Anyone here into math?

[Guest [ch...]]

...

Godel proved that logic can be either consistent or complete but not both.

We chose incompleteness over inconsistency.

...

We use subtraction on the natural numbers to construct/define integers.

We use division on the integers to construct/define rational numbers.

We use Dedekind cuts on rational numbers to construct/define the real numbers

...

Regarding computability I was talking about some numbers (pi in particular).

The real numbers are uncountable so most of them are not computable

but ALL algebraic numbers (like √2) and many transcendental numbers (like pi) are computable.

When we say a number is computable we mean that it can be computed

to within any desired precision by a terminating (finite) algorithm.

What you seem to be talking about is HYPERcomputation which appears to be impossible in our universe.

For that one needn't invoke Godel or Turing just Einstein and his cosmic speed limit (that of light).

 

Now I know infinity can be perplexing as nothing in nature is truly infinite,

so here is a finite compilation of mathematical facts about infinity:

 

1. Rational numbers are countably infinite.

2. Irrational numbers are uncountably infinite.

3. Whether there exists a set with cardinality between A0 and A1 remains an open problem.

4. There are countably infinite cardinal numbers.

5. An infinite sum of integers might not be an integer (1-1+1-1+...=1/2).

6. An infinite sum of rational numbers may be irrational (1-1/3+1/5-1/7+...=pi/4).

 

 

Good explanations, outis. I'm learning things. On your #5, that series is divergent - what definition of sum are you using?

 

I enjoy discussing math with someone knowledgeable and smart such as yourself. Thanks.

[Guest [ch...]]

if the dedekind cut (or ANY cut or algorithm) can generate an irrational, then why can i not express the irrational as a fraction by dividing it by 1? if you can generate an irrational, it ceases to be irrational.

BTW the flaw in this argument is that the "it" in the phrase "dividing it by 1" is not an integer. For instance, Pi = Pi / 1 is true, but that does not make Pi rational.

[Guest [ch...]]

let me know if you wish to take up the cantor challenge.

If by that you mean showing me your diagonalization proof that Pi cannot be an infinite sum of rationals... Sure, present it and I'll tell you what's wrong with it.

[Guest [ou...]]

5. An infinite sum of integers might not be an integer (1-1+1-1+...=1/2).

 

Good explanations, outis. I'm learning things.

On your #5, that series is divergent - what definition of sum are you using?

 

I enjoy discussing math with someone knowledgeable and smart such as yourself. Thanks.

 

First of all, thank you for the kind words.

Secondly, I don't know what the hell happened between you and Kpin

but I'm with Leslie on this; let us return to math.

Lastly, I may log out for a while so I'll get back to you and Kpin later.

PS: I've already discussed #5 with Kpin but I can elaborate further if you wish.

[[...]]

Jon Skeet's addition operator doesn't commute; it teleports to where he needs it to be

[[...]]

[Guest [Kp...]]

Some of the things you 'know' are false and

your unwillingness to learn more prevents you from seeing that.

yes this is possible. i could be wrong. have a little patience with me please -- i think i understand most of what you say in this post but please resolve the contradictions i point out below:

 

You are the one having difficulties, even with Godel.

He did NOT prove that logic is consistent and incomplete.

you are right. godel did not prove that peano axioms were consistent. i blame my treacherous memory.

 

He only proved that it can be either consistent or complete but not both.

We chose incompleteness over inconsistency.

again you are right. but where did you get the idea that we have to choose? are you quoting hawking verbatim from his paper? it has already been proven that peano arithmetic is consistent (though not by godel) -- https://en.wikipedia.org/wiki/Gentzen%27s_consistency_proof

 

Here you misunderstood every single thing I said.

i feel i did.

 

First of all, the Church–Turing thesis is just that, a thesis which could be false.

agreed. let us ignore it.

 

Did you see the word DEFINITION anywhere?

That's because this is NOT their definition, rather a consequence of it.

i see your point. then we do not disagree. but we should be consistent -- even if our usage is wrong.

 

We use subtraction on the natural numbers to construct/define integers.

We use division on the integers to construct/define rational numbers.

We use Dedekind cuts on rational numbers to construct/define the real numbers NOT compute them.

 

i get your point.

definition = order of the number

computation = compute the number.

