Gödel's Miracle and Why Conventionalism makes no sense in Computer Science

[Bruno Marchal]

Hi Philip, Hi Bruce, and possible others,

If I remind well, you have defended the idea that mathematics is a language, and that there is no mathematical reality others that convention of language. 

In my opinion, this does not make sense, even without the Church-Turing thesis. But I realise that with the Church-Thesis, it is easy to understand that the mathematical reality kicks back in a way enforcing some amount of realism. It helps also to understand that in computer science, there are many theorems which are “machine independent” (to use the computer science terminology), and this means that the results concerns all programming language. That gives examples of general truth which are independent of the formalism and language used to express them.

The reason is fundamental, and shows how much incompleteness by itself is an example where the mathematical, even the arithmetical, reality kicks back: quite a lot!

I very often present the reasoning which follows. Sometimes my motivation for it is more limited, to “just” show that the Church-Turing thesis (CT) is …a thesis, to oppose those who sometimes defend the idea that CT is only a definition (of what is computable).

Indeed, incompleteness is a direct consequence of CT.

I need you to accept that not set S can have a subset SS with more element that itself. SS included in S implies that the cardinality of SS is smaller or equal than the cardinality of S. This is obvious, but with infinite set, we have to be careful. I will not prove this, and refer you to cantor-Bernstein Theorem, from which it follows “easily”.

So, in particular, a subset of an enumerable set cannot be a non enumerable set. OK? I define a set S to be enumerable if it exists a bijection between S and N (N = the set of natural numbers).

CT is the thesis that a function f from N to N is computable iff f is computable by an expression in their language (claimed to be universal: they both agree and have proven that their language are equivalent.

Theorem: CT entails Incompleteness

I need a lemma. 

Lemma: The set of finite words build on a finite alphabet is enumerable.

Indeed, take a finite alphabet, like {a, b}, you can enumerate the set of all finite words by enumerating them simply by their length, and generate alphabetically the words which have the same length. On the alphabet {a, b}, you get in this way the bijection

0 ———————————— a

1 ———————————— b

2 ———————————— aa

3 ———————————— ab

4 ———————————— ba

5 ———————————— bb

6 ———————————— aaa

7 ———————————— aab

…

The same idea works for any finite set (alphabet), like a …z, aa, ab, … zz, aaa, aab, ...

Imposing a syntax on the words, like in the alphabet {a, b, =}, would only diminish the number of words, and an’t augment the cardinality of the set (of all words on some finite alphabet). That ordering on finite alphabet is called the lexicographical ordering.

By a (formal) language, I mean the given of an alphabet, and some easily checkable grammar. An expression is a grammatically correct word.

Now it is easy to give an intuitive, informal definition of what is a computable function defined on N and with values in N.

The idea is that a function from N to N is computable means that there is an expression, finite thus and written in some language, explaining how to compute its values in a finite time, with a finite result.

That makes the set of computable functions from N to N being enumerable.

Now, that looks eminently epistemological? Explaining to Who? How could we be sure that the functions computable in French are the same than the function computable by the English? They disagree already on the sign of 0. 

Here the idea is that we can decompose the explanation on elementary steps on which everyone agree that they are indeed quite simple.

The question is, is there a universal language which expressions exist to compute all computable functions from N to N?

CT asserts yes, and provides precise mathematical candidates. 

But even before betting on that answer, we can understand that IF such a universal language exists, it will have to compute much more than the computable functions from N to N. It will have to be able to compute also the functions defined only on subsets of N, and undefined out of that subset. And worse, the expressions denoting the computable function from N to N, will have to be ordered in a necessarily non computable way among the expressions in that universal language.

Imagine (to make a reduction ad absurdum) that L is a universal language enumerating only the computable functions from N to N. We get a correspondence, maybe with repetition of the computable functions from N to N, by enumerating the grammatically correct expression.We get a computable bijection between N and the computable functions from N to N.

Let us write f_n for the function from N to N computed by nth expression. 

Now, the function g defined by g(n) = f_n(n) + 1 is computable, and is defined on all N. So it is a computable function from N to N. It is computable because it each f_n is computable, “+ 1” is computable, and, vy our hypothesis it get all and only all computable functions from N to N.

But then, g has have itself an expression in that universal language, of course. There there is a number k such that g = f_k. OK?

But then we get that g_k, applied to k has to give f_k(k), as g = f_k, and f_k(k) + 1, by definition of g. 

So f_k(k) = f_k(k) +1.

And f_k(k) has to be number, given that we were enumerating functions from N to N. So we can subtract f_k(k) on both sides, and we get 0 = 1. CQFD.

If we keep our belief that a universal language exists, it has to compute other objects, and insert  the expressions of the computable functions in a non computable way (if not you can filter out those other objects, and enumerate again only (but all) computable functions.

