Anyone here into math?

[Guest [Kp...]]

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

We chose incompleteness over inconsistency.

 

i'm awfully sorry outis but we (you and chess on one side AND me) are having the same misunderstanding over and over. at least this is the impression i get.

 

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

 

my interpretation - again we are having the same issue as we did in my earlier post. because, godel actually proved that only logical systems with sufficient complexity can be either complete or consistent but not both.

 

this means that there can be "complete and consistent systems" but they will not be complex enough.

 

 

 

[Guest [Kp...]]

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.

 

i see. then my understanding is lacking. just confirm to me if what you have said is consistent with the following:

 

informal definition using a Turing machine as example[edit]

 

In the following, Marvin Minsky defines the numbers to be computed in a manner similar to those defined by Alan Turing in 1936; i.e., as "sequences of digits interpreted as decimal fractions" between 0 and 1:

 

"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)

The key notions in the definition are (1) that some n is specified at the start, (2) for any n the computation only takes a finite number of steps, after which the machine produces the desired output and terminates.

 

An alternate form of (2) – the machine successively prints all n of the digits on its tape, halting after printing the nth – emphasizes Minsky's observation: (3) That by use of a Turing machine, a finite definition – in the form of the machine's table – is being used to define what is a potentially-infinite string of decimal digits.

 

This is however not the modern definition which only requires the result be accurate to within any given accuracy. The informal definition above is subject to a rounding problem called the table-maker's dilemma whereas the modern definition is not.

 

if it is then i have to go back to understanding real numbers (not that i will actually do that.. i will just close the chapter).

[Guest [ch...]]

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

We chose incompleteness over inconsistency.

 

i'm awfully sorry outis but we (you and chess on one side AND me) are having the same misunderstanding over and over. at least this is the impression i get.

 

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

 

my interpretation - again we are having the same issue as we did in my earlier post. because, godel actually proved that only logical systems with sufficient complexity can be either complete or consistent but not both.

 

this means that there can be "complete and consistent systems" but they will not be complex enough.

Again outis was being imprecise. Any logical theory containing basic arithmetic cannot be complete and consistent. A logical theory has a domain (e.g. the set of integers), and it has statements you can make about members of the domain (e.g. "1+1=2"), and it has axioms and rules for proving things from those axioms. You can construct logical theories that are consistent and complete if you make them simple enough.  Any finite logic will do. For example, if the domain is {0,1,2,3,4,5,6} with operations + mod 7 and * mod 7, this is F7 (the finite field of order 7). The set of statements you can make about F7 is consistent and complete. More interestingly, the theory of arithmetic without multiplication (Presburger arithmetic) is consistent and complete. However with multiplication (Peano arithmetic) the theory becomes complex enough to be undecidable.

[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.

 

i see. then my understanding is lacking. just confirm to me if what you have said is consistent with the following:

 

informal definition using a Turing machine as example[edit]

 

In the following, Marvin Minsky defines the numbers to be computed in a manner similar to those defined by Alan Turing in 1936; i.e., as "sequences of digits interpreted as decimal fractions" between 0 and 1:

 

"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)

The key notions in the definition are (1) that some n is specified at the start, (2) for any n the computation only takes a finite number of steps, after which the machine produces the desired output and terminates.

 

An alternate form of (2) – the machine successively prints all n of the digits on its tape, halting after printing the nth – emphasizes Minsky's observation: (3) That by use of a Turing machine, a finite definition – in the form of the machine's table – is being used to define what is a potentially-infinite string of decimal digits.

 

This is however not the modern definition which only requires the result be accurate to within any given accuracy. The informal definition above is subject to a rounding problem called the table-maker's dilemma whereas the modern definition is not.

 

if it is then i have to go back to understanding real numbers (not that i will actually do that.. i will just close the chapter).

Yes what I said is consistent with that. Now will you please "close the chapter" as you repeatedly promised?

[Guest [Kp...]]

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

We chose incompleteness over inconsistency.

 

i'm awfully sorry outis but we (you and chess on one side AND me) are having the same misunderstanding over and over. at least this is the impression i get.

 

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

 

my interpretation - again we are having the same issue as we did in my earlier post. because, godel actually proved that only logical systems with sufficient complexity can be either complete or consistent but not both.

 

this means that there can be "complete and consistent systems" but they will not be complex enough.