 

right?

 

Regarding computability I was talking about some numbers (pi in particular).

The real numbers are uncountable so most of them are not computable

but ALL algebraic numbers (like √2) and many transcendental numbers (like pi) are computable.

 

wait. you just said that "most real numbers are not computable because they are uncountable" then you say "pi and root 2 are computable." how can you say this? isn't this a contradiction? inconsistency?

 

you also mean we can define pi AND compute it? are you sure?

 

When we say a number is computable we mean that it can be computed

to within any desired precision by a terminating (finite) algorithm.

are you saying pi is computable in the same sense turing meant by the word "computable?" "turing computable? "

 

What you seem to be talking about is HYPERcomputation which appears to be impossible in our universe.

 

you mean when turing was saying "turing compatible" he meant HYPERcomputation? are you sure? i will accept your yes/no and won't argue.

 

i will try understand the rest of your post after i understand the above.

 

by the way, thanks a lot for taking the time to respond. it will only help me learn and i am grateful to you for that.

 

 

[Guest [Kp...]]

let me know if you wish to take up the cantor challenge.

If by that you mean showing me your diagonalization proof that Pi cannot be an infinite sum of rationals... Sure, present it and I'll tell you what's wrong with it.

 

ok, i think i am the one who is misunderstanding. but, tell me. earlier you said, "There is also a notion of non-computable numbers which is a subset of the transcendentals. Pi is computable since there exists an algorithm to compute its digits, but there are non-computable numbers. See https://en.wikipedia.org/wiki/Computable_number"

 

i just checked that link and here is a text from it:

 

Countable but not computably enumerable

While the set of real numbers is uncountable, the set of computable numbers is only countable and thus almost all real numbers are not computable. That the computable numbers are at most countable intuitively comes from the fact that they are produced by Turing machines, of which there are only countably many. More precisely, assigning a Gödel number to each Turing machine definition produces a subset {\displaystyle S} S of the natural numbers corresponding to the computable numbers and identifies a surjection from {\displaystyle S} S to the computable numbers, which shows that the computable numbers are subcountable. Moreover, for any computable number {\displaystyle x,} x, the well ordering principle provides that there is a minimal element in {\displaystyle S} S which corresponds to {\displaystyle x} x, and therefore there exists a subset {\displaystyle S_{1}\subset S} {\displaystyle S_{1}\subset S} consisting of the minimal elements, on which the map is a bijection. The inverse of this bijection is an injection into the natural numbers of the computable numbers, proving that they are countable.

 

The set {\displaystyle S} S of these Gödel numbers, however, is not computably enumerable (nor consequently is {\displaystyle S_{1}} S_{1}), even though the computable reals are themselves ordered. This is because there is no algorithm to determine which Gödel numbers correspond to Turing machines that produce computable reals. In order to produce a computable real, a Turing machine must compute a total function, but the corresponding decision problem is in Turing degree 0′′. Consequently, there is no surjective computable function from the natural numbers to the computable reals, and Cantor's diagonal argument cannot be used constructively to demonstrate uncountably many of them.

 

now isn't the bolded text saying that computable numbers are countable. and aren't most reals uncountable?

[Guest [Kp...]]

outis and chessplayer,

 

i once again apologize for the arrogance i displayed in my earlier posts (which i deleted after a rethink). chessplayer, you read the posts addressed to you. outis, you did not get a chance to read i think 'cos you were offline. i had the impression that i understood reals and that you two did not but now i feel the truth was reverse. i got carried away earlier -- which is a bad character trait and which is probably why i am a bit closed to learning. oh well... i am human. the only difference is that unlike you two, i am mostly wrong, esp. in math. i think it is more important to admit you are wrong -- it does not increase your IQ but it comes with some redemption.

[Guest [Kp...]]

i have understood the source of my confusion and misinterpretation. turing computable does not mean computable. that is the old definition for computability. the new definition takes into account rounding off of numbers. then pi becomes computable. neat. chessplayer and outis, both of you are right. i should have read the links you guys gave! sorry for taxing your brain! i had better stay away from this thread henceforth. 

 

 

[Guest [ch...]]

let me know if you wish to take up the cantor challenge.