Accepting that the universal language enumerates more than the functions from N to N saves the situation, the proof above, made on the phi_i (the enumeration of the mix of functions from N to N and functions from subset of N to N), shows only that the corresponding function “g” is such that g(k) = phi_k(k) = phi_k(k) + 1, and in this case it means that phi_k(k) is not defined, and g is not a function from N to N, but from a subset of N to N. No contradiction, and CT remains consistent.

This means that, with CT, and thus with the mathematical definition of computable functions, obtained by CT,  the set of computable functions, although enumerable, is not recursively or computably enumerable. Like for provability, you have the choice to stay in the “total” domains, with functions everywhere defined, and extends them in the (constructive) transfinite, or you can transcend it, but then the partial (defined only on subset) is mixed in a non computable way, with the total functions, and you are in front of the unknown: you get situation where you don’t know what the machine is computing.

That shows that any theory capable of talking about those universal languages and their expressions are irremediably incomplete.

Gödel’s incompleteness mainly uses this showing the Turing universality of elementary arithmetic, and indeed of the necessary partialness of the arithmetical provability predicate. This means tat the realism implied by CT applies to arithmetic. That is what Kleene has made explicit with its Kleene predicate.

The truth about the behaviour of the expressions, that is the (halting or not) computations is far larger than what can be proved in any effective theory. Then, all this is true also for the machine with Oracles, and this makes possible to study mathematically the degrees of unsolvability of the problem in arithmetic, and in analysis (descriptive set theory).

There is a bit of more work to prove the unsolvability of the halting problem, and indeed being the code of a total function is of a higher degree of unsolvability than the halting problem. I can do that another day. 

Gödel, with his own admission, missed CT. After accepting it, he called miracle the closure of the formalism of computability for the diagonal procedure, transforming the paradoxes into non stopping machine. The miracle is the immunity of the notion of computation for the diagonal procedure. It is a bit like a sort of God has to put those codes in a very complex way among the digital machines. (In fact the set of such codes is PI_2 complete, equivalent with qG).

With Indexical Digital Mechanism, (YD + CT), we have a simple ontological realm (any model of any first order completely theory without induction axioms), and we get an explanation of why we will still need an infinity of axioms to get the internal phenomenologies, including the stable measurable sharable quanta appearances, and the non sharable first person singular qualia experiences.

The digital mechanist thesis makes the dream argument rigorous, and constructive, and testable, and rather well confirmed by contemporary physics, up to now. 

Strictly speaking invoking a god or a “material universe" at this stage is a bit premature. With mechanism, this would be like invoking a precise universal number with a precise oracle, and is at the antipode, needless to say, of the original “everything” philosophy of this list (supposedly). 

Anyone can ask precision. Even what is a function. The beauty of this is that you dot need any sophisticated mathematics: only finite numbers and words, and elementary succession of simple action/relation.

Bruno

[Bruce Kellett]

On Sun, May 31, 2020 at 2:26 AM Bruno Marchal <mar...@ulb.ac.be> wrote:

Hi Philip, Hi Bruce, and possible others,

Nothing that you have written below proves arithmetical realism. One can say all these things in a purely conventionalist framework.

Bruce

[Bruce Kellett]

On Sun, May 31, 2020 at 2:26 AM Bruno Marchal <mar...@ulb.ac.be> wrote:

Let us write f_n for the function from N to N computed by nth expression. 

Now, the function g defined by g(n) = f_n(n) + 1 is computable, and is defined on all N. So it is a computable function from N to N. It is computable because it each f_n is computable, “+ 1” is computable, and, vy our hypothesis it get all and only all computable functions from N to N.

But then, g has have itself an expression in that universal language, of course. There there is a number k such that g = f_k. OK?

But then we get that g_k, applied to k has to give f_k(k), as g = f_k, and f_k(k) + 1, by definition of g. 

That is a fairly elementary blunder. g_k applied to k, g_k(k) = f_n(k)+1, by definition of g_k. You do not get to change the function from f_n to f_k in the expression. It is only the argument that changes: in other words, f_n(n) becomes f_n(k). So you are talking nonsense.

Bruce

[Philip Thrift]

we will still need an infinity of axioms to get the internal phenomenologies, including the stable measurable sharable quanta appearances, and the non sharable first person singular qualia experiences

or

Consciousness as a Physical Process Caused by the Organization of Energy in the Brain

Robert Pepperell

https://ncbi.nlm.nih.gov/pmc/articles/PMC6225786/

information vs energy processing:

"Treating brains as neural information processors does not help us to understand consciousness as a physical process."

@philipthrift

[Bruno Marchal]

On 31 May 2020, at 07:44, Bruce Kellett <bhkel...@gmail.com> wrote:

On Sun, May 31, 2020 at 2:26 AM Bruno Marchal <mar...@ulb.ac.be> wrote:

Let us write f_n for the function from N to N computed by nth expression. 

Now, the function g defined by g(n) = f_n(n) + 1 is computable, and is defined on all N. So it is a computable function from N to N. It is computable because it each f_n is computable, “+ 1” is computable, and, vy our hypothesis it get all and only all computable functions from N to N.