Again outis was being imprecise. Any logical theory containing basic arithmetic cannot be complete and consistent. A logical theory has a domain (e.g. the set of integers), and it has statements you can make about members of the domain (e.g. "1+1=2"), and it has axioms and rules for proving things from those axioms. You can construct logical theories that are consistent and complete if you make them simple enough.  Any finite logic will do. For example, if the domain is {0,1,2,3,4,5,6} with operations + mod 7 and * mod 7, this is F7 (the finite field of order 7). The set of statements you can make about F7 is consistent and complete. More interestingly, the theory of arithmetic without multiplication (Presburger arithmetic) is consistent and complete. However with multiplication (Peano arithmetic) the theory becomes complex enough to be undecidable.

 

yes, i knew this. but i have only now understood as to how there exists a computable function that has the same meaning in the turing sense -- because the turing sense too assumes approximates then. it is a part of the definition of a real number that i was not aware of. this definition distinguishes the some non computable trans. from the other reals. the problem is no matter how you define the real or the turing machines, the universal turing machine will always have undecidable algorithms or non computable trans. will always exist (and always in the same "strength" in terms of spatial distance), even if you define reals a bit more "precisely" (dedekind cut+ another cut) as opposed to "imprecisely" (dedekind cut). and we are back to this scenario that i stated. and that "precisely" or "imprecisely" is precisely the godel theorem. phew scary.

[Guest [Kp...]]

this is why i keep forgetting cantor. it is because english is both relatively (incomplete and inconsistent) whereas math is relatively (less incomplete and less inconsistent). that is why there are no paradoxes in math. now i understand cantor... till i forget it. 

 

bye all. chapter closed.

[Guest [ou...]]

Turing computability refers to SETS not numbers just like countability.

 

isn't this a contradiction?

 

Outis was being imprecise.

 

My bad. Was thinking about countable sets at the time and jumped to computable sets.

 

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

We chose incompleteness over inconsistency.

 

only logical systems with sufficient complexity can be either complete or consistent but not both.

 

Again outis was being imprecise.

 

Not my bad. Had already clarified that:

 

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

Now that we know that a sufficiently complex system can only have one of those,

we choose to BELIEVE that math is consistent.

 

In any case, I consider this subject closed.

[Guest [Kp...]]

i edited my this post to make it more precise/imprecise. i am making this post to stop editing it, lol, because the subject is such that it never becomes "precise" no matter how much you edit it. 

 

it is a rounding issue because you are only making math easier to be able to juggle with cubes (or maybe 3d coordinate plane) as opposed to only squares (as opposed to "complete and consistent" systems of only powers of ones or only additions and subtractions). good luck mathematicians. thanks for teaching me.

[Guest [ou...]]

It is NOT a rounding issue because 'rounding' can be arbitrarily precise.

[Guest [ch...]]

I diagnose kpin as troll type #2 https://www.lifewire.com/types-of-internet-trolls-3485894 . It's best to stop responding.

[Guest [Kp...]]

yes, i think it is best you two stopped responding to me. i should not be arguing with harvard and applied physics. you guys understand cantor after all. thanks once again for un-teaching me yesterday what i had already learnt, lol, until i woke up this morning with day before's memory restored, lol.

 

carry on guys. let me not disturb you.

[Guest [ou...]]

While I agree with the part that says:

 

They believe they're right, and everyone else is wrong.

 

I don't believe he's trolling.

I think he's incorporated what he's learnt/misinterpreted in his 'identity' thereby making them beliefs.

And when you challenge one's beliefs they view it as a personal attack which is a biological response.

The above is evident by his admitting that he doesn't want to delve deeper into the real numbers.

[Guest [Kp...]]

I don't believe he's trolling.

I think he's incorporated what he's learnt/misinterpreted in his 'identity' thereby making them beliefs.

And when you challenge one's beliefs they view it as a personal attack which is a biological response.