If by that you mean showing me your diagonalization proof that Pi cannot be an infinite sum of rationals... Sure, present it and I'll tell you what's wrong with it.

 

ok, i think i am the one who is misunderstanding. but, tell me. earlier you said, "There is also a notion of non-computable numbers which is a subset of the transcendentals. Pi is computable since there exists an algorithm to compute its digits, but there are non-computable numbers. See https://en.wikipedia.org/wiki/Computable_number"

 

i just checked that link and here is a text from it:

 

Countable but not computably enumerable

While the set of real numbers is uncountable, the set of computable numbers is only countable and thus almost all real numbers are not computable. That the computable numbers are at most countable intuitively comes from the fact that they are produced by Turing machines, of which there are only countably many. More precisely, assigning a Gödel number to each Turing machine definition produces a subset {\displaystyle S} S of the natural numbers corresponding to the computable numbers and identifies a surjection from {\displaystyle S} S to the computable numbers, which shows that the computable numbers are subcountable. Moreover, for any computable number {\displaystyle x,} x, the well ordering principle provides that there is a minimal element in {\displaystyle S} S which corresponds to {\displaystyle x} x, and therefore there exists a subset {\displaystyle S_{1}\subset S} {\displaystyle S_{1}\subset S} consisting of the minimal elements, on which the map is a bijection. The inverse of this bijection is an injection into the natural numbers of the computable numbers, proving that they are countable.

 

The set {\displaystyle S} S of these Gödel numbers, however, is not computably enumerable (nor consequently is {\displaystyle S_{1}} S_{1}), even though the computable reals are themselves ordered. This is because there is no algorithm to determine which Gödel numbers correspond to Turing machines that produce computable reals. In order to produce a computable real, a Turing machine must compute a total function, but the corresponding decision problem is in Turing degree 0′′. Consequently, there is no surjective computable function from the natural numbers to the computable reals, and Cantor's diagonal argument cannot be used constructively to demonstrate uncountably many of them.

 

now isn't the bolded text saying that computable numbers are countable. and aren't most reals uncountable?

I think outis already corrected you on this mistake, but: "countable" applies to *sets* of numbers, not individual numbers. The *set* of computable numbers is countable. The *set* of reals is uncountable. Individual numbers like Pi cannot be described as either countable or uncountable since countable is a property of sets.

[Guest [Kp...]]

The *set* of computable numbers is countable. The *set* of reals is uncountable. Individual numbers like Pi cannot be described as either countable or uncountable since countable is a property of sets.

yes, the set of computable numbers has to be countable because we can express them with rounding. once we do that, the term "countable number" becomes inconsistent because pi belongs in both sets. i realize that now.

[Guest [Kp...]]

it baffles me though why none of you pointed out, whenever i said "an irrational cannot be written," that it could be written with rounding! oh well... the mysteries of mind and nature!

[Guest [ou...]]

He only proved that it can be either consistent or complete but not both.

We chose incompleteness over inconsistency.

 

you are right. but where did you get the idea that we have to choose?

it has already been proven that peano arithmetic is consistent.

 

Before Godel we took completeness for granted, worrying only about consistency.

Now that we know that a sufficiently complex system - Peano arithmetic is relatively 'weak' -

can only have one of those, we choose to BELIEVE that math is consistent,

because we can live with incompleteness whereas inconsistency is unacceptable.

 

definition = order of the number

 

right?

 

Definition/Construction = we can prove that x^2=2 has a unique positive root

 

you just said that "most real numbers are not computable because they are uncountable"

then you say "pi and root 2 are computable." isn't this a contradiction?

 

you also mean we can define pi AND compute it?

 

I said that real numbers are uncountable so MOST (not ALL) of them are not computable.

Pi is one those exceptions. Again here is the algorithm to compute any of its digits.

 

When we say a number is computable we mean that it can be computed

to within any desired precision by a terminating (finite) algorithm.

 

are you saying pi is computable in the same sense turing meant by the word "computable?"

"turing computable?"

 

Turing computability refers to SETS not numbers just like countability.

[Guest [ch...]]

ARGH! kpin, you're driving me crazy with your endless argumentation. Does anyone have any Valium? (KIDDING!)

 

[Guest [ou...]]

1. There's nothing wrong with asking why, quite the opposite in fact.

So long as you're open to be convinced and you don't resort to insults if you're not.