But then, g has have itself an expression in that universal language, of course. There there is a number k such that g = f_k. OK?

But then we get that g_k, applied to k has to give f_k(k), as g = f_k, and f_k(k) + 1, by definition of g. 

That is a fairly elementary blunder. g_k

What is g_k? The only enumeration here is the f_k, then we have define a precise, single, function g such that

g(n) = f_n(n) + 1.  (f_n(n) is the diagonal term, you can see this by making the table (the infinite matrice) with the number in the top row, and the f_i in a column):

0 1 2 3 ...

f_0  f_0(0) f_0(1) f_0(2) f_0(3)

f_1 f_1(0) f_1(1) f_1(2) f_1(3)

f_2 f_2(0) f_2(1) f_2(2) f_2(3)

…

Here the underlining means “+1”.

applied to k, g_k(k) = f_n(k)+1,

There are no g_k. 

g is the function defined by diagonalisation. g(x) = f_x(x) + 1, that g(0) = f_0(0) + 1, g(1) = f_1(1) + 1, g(2) = f_2(2) + 1, ...

by definition of g_k.

The only enumeration was the enumeration of the functions f_k

You do not get to change the function from f_n to f_k in the expression.

We do. 

It is only the argument that changes: in other words, f_n(n) becomes f_n(k).

This makes no sense. What is g(2) ? f_n(2) + 1 ? What is n then? 

So you are talking nonsense.

You miss the diagonal. Read again.

Bruno

Bruce

So f_k(k) = f_k(k) +1.

And f_k(k) has to be number, given that we were enumerating functions from N to N. So we can subtract f_k(k) on both sides, and we get 0 = 1. CQFD.

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[Bruce Kellett]

On Sun, May 31, 2020 at 5:21 PM Bruno Marchal <mar...@ulb.ac.be> wrote:

On 31 May 2020, at 07:44, Bruce Kellett <bhkel...@gmail.com> wrote:

On Sun, May 31, 2020 at 2:26 AM Bruno Marchal <mar...@ulb.ac.be> wrote:

Let us write f_n for the function from N to N computed by nth expression. 

Now, the function g defined by g(n) = f_n(n) + 1 is computable, and is defined on all N. So it is a computable function from N to N. It is computable because it each f_n is computable, “+ 1” is computable, and, vy our hypothesis it get all and only all computable functions from N to N.

But then, g has have itself an expression in that universal language, of course. There there is a number k such that g = f_k. OK?

But then we get that g_k, applied to k has to give f_k(k), as g = f_k, and f_k(k) + 1, by definition of g. 

That is a fairly elementary blunder. g_k

What is g_k?

That is your notation: "But then we get that g_k, applied to k has to give f_k(k),"

The only enumeration here is the f_k, then we have define a precise, single, function g such that

g(n) = f_n(n) + 1.  (f_n(n) is the diagonal term, you can see this by making the table (the infinite matrice) with the number in the top row, and the f_i in a column):

0 1 2 3 ...

f_0  f_0(0) f_0(1) f_0(2) f_0(3)

f_1 f_1(0) f_1(1) f_1(2) f_1(3)

f_2 f_2(0) f_2(1) f_2(2) f_2(3)

…

Here the underlining means “+1”.

applied to k, g_k(k) = f_n(k)+1,

There are no g_k.

You defined g_k!!  g_k applied to k is f_k(k), and that is your error.

g is the function defined by diagonalisation. g(x) = f_x(x) + 1, that g(0) = f_0(0) + 1, g(1) = f_1(1) + 1, g(2) = f_2(2) + 1, ...

But that is not what you said before.

 

by definition of g_k.

The only enumeration was the enumeration of the functions f_k

You do not get to change the function from f_n to f_k in the expression.

We do. 

It is only the argument that changes: in other words, f_n(n) becomes f_n(k).

This makes no sense. What is g(2) ? f_n(2) + 1 ? What is n then? 

n is the number of the function in the ordered list of all functions from N to N. Adding 1 to f_n(n) gives a different function. Diagonalization does not help you here.

So g(n) = f_n(n)+1 is a different function. It is NOT f_n() with some different argument. So your attempt to make them the same function is invalid.

Bruce

[Bruno Marchal]

On 31 May 2020, at 10:08, Bruce Kellett <bhkel...@gmail.com> wrote:

On Sun, May 31, 2020 at 5:21 PM Bruno Marchal <mar...@ulb.ac.be> wrote:

On 31 May 2020, at 07:44, Bruce Kellett <bhkel...@gmail.com> wrote:

On Sun, May 31, 2020 at 2:26 AM Bruno Marchal <mar...@ulb.ac.be> wrote:

Let us write f_n for the function from N to N computed by nth expression. 

Now, the function g defined by g(n) = f_n(n) + 1 is computable, and is defined on all N. So it is a computable function from N to N. It is computable because it each f_n is computable, “+ 1” is computable, and, vy our hypothesis it get all and only all computable functions from N to N.

