In article <50b4624a-fd59-42b4-a988-64987b8637a8@googlegroups.com>,
Julio Di Egidio <julio@diegidio.name> wrote:
On Tuesday, June 30, 2015 at 7:20:49 AM UTC+1, Virgil wrote:
In article <7cdade36-3dd3-4b73-a5f5-dc56d7fce042@googlegroups.com>,<snip>
Julio Di Egidio <j***@diegidio.name> wrote:
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. \
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
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
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
have a set of all naturals, much less a set of all rationals or a set of
all reals.
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.
Prove that measure([0,1]) = 1.
On 06/30/2015 06:24 PM, Virgil wrote:
In article <50b4624a-fd59-42b4-a988-64987b8637a8@googlegroups.com>,
Julio Di Egidio <julio@diegidio.name> wrote:
On Tuesday, June 30, 2015 at 7:20:49 AM UTC+1, Virgil wrote:
In article <7cdade36-3dd3-4b73-a5f5-dc56d7fce042@googlegroups.com>,<snip>
Julio Di Egidio <j***@diegidio.name> wrote:
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. \
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
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
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
have a set of all naturals, much less a set of all rationals or a set of
all reals.
On 07/02/2015 09:36 AM, Ross A. Finlayson wrote:
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.
On 07/14/2015 06:23 AM, Ross A. Finlayson wrote:
Prove that measure([0,1]) = 1.
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