diagonal argument. The other proofs have been refuted separately.
> > Why would the diagonal argument as given by Cantor support the claim
> > that the reals are uncountable, rather than the representations
> > of reals? It doesn't.
>
> There is another proof of the uncountability of the real numbers
Regards, WM
WM
Jul 4, 2015, 8:12:31 AM7/4/15
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On Friday, 3 July 2015 21:11:08 UTC+2, Ben Bacarisse wrote:
> WM <muec...@rz.fh-augsburg.de> writes:
>
> > This very confusion, as shown by you here, is the
> > basis of set theory.
>
> No. The axioms are the basis of set theory.
Which axiom says that infinity is actual, i.e., that the number of natural numbers is a fixed quantity larger than every natural number?
> WM <muec...@rz.fh-augsburg.de> writes:
>
> > This very confusion, as shown by you here, is the
> > basis of set theory.
>
> No. The axioms are the basis of set theory.
Regards, WM
WM
Jul 4, 2015, 8:13:21 AM7/4/15
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On Friday, 3 July 2015 20:32:38 UTC+2, Martin Shobe wrote:
> > The proof is done in mathematics. It is enough to show that set theory is incompatible with mathematics.
>
> It doesn't even touch set theory,
My proof uses the infinite set {1, 22, 333, 4444, ...} of natural numbers. My proof shows by a special arrangement that this set has not aleph_0 elements.
> > The proof is done in mathematics. It is enough to show that set theory is incompatible with mathematics.
>
> It doesn't even touch set theory,
Regards, WM
WM
Jul 4, 2015, 8:13:30 AM7/4/15
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On Friday, 3 July 2015 19:46:46 UTC+2, Virgil wrote:
> > I have not applied any "theory" other than the mathematical truth that there
> > is no infinite natural number.
>
> No one claims otherwise, But the number of natural numbers, not being
> itself a natural number, IS infinite!
Never in the following sequence:
> > I have not applied any "theory" other than the mathematical truth that there
> > is no infinite natural number.
>
> No one claims otherwise, But the number of natural numbers, not being
> itself a natural number, IS infinite!
1
1
22
3
31
322
3
31
322
4444
5
53
531
5322
54444
...
WM
Jul 4, 2015, 8:32:07 AM7/4/15
to
On Saturday, 4 July 2015 00:27:23 UTC+2, Zeit Geist wrote:
>
>
There are names, which are finite strings of symbols, for real numbers; and ALL of them have more than one name.
Eadch one has infinitely many names.
> Now, if we take the set of names for real numbers and mod out over names that refer to the same real number, then we find that IN ANY REALITY, there is no one-to-one function from N ONTO those equivalence classes. Except, maybe in your reality, because your reality is inconsistent!
Every set has a definition, in fact every set has infinitely many definitions, but the set of all sets does not exist.
Same holds for real numbes. Same holds for natural numbers. Same holds for every infinite set.
Regards, WM
Ben Bacarisse
Jul 4, 2015, 9:22:51 AM7/4/15
to
every set X, there does not exist a bijective function from X to P(X).
What are your axioms, and why does that proof not work with yours?
--
Ben.
Martin Shobe
Jul 4, 2015, 10:06:00 AM7/4/15
to
On 7/4/2015 7:13 AM, WM wrote:
> On Friday, 3 July 2015 20:32:38 UTC+2, Martin Shobe wrote:
>
>
>>> The proof is done in mathematics. It is enough to show that set theory is incompatible with mathematics.
>>
>> It doesn't even touch set theory,
>
> My proof uses the infinite set {1, 22, 333, 4444, ...} of natural numbers.
Not really, there's nothing in it that requires that the referents of
> On Friday, 3 July 2015 20:32:38 UTC+2, Martin Shobe wrote:
>
>
>>> The proof is done in mathematics. It is enough to show that set theory is incompatible with mathematics.
>>
>> It doesn't even touch set theory,
>
> My proof uses the infinite set {1, 22, 333, 4444, ...} of natural numbers.
the symbols you use form a set.
> My proof shows by a special arrangement that this set has not aleph_0 elements.
Martin Shobe
Virgil
Jul 4, 2015, 11:24:46 AM7/4/15
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In article <fa261872-c01a-46d6...@googlegroups.com>,
One can map each member of that sequence to its successor, thus mapping
the set of the to a proper subset of itself, thus proving its actual
infiniteness according the standard definition of actual infinitenesss.
At least one can do so everywhere outside of WM's worthless world of
WMytheology.
the set of the to a proper subset of itself, thus proving its actual
infiniteness according the standard definition of actual infinitenesss.
At least one can do so everywhere outside of WM's worthless world of
WMytheology.