But then, g has have itself an expression in that universal language, of course. There there is a number k such that g = f_k. OK?

But then we get that g_k, applied to k has to give f_k(k), as g = f_k, and f_k(k) + 1, by definition of g. 

That is a fairly elementary blunder. g_k

What is g_k?

That is your notation: "But then we get that g_k, applied to k has to give f_k(k),”

That was a typo error. You need to read “g applied to k”. Sorry. But you should have just ask “what is g_k?”. 

I mean it is not a blunder, but a typo error (probably among many). The typo error is detectable from just the few words which go after. 

The only enumeration here is the f_k, then we have define a precise, single, function g such that

g(n) = f_n(n) + 1.  (f_n(n) is the diagonal term, you can see this by making the table (the infinite matrice) with the number in the top row, and the f_i in a column):

0 1 2 3 ...

f_0  f_0(0) f_0(1) f_0(2) f_0(3)

f_1 f_1(0) f_1(1) f_1(2) f_1(3)

f_2 f_2(0) f_2(1) f_2(2) f_2(3)

…

Here the underlining means “+1”.

applied to k, g_k(k) = f_n(k)+1,

There are no g_k.

You defined g_k!!  g_k applied to k is f_k(k), and that is your error.

“g_k" just has no meaning at all. Nowhere is a sequence g_k defined. 

g is the function defined by diagonalisation. g(x) = f_x(x) + 1, that g(0) = f_0(0) + 1, g(1) = f_1(1) + 1, g(2) = f_2(2) + 1, ...

But that is not what you said before.

It is. Read again, avoid the typo error when I apply g on its code k. 

 

by definition of g_k.

The only enumeration was the enumeration of the functions f_k

You do not get to change the function from f_n to f_k in the expression.

We do. 

It is only the argument that changes: in other words, f_n(n) becomes f_n(k).

This makes no sense. What is g(2) ? f_n(2) + 1 ? What is n then? 

n is the number of the function in the ordered list of all functions from N to N. Adding 1 to f_n(n) gives a different function. Diagonalization does not help you here.

It defines g. 

g(n) = f_n(n) + 1

To compute g, enumerate the f_i up to f_n, and compute f_n on n, and add one. Of course this will lead to a contradiction, which shows that the bijection n —> f_n is not computable. (But that does not destroy CT, it only make the f_n mixed with partial (defined on a subset of N) computable function, and that mowing has to be non computable (if not we can filter out the partial and get a computable enumeration of the f_n, and get again the contradiction).

So g(n) = f_n(n)+1 is a different function.

Different from what? If it is different from all f_n, the language is no more universal. It means that if the language is universal, g is not computable, and what is not computable is not the f_n, nor “+ 1”, so it can only be the enumeration f_n itself. That is the point against conventionalism: computable entails the existence of non computable well defined entities.

It is NOT f_n() with some different argument. So your attempt to make them the same function is invalid.

?

Are you saying that you believe that we can enumerate computably the computable functions from N to N? If that is the case, you have missed the argument.

Apology for the typo error. But now that it is has been corrected, you might read more cautiously the argument; I just do not understand you last remark. I will re-explain, if necessary. This post is the minimal think to proceed. It ex^lmains also why the universal dovetailer has to … dovetail, necessarily.

Bruno

Bruce

So you are talking nonsense.

You miss the diagonal. Read again.

Bruno

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[Bruce Kellett]

On Sun, May 31, 2020 at 8:56 PM Bruno Marchal <mar...@ulb.ac.be> wrote:

On 31 May 2020, at 10:08, Bruce Kellett <bhkel...@gmail.com> wrote:

That is your notation: "But then we get that g_k, applied to k has to give f_k(k),”

That was a typo error. You need to read “g applied to k”. Sorry. But you should have just ask “what is g_k?”. 

I mean it is not a blunder, but a typo error (probably among many). The typo error is detectable from just the few words which go after. 

I am not a mind reader. I try to understand what you write, but I can't always differentiate between typos and straight blunders.

Your attempted diagonalization argument just does not work here.

f_n(n)+1 has not been shown not to be among the other enumerated functions. f_n(n) is just a number, as is f_n(n)+1, so it is clearly among the functions that map N to N, it is just not f_n. Remember that N is not finite.

Bruce

[Brent Meeker]

On 5/30/2020 10:44 PM, Bruce Kellett wrote:

On Sun, May 31, 2020 at 2:26 AM Bruno Marchal <mar...@ulb.ac.be> wrote:

Let us write f_n for the function from N to N computed by nth expression. 

Now, the function g defined by g(n) = f_n(n) + 1 is computable, and is defined on all N. So it is a computable function from N to N. It is computable because it each f_n is computable, “+ 1” is computable, and, vy our hypothesis it get all and only all computable functions from N to N.

But then, g has have itself an expression in that universal language, of course. There there is a number k such that g = f_k. OK?

