Re: Galileo's Paradox and the Project of the Reals

From: Six Letters
The bijection between N and S is an artefact of infinity.
Not necessarily so. For example, consider the following bijection
between the positive integers and the positive even integers:
n=0
lp: n=n+1
output pair(n, 2*n)
go lp
The pairs output by that program define a bijection.
Every positive integer is a first-element of exactly one pair that
is output, and vice versa.
Every positive even integer is a second-element of exactly one pair
that is output, and vice versa.
Whch proves what exactly?

That in a finite number of characters, using a well-defined
language, I've specified a bijection between the positive integers
N and the positive even integers S. Nothing artefact about that.
It's right there for you to see in four lines of code.

What I want you to try and do is to take seriously the idea that
the infinite never ends or stops.
What you just said has no meaning whatsoever.
On the contrary, it's very simple. You just aren't prepared to
consider the ramifications.

It's more like *you* aren't prepared to say anything meaningful,
preferring instead to mumble words together randomly.
Perhaps you have some idea in your mind, but you just aren't
capable of making a sentence to express it. Or perhaps you don't
have an idea and are just trying to "snow" us. It ain't working. I
ain't snowed by your gibberish.

Counting is unfinishable.
That statement is not true in general.
It depends very much on what you are counting.
Some counting tasks are finishable and some are not.
You know what I mean.

Nope. You need to learn how to communicate precisely, say what you
mean instead of saying something else and expecting us to gusss
what you really mean. When you say "Counting is unfinishable." it
seems to mean "Every act of counting is unfinishable.", which is
false, as I pointed out. Now if you mean something else, such as
"Some acts of counting are unfinishable, others are't." then you
need to say that clearly. If you mean something else entirely, then
you haven't even begun to communicate.

Cantor's Diagonal Argument does not work
Yes it does.
Oh no it doesn't.

Cantor's diagonal argument shows that if you assume a bijection
between the positive integers and the reals in the interval [0,1],
you get a contradiction. Do you disagree that such a contradiction
is achieved under that assumption, or do you somehow claim that's
"not working" despite the contradiction having been achived?
Achiving that contradiction sure sounds like "working" to me.
It's just like Euclid's proof that no rational number can satisfy
the equation x^2 = 2, by assuming such a rational number solution
and getting a contradiction. You got a problem with showing
something false by deriving a contradiction from it?

You go to the right of the decimal point, you get reals; you go
to the left of the decimal point, you get infinite integers.

The way you get reals by a neverending process of generating digits
to the right of a decimal point is by explicitly constructing a
neverending chain of nested intervals. For example:
print ".";
while true do { print "1"; print "4"; print "2";
print "8"; print "5"; print "7"; }
constructs a neverending sequence of digits to the right of the
decimal point whose leftmost 73 characters (1 decimal point and 72
digits) look like:
.142857142857142857142857142857142857142857142857142857142857142857142857
and which thereby constructs a sequence of nested intervals whose
first eleven elements are:
[0,1], [.1,.2], [.14,.15], [.142,.143], [.1428],.1429], [.14285],.14286],
[.142857],.142858], [.1428571],.1428572], [.14285714],.14285715],
[.142857142],.142857143], [.1428571428],.1428571429]
The upper and lower bounds of those intervals define a Cauchy
sequence which thereby define a real number, and the sets of
rationals eventually excluded below and above the intervals
respectively define the two halves of a Dedekind cut which thereby
define a real number, and those are in fact equal numbers in the
sense of the usual mapping between Cauchy sequence equivalance
classes and Dedekind cuts. Thus the sequence of nested intervals
define a real number in either sense of "real number" (Cauchy or
Dedekind construction). In fact the particular "real number"
constructed there is equal to the rational number 1/7 via the usual
mapping between rational numbers and "rational reals", a subset of
the reals.

In what sense do your "neverending sequence of digits to left of decimal
point" generate anything that can meaningfully be called an "infinite
integer"? If you use a metric whereby such sequences generate a Cauchy
sequence, you don't get anything that is either "infinite" in any
meaningful sense, nor anything which is an "integer" in any
meaningful sense. For example, if you use the eleven-adic metric,
with digits selected from the set {0,1,2,3,4,5,6,7,8,9,A} (with 'A'
having the usual hexadecimal-digit meaning as "ten" just as '9' has
the meaning "nine"), then you can get such sequences as:
print "."; printToLeft "4";
while true do { printToLeft "7"; printToLeft "3"; }
That particular one converges to the rational number 1/3 in the
eleven-adic metric. Nothing infinite nor integer about that!!
So obviously either you're mistaken when you claim such a value is
infinite, or you're mistaken when you claim such a value is in any
sense "integer", or you're not talking about a p-adic metric to
give meaning to your neverending sequence of digits to left of
decimal point". I'll give you the benefit of doubt and presume
you do *not* mean any p-adic metric, but have some other idea how
to give meaning to such a neverending-leftward sequence of digits
in the context of considering them "numbers". So please tell us
what your proposed rule is for adding and multiplying such
sequences.

Note that arithmetic (add/mul) on nevereding-rightward sequences of
digits, using the real (absolute-value) metric/norm, have a
well-defined meaning:
- As nested intervals: Use interval arithmetic on the sequence of
sums or products of corresponding terms.
- As Cauchy sequences: Use all four possible combiations of
endpoints of corresponding intervals, sum or product thereof.
- As Dedekind cuts: Use the nested-interval definition above, and
convert to upper/lower Dedekind cut via upper/lower exclusion
from interval.
All three definitions give the same result.