Virgil
Jul 4, 2015, 11:30:20 AM7/4/15
to
In article <e08ccc7d-6f2e-4510...@googlegroups.com>,
> Every set has a definition
The standard field of real numbers has a definition from which the set
of real numbers is derived, and one cannot satisfy the definition of
that field with any merely countable set of reals!
WM <muec...@rz.fh-augsburg.de> wrote:
> On Saturday, 4 July 2015 00:27:23 UTC+2, Zeit Geist wrote:
>
> >
> There are names, which are finite strings of symbols, for real numbers; and
> ALL of them have more than one name.
>
> Eadch one has infinitely many names.
>
> > Now, if we take the set of names for real numbers and mod out over names
> > that refer to the same real number, then we find that IN ANY REALITY,
> > there is no one-to-one function from N ONTO those equivalence classes.
> > Except, maybe in your reality, because your reality is inconsistent!
>
> Simple resolution. Like the set of all sets the set of real numbers is not a
> set.
It is a set everywhere outside of WM's worthless world of WMytheology.
> On Saturday, 4 July 2015 00:27:23 UTC+2, Zeit Geist wrote:
>
> >
> There are names, which are finite strings of symbols, for real numbers; and
> ALL of them have more than one name.
>
> Eadch one has infinitely many names.
>
> > Now, if we take the set of names for real numbers and mod out over names
> > that refer to the same real number, then we find that IN ANY REALITY,
> > there is no one-to-one function from N ONTO those equivalence classes.
> > Except, maybe in your reality, because your reality is inconsistent!
>
> Simple resolution. Like the set of all sets the set of real numbers is not a
> set.
> Every set has a definition
The standard field of real numbers has a definition from which the set
of real numbers is derived, and one cannot satisfy the definition of
that field with any merely countable set of reals!
Virgil
Jul 4, 2015, 11:39:01 AM7/4/15
to
In article <69ee5b44-d004-4dbe...@googlegroups.com>,
That set, with the well-ordering shown, admits an injection to a proper
subset by mapping from each member to the next, thus satisfying the
definition of a set being actually infinite.
And, at least everywhere outside WM's worthless world of WMytheology.
every and any well-ordered set with no last member is similarly actually
infinite by reason of being injectable into a proper subset of itself!
subset by mapping from each member to the next, thus satisfying the
definition of a set being actually infinite.
And, at least everywhere outside WM's worthless world of WMytheology.
every and any well-ordered set with no last member is similarly actually
infinite by reason of being injectable into a proper subset of itself!
Virgil
Jul 4, 2015, 11:46:00 AM7/4/15
to
In article <3b5ec51a-d10b-4472...@googlegroups.com>,
In proper set theories, a set is finite if and only if there is no
injection from that set to any of its proper subsets, and not finite (or
infinite) whenever there does exist a mapping from that set to a proper
subset.
By those definitions, any non-empty well-ordered set without a last
member, like |N, is actually infinite!
injection from that set to any of its proper subsets, and not finite (or
infinite) whenever there does exist a mapping from that set to a proper
subset.
By those definitions, any non-empty well-ordered set without a last
member, like |N, is actually infinite!
Virgil
Jul 4, 2015, 11:52:02 AM7/4/15
to
The standard definition says that a set is infinite if and only if there
exists an injection of the set to a proper subset.
By this definition, every Peano set, including |N, is actually infinite.
That WM prefers some other definition inside his own wild weird wacky
worthless world of WMytheology is irrelevant in proper mathematics!
Virgil
Jul 4, 2015, 11:57:18 AM7/4/15
to
In article <b1a7700f-aa8f-493d...@googlegroups.com>,
for which Cantor provided two proofs, neither of which WM can invalidate
without special assumptions acceptable nowhere outside of his own wild
weird wacky worthless world of WMytheology.
Here we are concerned with the diagonal argument. Which OUTSIDE of WM's
worthless world of WMytheology, WM cannot refute!
WM <muec...@rz.fh-augsburg.de> wrote:
> On Saturday, 4 July 2015 00:13:55 UTC+2, Zeit Geist wrote:
>
>
> > > Why would the diagonal argument as given by Cantor support the claim
> > > that the reals are uncountable, rather than the representations
> > > of reals?
> >
> On Saturday, 4 July 2015 00:13:55 UTC+2, Zeit Geist wrote:
>
>
> > > Why would the diagonal argument as given by Cantor support the claim
> > > that the reals are uncountable, rather than the representations
> > > of reals?
> >
> > There is another proof of the uncountability of the real numbers
>
> Don't chnge the topic
The topic is the validity of the claimed uncountability of the reals,
>
> Don't chnge the topic
for which Cantor provided two proofs, neither of which WM can invalidate
without special assumptions acceptable nowhere outside of his own wild
weird wacky worthless world of WMytheology.
Here we are concerned with the diagonal argument. Which OUTSIDE of WM's
worthless world of WMytheology, WM cannot refute!
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