But then we get that g_k, applied to k has to give f_k(k), as g = f_k, and f_k(k) + 1, by definition of g. 

That is a fairly elementary blunder. g_k applied to k, g_k(k) = f_n(k)+1, by definition of g_k. You do not get to change the function from f_n to f_k in the expression. It is only the argument that changes: in other words, f_n(n) becomes f_n(k). So you are talking nonsense.

No, I think that's OK.  It's a straight substitution n->k.  The trick is that g(n) is not some well defined specific function because n has infinite range.  So none of this works in a finite world.  But it's not surprising that there is incompleteness in an infinite theory.

Brent

Bruce

So f_k(k) = f_k(k) +1.

And f_k(k) has to be number, given that we were enumerating functions from N to N. So we can subtract f_k(k) on both sides, and we get 0 = 1. CQFD.

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[Bruce Kellett]

On Mon, Jun 1, 2020 at 3:12 AM 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 5/30/2020 10:44 PM, Bruce Kellett wrote:

On Sun, May 31, 2020 at 2:26 AM Bruno Marchal <mar...@ulb.ac.be> wrote:

Let us write f_n for the function from N to N computed by nth expression. 

Now, the function g defined by g(n) = f_n(n) + 1 is computable, and is defined on all N. So it is a computable function from N to N. It is computable because it each f_n is computable, “+ 1” is computable, and, vy our hypothesis it get all and only all computable functions from N to N.

But then, g has have itself an expression in that universal language, of course. There there is a number k such that g = f_k. OK?

But then we get that g_k, applied to k has to give f_k(k), as g = f_k, and f_k(k) + 1, by definition of g. 

That is a fairly elementary blunder. g_k applied to k, g_k(k) = f_n(k)+1, by definition of g_k. You do not get to change the function from f_n to f_k in the expression. It is only the argument that changes: in other words, f_n(n) becomes f_n(k). So you are talking nonsense.

No, I think that's OK.  It's a straight substitution n->k.  The trick is that g(n) is not some well defined specific function because n has infinite range.  So none of this works in a finite world.  But it's not surprising that there is incompleteness in an infinite theory.

Yes, I had misunderstood what g(n) was supposed to be -- it is simply a representation of the diagonal elements of the array, plus 1. But Bruno's attempt to use the diagonal argument here fails, because  he has to show that f_n(n)+1 is not contained in the infinite list. He has failed to do this.

Bruce

[Brent Meeker]

All computable functions are in the list ex hypothesi.

Brent

Bruce

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[Bruce Kellett]

On Mon, Jun 1, 2020 at 8:31 AM 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 5/31/2020 3:23 PM, Bruce Kellett wrote:

On Mon, Jun 1, 2020 at 3:12 AM 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 5/30/2020 10:44 PM, Bruce Kellett wrote:

On Sun, May 31, 2020 at 2:26 AM Bruno Marchal <mar...@ulb.ac.be> wrote:

Let us write f_n for the function from N to N computed by nth expression. 

Now, the function g defined by g(n) = f_n(n) + 1 is computable, and is defined on all N. So it is a computable function from N to N. It is computable because it each f_n is computable, “+ 1” is computable, and, vy our hypothesis it get all and only all computable functions from N to N.

But then, g has have itself an expression in that universal language, of course. There there is a number k such that g = f_k. OK?

But then we get that g_k, applied to k has to give f_k(k), as g = f_k, and f_k(k) + 1, by definition of g. 

That is a fairly elementary blunder. g_k applied to k, g_k(k) = f_n(k)+1, by definition of g_k. You do not get to change the function from f_n to f_k in the expression. It is only the argument that changes: in other words, f_n(n) becomes f_n(k). So you are talking nonsense.

No, I think that's OK.  It's a straight substitution n->k.  The trick is that g(n) is not some well defined specific function because n has infinite range.  So none of this works in a finite world.  But it's not surprising that there is incompleteness in an infinite theory.

Yes, I had misunderstood what g(n) was supposed to be -- it is simply a representation of the diagonal elements of the array, plus 1. But Bruno's attempt to use the diagonal argument here fails, because  he has to show that f_n(n)+1 is not contained in the infinite list. He has failed to do this.

All computable functions are in the list ex hypothesi.

That is what the diagonal argument is all about: you hypothesize that all bit strings (for example) are in your infinite list. Then you flip the diagonal bit of each string and form a new string from all the diagonal elements. And lo, that new string is not in the initial list. Therefore your hypothesis that all bit strings are in the list is disproven.

Bruno has attempted toride to glory on this argument, and has failed miserably!

Bruce

[Brent Meeker]

That's a general problem with reductio arguments.  When you get to end you don't know which premise was wrong.  Bruno, isn't changing the hypothetical list though, so he's saying the premise that you can order the total functions is wrong.  You can order the functions (say lexigraphically) but you can't know which are total.

ISTM the result, that there's an incompleteness theorem for the set of all functions, is quite intuitive.  But Bruno seems to be saying this is all finitist because he doesn't assum and axiom of infinity.  Yet the "diagonalization" doesn't work in a finite world.