The above is evident by his admitting that he doesn't want to delve deeper into the real numbers.

no outis, i am not trolling. nothing you or chessplayer said was wrong. there are things that i said that were clearly wrong but my understanding was correct. i realized this yesterday. the reason i was using wrong terms like "uncountable number" was because i was constantly trying to understand numbers and cantor in terms of turing computable. but the problem was that turing's theorem is in a logical order that is different from that of cantor's and thus a turing machine does not map directly to a number or the diagonal slash. to map turing's theorem, you need to recast turing's theorem in a godel logical order where truth value and proof are separate entities. so in a loose sense, turing's theorem is in english while our numbers are in godel logic. thus turing theorem's is phrased as a paradox while cantor's and godel's are not. now i am sure there are terms in math that explain all this in two lines using mathematical rhetoric that you are familiar with but i am not.  but even if you had used those terms, i would not have understood because i am not a math savvy guy and i would have always resorted to turing to understand because turing speaks in english.

 

now turing's paradox and the church turing thesis always remain. this is what i realized yesterday after understanding dedekind cuts and non computable trans numbers. the paradox is the question, "how many digits are there in the decimal expansion of any real once we accept that there are two infinities?" so even if we invent dedekind cuts to avoid decimals and appropriate some irrationals, or use approximations to circumvent the above paradox, we get a new paradox that is the same as the original. in other words, an irrational terminates at an indeterminate point. let us make it determinate using dedekind cuts. but now we have new irrationals that end at indeterminate points (non computable trans.). this is why we still publish two definitions of computability, informal and formal (2 definitions in 2 diff. logical orders), and the church turing thesis despite it having no proof. i think you understand my layman language... even if you don't it is ok. i may still be making mistakes but i hope you will excuse me. i was only interested in understanding the concept and never the math. and i tried to summarize all this in my last two posts (in layman speak) yesterday -- but i can perfectly understand why you would not understand me and think i am wrong. i admit you and chessplayer have a superior knowledge of the subject because you two are mathematicians -- it is just that a layman like me wished to understand it in laymam terms and which you were loathe to use.

[Guest [ou...]]

Dedekind cuts are used to construct and thereby define (not compute) real numbers.

The fact that the set of irrational numbers is R-Q is just a consequence of that definition.

 

Here's how we do math in layman terms:

1. We define stuff.

2. We agree on a list of axioms considered self-evident.

3. We build from there.

 

Here's how we use Dedekind cuts to construct √2 in layman terms:

 

1. We divide rational numbers into 2 sets.

The former contains those numbers that are negative or have a square <2.

The latter contains those numbers that are positive and have a square >2.

 

2. This cut represents a positive number whose square is 2, the definition of √2.

 

Now we build. For example, we can prove (by contradiction) that √2 is irrational.

If you wish to compute its value, there exist algorithms that can do so with arbitrary precision,

that is I can give you any digit in a finite amount of time.

By that definition √2 is computable. Same goes for Pi.

 

Unfortunately, this is as simple as it gets.

The only way to go from here is to Wikipedia or open an up-to-date book.

[Guest [Kp...]]

Dedekind cuts are used to construct and thereby define (not compute) real numbers.

oop. i am correcting my mistake in my earlier post.

 

so even if we invent dedekind cuts to avoid decimals and appropriate some irrationals, or use approximations to circumvent the above paradox, we get a new paradox that is the same as the original. in other words, an irrational terminates at an indeterminate point. let us make it determinate using dedekind cuts. but now we have new irrationals that end at indeterminate points (non computable trans.). this is why we still publish two definitions of computability, informal and formal (2 definitions in 2 diff. logical orders), and the church turing thesis despite it having no proof.

 

what i did not realize was that we probably cannot invent a logical order higher than peano arithmetic - so the scenario i present is mythical. the non computable trans. are just that -- non computable. and the dedekind cuts do not connote to an order higher than peano arithmetic -- it is in the same axiomatic system and same order.

 

apologies for troubling you man!

[Guest [Kp...]]

@[ o...]chaitin and uncertainty interesting) -- i am curious to hear from you guys as to what you think about all this:

 

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

 

-------------------------------------------------------------------

question # 2: do you guys think, a) randomness exists in nature, or, b) it is hard for us to prove or disprove it does (consistent/complete dilemma), c) no clue.

 

[Guest [ch...]]

On #1, I don't know enough about quantum physics to comment.

 

On #2, the mathematical definition of a random sequence applies only to infinite sequences, so your question is more or less equivalent to asking if I think the universe is finite or infinite. I personally think it's finite (but large). Therefore I think random sequences(in the mathematical sense) do not exist. However the universe may be random in the more informal sense of the word.

[Guest [ou...]]