 

2. Do you still want me to elaborate on the series 1-1+1-1+...?

 

3. Did you get a chance to take a look at my most recent problem?

 

PS: I have some Valium. (unfortunately not kidding)

[Guest [ch...]]

>1. There's nothing wrong with asking why, quite the opposite in fact.

 

You have more patience than I do.

 

> 2. Do you still want me to elaborate on the series 1-1+1-1+...?

 

No thanks, I looked it up and found articles on Grandi's series and Casaro sum.

 

3. Did you get a chance to take a look at my most recent problem?

 

The one about computing the root of e^x+x-2=0 to arbitrary precision epsilon? I suppose I would just write a method f(x) that computes e^x+x-2, and then binary search between 0 and 1 to zoom in on the answer. By binary search, I mean try 0.5, and depending on whether f(0.5) is positive or negative, next try 0.25 or .75, etc. Is there a better algorithm?

 

PS: I have some Valium. (unfortunately not kidding)

 

In seriousness, I threw mine out and now shudder at the thought of ever taking a benzo again. The withdrawal was so bad, and took so long. (I remember reading in BB that you should cut 5% of current dose every 2 weeks, and thinking like a mathematician "the series .95^N converges very slowly and will take infinitely long to actually reach zero"...)

[Guest [ch...]]

P.S. I'm not going to actually code the programming problem... my coding for my job takes precedence.

[Guest [ou...]]

1. You have more patience than I do.

 

2. No thanks, I looked it up and found articles on Grandi's series and Casaro sum.

 

3. I suppose I would just write a method f(x) that computes e^x+x-2, and then binary search between 0 and 1 to zoom in on the answer. By binary search, I mean try 0.5, and depending on whether f(0.5) is positive or negative, next try 0.25 or .75, etc. Is there a better algorithm?

 

I remember reading in BB that you should cut 5% of current dose every 2 weeks, and thinking like a mathematician "the series .95^N converges very slowly and will take infinitely long to actually reach zero"

 

1. I'm a teacher, remember?

 

2. I'm glad. Though I'd like to mention that Cesaro summation is just an extension of

'standard' summation that is 1/2+1/4+1/8+... still equals 1.

 

3. Bolzano's dichotomy method was my 1st thought as well but as it turns out

the Newton-Raphson method is faster albeit more complicated.

 

PS: Finally, someone who understands benzo-math!

[Guest [ou...]]

I'm not going to actually code the programming problem... my coding for my job takes precedence.

 

Let me do the coding then since withdrawal has cost me my job.

I'll be using the 'binary search' method. You can point out any possible mistakes.

After that, I won't be posting any more problems as I've run out... for now!

 

program Bolzano;

 

var a,b,c,err:real;

 

function f(x:real):real;

begin

  f:=exp(x)+x-2;

end;

 

begin

  readln(err);

  a:=0;b:=1;

  c:=1/2;

  while abs(f©)>err do begin

    if f(a)*f©>0 then

      a:=c

    else

      b:=c;

    c:=(a+b)/2;

  end;

  writeln©;

end.

[[...]]

Tell me, are there mathematical ideas that can not be expressed by computer programing ?

python, scypy, numpy, pandas, etc

[[...]]

I am actually quite interested in this thread, and would really appreciate some "thought experiments" to explain what you guys are actually talking about.

 

[Guest [ch...]]

That Bolzano program looks correct!

[Guest [Kp...]]

are you saying pi is computable in the same sense turing meant by the word "computable?"

"turing computable?"

 

Turing computability refers to SETS not numbers just like countability.

 

isn't this a contradiction?

 

https://en.wikipedia.org/wiki/Computable_number

 

"A computable number [is] one for which there is a Turing machine which, given n on its initial tape, terminates with the nth digit of that number [encoded on its tape]." (Minsky 1967:159)

 

i think you cannot use turing and computable in the same sentence unless you specify n... right?

[Guest [ch...]]

Outis was being imprecise. The Turing-Church thesis is about computable functions, where a function is defined as a mapping  f: N^k->N. There is also the notion of a computable number, which is a number whose digits can be computed by a computable function. The two are obviously related. There is a single notion of computable and it has nothing to do with rounding.