Brent

Bruce

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[Bruce Kellett]

On Mon, Jun 1, 2020 at 9:05 AM 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 5/31/2020 3:49 PM, Bruce Kellett wrote:

On Mon, Jun 1, 2020 at 8:31 AM 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 5/31/2020 3:23 PM, Bruce Kellett wrote:

On Mon, Jun 1, 2020 at 3:12 AM 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 5/30/2020 10:44 PM, Bruce Kellett wrote:

On Sun, May 31, 2020 at 2:26 AM Bruno Marchal <mar...@ulb.ac.be> wrote:

Let us write f_n for the function from N to N computed by nth expression. 

Now, the function g defined by g(n) = f_n(n) + 1 is computable, and is defined on all N. So it is a computable function from N to N. It is computable because it each f_n is computable, “+ 1” is computable, and, vy our hypothesis it get all and only all computable functions from N to N.

But then, g has have itself an expression in that universal language, of course. There there is a number k such that g = f_k. OK?

But then we get that g_k, applied to k has to give f_k(k), as g = f_k, and f_k(k) + 1, by definition of g. 

That is a fairly elementary blunder. g_k applied to k, g_k(k) = f_n(k)+1, by definition of g_k. You do not get to change the function from f_n to f_k in the expression. It is only the argument that changes: in other words, f_n(n) becomes f_n(k). So you are talking nonsense.

No, I think that's OK.  It's a straight substitution n->k.  The trick is that g(n) is not some well defined specific function because n has infinite range.  So none of this works in a finite world.  But it's not surprising that there is incompleteness in an infinite theory.

Yes, I had misunderstood what g(n) was supposed to be -- it is simply a representation of the diagonal elements of the array, plus 1. But Bruno's attempt to use the diagonal argument here fails, because  he has to show that f_n(n)+1 is not contained in the infinite list. He has failed to do this.

All computable functions are in the list ex hypothesi.

That is what the diagonal argument is all about: you hypothesize that all bit strings (for example) are in your infinite list. Then you flip the diagonal bit of each string and form a new string from all the diagonal elements. And lo, that new string is not in the initial list. Therefore your hypothesis that all bit strings are in the list is disproven.

Bruno has attempted toride to glory on this argument, and has failed miserably!

That's a general problem with reductio arguments.  When you get to end you don't know which premise was wrong.  Bruno, isn't changing the hypothetical list though, so he's saying the premise that you can order the total functions is wrong.  You can order the functions (say lexigraphically) but you can't know which are total.

ISTM the result, that there's an incompleteness theorem for the set of all functions, is quite intuitive.  But Bruno seems to be saying this is all finitist because he doesn't assum and axiom of infinity.  Yet the "diagonalization" doesn't work in a finite world.

Take all bit strings of length N (finite) and apply the diagonal argument. The string resulting from putting all the flipped diagonal bits together is not in the original list, contradicting the assumption that the list is complete. Of course, the list of all strings of length N contains more than N elements, so the diagonal argument does not apply. The set of all strings of infinite length is certainly infinite, so one might work the diagonal argument there -- if one doesn't worry too much about cardinality issues......

I think Bruno should rephrase his argument -- it might be sensible, but as presented it was clearly invalid.

Bruce

[Brent Meeker]

On 5/31/2020 4:32 PM, Bruce Kellett wrote:

On Mon, Jun 1, 2020 at 9:05 AM 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 5/31/2020 3:49 PM, Bruce Kellett wrote:

On Mon, Jun 1, 2020 at 8:31 AM 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 5/31/2020 3:23 PM, Bruce Kellett wrote:

On Mon, Jun 1, 2020 at 3:12 AM 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 5/30/2020 10:44 PM, Bruce Kellett wrote:

On Sun, May 31, 2020 at 2:26 AM Bruno Marchal <mar...@ulb.ac.be> wrote:

Let us write f_n for the function from N to N computed by nth expression. 

Now, the function g defined by g(n) = f_n(n) + 1 is computable, and is defined on all N. So it is a computable function from N to N. It is computable because it each f_n is computable, “+ 1” is computable, and, vy our hypothesis it get all and only all computable functions from N to N.

But then, g has have itself an expression in that universal language, of course. There there is a number k such that g = f_k. OK?

But then we get that g_k, applied to k has to give f_k(k), as g = f_k, and f_k(k) + 1, by definition of g. 

That is a fairly elementary blunder. g_k applied to k, g_k(k) = f_n(k)+1, by definition of g_k. You do not get to change the function from f_n to f_k in the expression. It is only the argument that changes: in other words, f_n(n) becomes f_n(k). So you are talking nonsense.

No, I think that's OK.  It's a straight substitution n->k.  The trick is that g(n) is not some well defined specific function because n has infinite range.  So none of this works in a finite world.  But it's not surprising that there is incompleteness in an infinite theory.

