Logik wrote: ↑Mon Mar 18, 2019 10:25 am
Scott Mayers wrote: ↑Mon Mar 18, 2019 9:49 am
You responded before you read the end, I'm guessing. My point about you missing it is about thinking that the theorems were asserting that reality itself was finite. That is a misrepresentation of the theorems. If you read
Godel, Escher, and Bach: An Eternal Braid, as you said you had, AND understood it, this was the point about his own explanation. The empty complement to a finite universe/universal, is only a one way reality. That is, if you complement X to become not-X, the complement of not-X, in reality is not-X, but X PLUS not-X. [See page 71(paperback version) figure 18 in the chapter on Figure and Ground. The 'ground' is that complement to the posited 'figure' of a finite known logical universal.]
And if you were actually familiar with logic, its theory and inner workings you wouldn't have to use an entire, verbose paragraph to say what you just said.
It's called "completeness'.
https://en.wikipedia.org/wiki/Completeness_(logic)
The book you appeal to is 40 years old. It merely popularised that which was understood in the 70s. Things have happened since.
Maybe you need a break?
I appear to be angering you and I am not meaning to. I can explain better if you don't follow but require MORE effort that I'm guessing is only going to piss you off more. So which is it?
logik wrote:
Scott Mayers wrote: ↑Mon Mar 18, 2019 9:49 am
A Turing Complete machine is any GENERAL COMPUTER now but WITH an infinite memory (ideal). The machine itself is finite, the memory infinite. You can't use Cantor's final representation to his proof without this factor. Since the machine represents logic, the data and its spaces represent the input/outputs of reality.
Cantor is a set-theorist. All my arguments are based on the rejection of set theory as the foundation of logic. I also reject all the attempts to rescue set theory e.g ZFC.
My foundation is type theory. And until you recognize this disparity we are speaking from different paradigms and hence: different languages.
Why are you stepping beyond our common domain. If you disagree with ZFC nor Cantor's naive form of set theory that was used to express the proofs of Turing and Godel and Church, then who are you to accept them when these incompleteness theorems are DEPENDENT upon them?
[As to your 'Type theory' I am more familiar with the original "Type theory" discussed by Russell. All that does is it distinctly separates meaning from the container that holds it. The liars paradox, for instance, is cheaply resolved by separating the types of classes defined to speak about what is in the sets from the actual members to avoid the circularity. But it doesn't fix it and why Godel proved it using Russell's own kind of reasoning to dislodge it as inescapable.
I'm not against your extended interest but just can't argue that one way or the other. But if you actually understand the underlying incompleteness theorems, this is what we have to at least come to a common understanding of first. Maybe that 'theory of types" is itself at fault for misunderstanding these theorems?
logik wrote:
Scott Mayers wrote: ↑Mon Mar 18, 2019 9:49 am
Yes. He needed to have a general computer that can hold an ideal infinite memory set. The proof was to show that the programs placed in the memory to create any particular program (Turing machine) cannot have one of them specifically designed to solve the problem: "list all the programs (Turing machines) that cannot 'hang', using our terminology today." It can't do so because if you make a list of all possible programs even if you could exhaust all infinite possibilities, you will still find that there is another program that is not on that list....Infinity + 1. He then uses Cantor's proof of this to show the list cannot be exhausted, thus the VERY program (particular Turing machine) used to seek a problem WITHIN its domain is not able to....and thus, incomplete.
You are mistaken.
1. You think a program is a Turing machine - it's not. That which EVALUATES the program is the Turing machine.
That which evaluates "A and B is True" is your mind. The hardware.
The logic (language, rules for evaluating things) is the software.
2. That is PRECISELY THE POINT of the halting problem. The consequence of the halting problem is a rejection of infinities!
Because (100th time I am saying this and I am getting ruther frustrated with the dumb)
Logic is the laws of THOUGHT, NOT the laws of the universe.
Logical theorems/axioms etc. are about THOUGHT, not REALITY.
How you apply thought TO reality? Entirely separate issue!
NO. The rejection is not of infinities but the proof that you cannot exhaustively cover an infinitesimal (a bounded infinite) precisely because there are always another infinity beyond those boundaries. The machine used IN that boundary can solve all problems, inside and out, but because most solutions lie outside that boundary, it cannot solve all problems in reality. If reality is greater than the calculator, reality is like the domain of a greater infinity outside it. The perfect ideal calculator (as an ideal logic) is unable to solve all problems without becoming greater itself than the universe it is speaking about.
So the theorems are about setting limits to the capacity of solving all real problems
ideally through any logic machine or calculator. Logic is still real, just as the fact you can hold a real smart phone that holds a potential general computing chip that is 'complete'. [they do use a general computer but more often locks us out because the operating system is embedded and limits the chip's capacity]
You can also prove something universally ABOUT totality outside that is itself a universal theory or theorem. You just can't solve ALL particular problems without literally trying each one out one-by-one.
logik wrote:
Scott Mayers wrote: ↑Mon Mar 18, 2019 9:49 am
We can determine whether a finite boundary exists for discovering MORE but without that evidence, we are permanently unable to assertively claim a finite end when an infinite set of possibilities is the only option to asserting a unique and special one without proof. The default to either nothing or everything, which inclusively contains nothing, means that only nothing can be an origin but we can never get there, just approach it infinitely.
But I can permanently claim that your mind is finite. And for as long as your mind is finite and it's SMALLER than the universe you will always be working with the map not the teritory.
e.g Kolmogorov complexity. a.k.a compression.
Then you can't accept that reality is itself a type of real calculator?
The 'map' is real too, though. The map is all we have to deal with. But you can determine the formula for a finite general machine that can speak about general truths about the whole, just not about all of the specific cases in it. My mind can create a proper 'map' of Totality and its
general logic but not its complete set of EACH sub-logic system it contains. I can't even know all my own mind's component logic but can discover its general forms. I cannot make the neuron I want to determine what it is made up of reflectively because the information to observe it is greater than its capacity to hold it.
So we understand each other on this. You just interpret that I cannot make a universal machine that cannot determine itself as incomplete, which is not the case. How else could you hold a limit of our mind's map to be incomplete as a certainty without believing that map determined this fact?
logik wrote:
Scott Mayers wrote: ↑Mon Mar 18, 2019 9:49 am
No, the meaning is like asking how many real numbers are between the bounded integers 0 and 1. There are an infinity of them!
No there aren't. The number of real numbers between 0 and 1 is a function of amount-of-time, the rate at which you can generate new numbers and your range-precision trade-off function.
I will spell out range-precision trade-off in the easiest way possible for you. ARBITRARY CHOICE.
infinite precision requires infinite numbers between any range.
Be it 0 and 1, or 0 and 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000..........1
So you will spend an infinite amount of time BEFORE you can even utter a single real number!
Infiinties are stupid.
And so you STILL concluded with closure something universal about reality. Thus you cannot deny that some map (a logic) can determine A universal theory/theorem about reality itself.
[Now I need a break even though I'm not wanting to: my neck and back are killing me. If you have more, I'll wait to respond. We've covered a lot anyways and it might help to have some time to absorb it. Goodnight/morning.]