Likewise arithmetic on neverending-leftward sequences of digits,
using p-adic metric/norm, have a well-defined meaning:
- As Cauchy sequences: Use all four possible combiations of
endpoints of corresponding intervals, sum or product thereof.
- As digit sequences: Simply compute the successive digits by the
standard grade-school algorithm, extended to the left forever,
for example with eleven-adic norm, addition is easy:
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :carries
3737373737373737373737373737373737373737373737373737374. :first addend (1/3)
1919191919191919191919191919191919191919191919191919192. :second addend (1/6)
---------------------------------------------------------
5555555555555555555555555555555555555555555555555555556. :sum (1/2)

Eleven-adic multiplication isn't much harder:
3737373737373737373737373737373737373737373737373737374. :multiplicand (1/3)
5555555555555555555555555555555555555555555555555555556. :multiplier (1/2)
---------------------------------------------------------
0000000000000000000000000000000000000000000000000000002. :multiplicand * 6.
373737373737373737373737373737373737373737373737373739 :multiplicand * 50.
73737373737373737373737373737373737373737373737373739 :multiplicand * 500.
3737373737373737373737373737373737373737373737373739 :multiplicand * 5000.
737373737373737373737373737373737373737373737373739 :multiplicand * 50000.
-----
19192. :partial sum so-far
Continuing to generate :multiplicand * 500...00. and adding, the
partial sum eventually looks like:
1919191919191919191919191919191919191919191919191919192. :partial sum so-far
Anyway, it's clear you get the 11-adic representation of 1/6.

So if you use similar-looking neverending-leftward sequences of
digits to represent some new sort of number you invented, how
exactly do *you* define addition and multiplication on them??

Of course, going right we get closer and closer to some finite
value, or rather range of values in most cases -- that is the
point.

Actually at any point along the sequence of nested intervals, you
get an interval of values. But there's only a single real number
that is in every one of those intervals. You *never* get an
interval or as you call it "range" of values in the intersection of
all the intervals in the sequence. Every real number except that
one is eventually eliminated from some interval in the sequece.
There's not a single case where there's a range of values in the
intersection of the nested intervals. So your "most cases" is a
complete lie, a fucking lie, and you should stop posting such lies.

I'm not sure why people stopped worrying about sq. rt of 2 being
irrational. I feel it has something to tell us about number.

Of course it has something to tell you about that number: It's not
rational. So you have to **extend** the rationals in some way to
include it. It isn't *already* in the rationals you started with.
If you have an equation in rationals x^2=2, it has no solution
there. It's only in an extended ring beyond the rationals that you
can artificially construct a solution to it. It's just like 1/2
which isn't in the integers. If you have a equation 2*x=1 in the
integers, it has no solution. It's only by extending the integers
that you can artificially construct a number 1/2 which is a
solution to that equation.

People stopped worrying because worrying causes acid indigestion
and "heartburn" and ulcers and high blood pressure and heart
attacks and strokes. It's stupid to worry about the fact that 2*x=1
has no solution in integers, or that x*x=2 has no solution in
rationals or in 2-adics either, or that x*x=-1 has no solution in
reals.

Or take pi. Is there some deep incommensurability between the
curvilinear measure of the circle and its rectilinear diameter,
hiding behind the irrationality of the ration?

It's not hiding. It's right there for everyone to see.
There's no polynomial equation with rational coefficients, except
the trivial kind such as 5=5, which are satisfied by pi.
If 1/2 doesn't satisfy any equation of the form
x + n = m (m,k integer),
and if sqrt(2) doesn't satisfy any equation of the form
k*x + n = m (k,n,m rational, k nonzero),
and if cuberoot(2) doesn't satisfy any equation of the form
m*x**2 + n*x = k (m,n,k rational, m nonzero),
then what's "hiding" about the fact that there's a real number that
doesn't satisfy *any* nontrivial polynomial equation (with rational
coefficients) whatsoever? Is there any reason you would expect that
for *every* number r which is the limit of a sequence there'd be
some nontrivial polynomial that it satisfies?? There's certainly no
way to construct such a polynomial or even come close.
For example, what polynomial do you think would be satisfied by the
limit of this simple-looking series: 1 - 1/2 + 1/3 - 1/4 + 1/5 etc.
For example, pick a prime p, and construct the p-adic
representation of each partial sum. See if you can find a pattern
in the digit sequences that leads you to a polynomial that the
limit satisfies. Ain't gonna happen.

any particular real that can be instantiated geometrically or
algebraicly or however is not the problem. It's the vast bottomless
sea of irrationals that are interminable and indeterminate in
exactly the way infinite integers are.

The "depth" of that "sea" depends very much on whether you accept
the Axiom of Choice, or a Turing Oracle, or a Goedel Oracle, to
posit the existance of uncomputable real numbers. Any computable
real number is neither interminable nor indeterminate, so it's the
"other" real numbers, if you posit any, which you must be talking
about??

There was a young chap called Maas,
Who could not infinity parse,
All of his words,
Were like so many turds,
That came from his flatulent Arse.

That doesn't even rhyme. And I can parse infinity just fine if the
grammar is appropriate.

There once was a guy named Six Letters,
Who let numbers get him the betters.
He couldn't grok Canter,
Nor stop his dumb banter.
He knows as much math as some Setters.
<http://www.esaa.com/>
.

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