Yes, I had misunderstood what g(n) was supposed to be -- it is simply a representation of the diagonal elements of the array, plus 1. But Bruno's attempt to use the diagonal argument here fails, because  he has to show that f_n(n)+1 is not contained in the infinite list. He has failed to do this.

All computable functions are in the list ex hypothesi.

That is what the diagonal argument is all about: you hypothesize that all bit strings (for example) are in your infinite list. Then you flip the diagonal bit of each string and form a new string from all the diagonal elements. And lo, that new string is not in the initial list. Therefore your hypothesis that all bit strings are in the list is disproven.

Bruno has attempted toride to glory on this argument, and has failed miserably!

That's a general problem with reductio arguments.  When you get to end you don't know which premise was wrong.  Bruno, isn't changing the hypothetical list though, so he's saying the premise that you can order the total functions is wrong.  You can order the functions (say lexigraphically) but you can't know which are total.

ISTM the result, that there's an incompleteness theorem for the set of all functions, is quite intuitive.  But Bruno seems to be saying this is all finitist because he doesn't assum and axiom of infinity.  Yet the "diagonalization" doesn't work in a finite world.

Take all bit strings of length N (finite) and apply the diagonal argument. The string resulting from putting all the flipped diagonal bits together is not in the original list, contradicting the assumption that the list is complete.

If N is finite then there are only 2^N possible bit strings and the list will include all of them.  So when you flip a bit in each one you get the same list, just reordered.

Brent

Of course, the list of all strings of length N contains more than N elements, so the diagonal argument does not apply. The set of all strings of infinite length is certainly infinite, so one might work the diagonal argument there -- if one doesn't worry too much about cardinality issues......

I think Bruno should rephrase his argument -- it might be sensible, but as presented it was clearly invalid.

Bruce

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[Bruno Marchal]

Where?

I repeat the argument. If f_n is an enumeration of *al*l computable functions from N to N, then we can define the function g by saying that on each n in N, g(n) = f_n(n) + 1. (Or if you prefer g = [n](f_n(n) + 1) if you dislike (like me) to use f(x) to describe a function. But most people fear the lambda notation ([n]), so I will use the secondary usual denotation of function, with the implicit meaning that x or n are the variable/input.

But now we have a contradiction if we assume f_n mechanically (recursively, computably) enumerable. 

In that case g is a computable defined on all n in N, and thus has to be in the enumeration of all computable function f_n. That means that it exists a number k such g is f_k, but then:

g(k) = f_k(k) (by g being a function computable from N to N)

g(k) = f_k(k) + 1 (by definition of g)

And thus

f_k(k) = f_k(k) + 1

And f_k(k) is a number (by definition of what is a function from N to N), so we can subtract it at both sides, and we get

0 = 1.

Conclusion: the set of the computable functions from N to N, which is obviously enumerable (as explained in my preceding post) is not mechanically enumerable. Any bijection or surjection associating n to f_n (with possible repletion) is a non computable function.

At first sight this seems to contradict Church thesis, but it implies only that if there exist a universal language (CT), i.e. capable of defining all computable functions from N to N, then the enumeration of the expressions (among those computing notably all functions from N to N), there will be other objects, and indeed, some undefined functions on some, i.e. functions from subset of N (called the domain of f) in N.  That gives the phi_n, and their domain w_n.

We can define g in the universal language, and the diagonal argument, on those phi_n will go through up to

phi_k(k) = phi_k(k) + 1

But this does not lead to a contradiction now, as we can derive from this that phi_k(k) is not defined, and can no more be subtracted at the left and right hand side.

This entails a strong form of incompleteness, called by Tarski: Essential Incompleteness: it means that *all* effective theory (theory such that we can check mechanically if a proof is valid or not) are incomplete, and cannot be completed by adding axioms (any effective number of axioms).

It means that concerning the finite machines, there will always be true, and well determined proposition, which cannot be proved in any theory about them. Indeed if a theory could effectively prove if a phi_n is from N to N or not, then we could filter out the partial functions (not defined on all n in N), and get a computable enumeration  of the f_n, and in this case, the diagonal argument lead (again) to 0 = 1.

Brent suggest that we might recover completeness by restricting N to a finite domain. That is correct, because all finite function are computable, but then, we have incompleteness directly with respect to the computable functions, even limited on finite but arbitrary domain. In fact, that moves makes the computer simply vanishing, and it makes Mechanism not even definable or expressible. On the contrary, what the diagonal argument shows here is that there are FINITE machines, whose behaviour is not computable, nor most of its attribute.

Which one? Well, given that phi_n is mechanically enumerable, and given that we can build a computable bijection between NxN and n, we get a universal machine u, which is such that for all n, m we have phi_u(n, m) = phi_n(m). u is called universal digital machine/number, or computer/intepreter. The computer has been discovered in this way, before its implementation (excepting Babbage machine, never completely implemented though).