On #2, the mathematical definition of a random sequence applies only to infinite sequences, so your question is more or less equivalent to asking if I think the universe is finite or infinite. I personally think it's finite (but large). Therefore I think random sequences(in the mathematical sense) do not exist. However the universe may be random in the more informal sense of the word.

 

I've already had this conversation with Kpin before you joined the discussion.

Before I comment on your reply, here's a bit of pseudocode for you:

 

If he_remembers then

  I think you were right to believe he's 'trolling' in some way

else

  he still tried to steer the conversation back to incompleteness, Turing, ...

 

Now for my response.

Current data does indeed suggest that the universe is expanding but finite.

However, it does not appear to be bound by time.

 

As I previously told Kpin, an event is random if its outcome cannot be predicted.

The outcome of a coin flip can be predicted if we know the initial conditions precisely.

However, our current understanding of QM suggests that nature is (at its core) truly probabilistic.

[Guest [ch...]]

You're right. Time is unbounded and therefore we are doomed to repeat this loop forever:

 

while True:

    kpin reopens discussion of incompleteness etc

    we answer the same questions again and again

 

However we can enclose it in a try..catch like so:

 

try:

  while True:

        kpin reopens discussion of incompleteness

        raise YouAlreadyAskedThatException

catch BaseException ex

  print 'Stop!'

  sys.exit(-1)

 

Phew!

[[Be...]]

John Hagelin, PhD, quantum physicist.  Natural Law Party, third-party presidential candidate, 2000 and 2004.  Claims he's solved the Unified Field Theory.  http://www.hagelin.org/  Had a few of his published papers before computers, and made-up T-shirts for his Presidential campaign.  IQ well over 200.  Consciousness is the unified field?

 

Amit Goswami, PhD. Quantum Physicist.  http://www.amitgoswami.org/  Have a couple of his books.

 

These guys know quantum physics. 

[Guest [Kp...]]

You're right. Time is unbounded and therefore we are doomed to repeat this loop forever:

 

while True:

    kpin reopens discussion of incompleteness etc

    we answer the same questions again and again

 

However we can enclose it in a try..catch like so:

 

try:

  while True:

        kpin reopens discussion of incompleteness

        raise YouAlreadyAskedThatException

catch BaseException ex

  print 'Stop!'

  sys.exit(-1)

 

Phew!

 

omg!!  :laugh:

[Guest [ou...]]

we can enclose it in a try..catch

 

Nice try but I think he'll find a catch

[Guest [ou...]]

John Hagelin, PhD, quantum physicist. Claims he's solved the Unified Field Theory.

 

His model has been rejected.

 

Consciousness is the unified field?

 

Last time we invoked anthropocentrism we ended up with the geocentric model.

 

IQ well over 200.

 

IQ is a necessary but not sufficient condition.

 

These guys know quantum physics.

 

Paraphrasing Feynman: If one thinks they understand QM they do not.

[Guest [Kp...]]

exercise in logic

 

can we refine this?

 

Benzo Withdrawal (BW):

 

In BW reality gets altered: sensory inputs get altered. our responses get altered. our physical selves get altered (psychosomatic disorder).

A) BW lasts long enough to suggest to us that normal people are faking it

B) BW occurs late enough in life to suggest to us that it is not a normal state

A & B co-exist till they become a paradox

faith helps dismiss the paradox

BW affects past, future and present. It changes them.

It seems to be a higher entropy state than normalcy (?) it means there are more states present in BW that cannot be expressed using language, or, there is lesser information about the total BW state (?) any state that cannot be expressed using language is non-computable.

what distinguishes BW from a drug's effect is its prolonged and non-linear nature.

 

 

[[Be...]]

John Hagelin, PhD, quantum physicist. Claims he's solved the Unified Field Theory.

 

His model has been rejected.

 

Consciousness is the unified field?

 

Last time we invoked anthropocentrism we ended up with the geocentric model.

 

IQ well over 200.

 

IQ is a necessary but not sufficient condition.

 

These guys know quantum physics.

 

Paraphrasing Feynman: If one thinks they understand QM they do not.

 

No human being knows everything, and our thinking is limited and bounded by our physical bodies and minds.  Some people understand a few things about quantum mechanics and some people understand a great deal about quantum mechanics.  Can't throw the baby out with the bath water.