Definition: I call “universal enumeration” or “universal machinery” the enumeration of the phi_n, (or of the corresponding expressions). I call “universal machine” the number u as defined above. It shows that one finite machine (u) can, given the expression (encoded by its number in the enumeration of expression) computing it on any argument m, on which phi_n is defined.

Note that when we have a computable enumeration phi_n, i.e. a universal machinery, we will have all universal machine defined in it. Inversely, give a universal machine u, we get a universal machinery canonically associated with it:

phi_u(0, n), phi_u(1, n), phi_u(2, n), phi_u(3, n), phi_u(4, n), phi_u(5, n), phi_u(6, n), …

So, universal machinery and universal machine are always canonically associated together. It is but the difference between universal language (something infinite) and the universal machine (which is a finite entity, understanding that language).

It is the difference between saying, say, that the notion of Turing machine formalism is Turing universal, and saying this precise universal machine is a computer (universal machine, or universal number).

Does this prove Gödel’s original incompleteness theorem? Almost. Gödel’s original incompleteness was about an ultra powerful theory (Principia Mathematica), and all you have to show is to show that his powerful theory can enumerate mechanically some universal machinery. Now, Gödel did much more, as it showed that elementary arithmetic can already enumerate the phi_i and compute the phi_u, and is thus Turing universal. This is much longer and tedious to show, and contains some difficulties, although it becomes simple (but still tedious), if you add the exponential axioms:

Exp(n, 0) = 1

Exp(n s(m)) = n * Exp(n, m)

What is not entirely obvious here is to proves those axioms in PA (which has only the axioms of addition and multiplication).

In that case you can no more represent a list by using the factorisation theorem, and you need to use a bit more of elementary number theory. Gödel famously used the Chinese Lemma.

What is terribly more difficult is to shows that Exp can be defined by a diophantine polynomial. This has taken 50 years, starting with Davis and Putnam, quite well developed by Julia Robinson, and transformed by Matiyasevic. This shows that not only arithmetic is Turing universal, but that the diophantine polynômes already are enough. 

Now, the human are obviously Turing universal. The mechanist hypothesis (well just CT) asserts that we are not *more* than universal machine, with respect to our computability hypothesis. It is easy to show that such a universal machine cannot distinguish between being emulated by this or that universal machine/number. That is why we have to extract physics from a statistic on all computations, as we cannot “localise” ourself in any universal dovetailing, which is emulated in all model of any theory in which we can define the phi_n, like arithmetic as Gödel showed already in his 1931 paper. I explain this with all details in my papers and books.

Conclusion: to save physicalism, you need a non mechanist theory of mind. With mechanism, you have “only” to derive the laws of physics (the mathematic of the observable, []p & <>t, p sigma_1 + oracle) from a measure on all (relative) computations. (The need of the oracles is due to step 2: the first person is invariant for the number of step done by the Universal Dovetailer to access their state: then with ZFC + DP, the measure has been proved to exist, and I have already shown that if the measure exists, it obeys quantum logic, with a semantic made in terms of alternate consistent histories: this generalise feynmann integral, and normally should get it exactly, but we still have no precise hamiltonian or Lagrangian, or “action”).

Bruno

Bruce

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[Lawrence Crowell]

It took a while for me to get to this. The formula f_k(k) = f(k)+1 and g such that you get 0 = 1 appears to be the contradiction one gets with the enumeration of all Gödel numbers. This illustrates in Gödel's first theorem that no axiomatic system can determine its consistency. 

I am not sure about the claims with respect to the ontology of mathematics. Gödel made statements about how his theorems indicated there is an objective reality to mathematics. I do not disagree with mathematics as objective, but on the other hand I am not sure this is a clear demonstration of that. The objective or epistemological nature of mathematics is something we endlessly debate. 

LC

On Saturday, May 30, 2020 at 11:26:08 AM UTC-5, Bruno Marchal wrote:

[Bruce Kellett]

On Mon, Jun 1, 2020 at 7:08 PM Bruno Marchal <mar...@ulb.ac.be> wrote:

On 1 Jun 2020, at 00:23, Bruce Kellett <bhkel...@gmail.com> wrote:

On Mon, Jun 1, 2020 at 3:12 AM 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 5/30/2020 10:44 PM, Bruce Kellett wrote:

On Sun, May 31, 2020 at 2:26 AM Bruno Marchal <mar...@ulb.ac.be> wrote:

Let us write f_n for the function from N to N computed by nth expression. 

Now, the function g defined by g(n) = f_n(n) + 1 is computable, and is defined on all N. So it is a computable function from N to N. It is computable because it each f_n is computable, “+ 1” is computable, and, vy our hypothesis it get all and only all computable functions from N to N.

But then, g has have itself an expression in that universal language, of course. There there is a number k such that g = f_k. OK?

But then we get that g_k, applied to k has to give f_k(k), as g = f_k, and f_k(k) + 1, by definition of g. 

That is a fairly elementary blunder. g_k applied to k, g_k(k) = f_n(k)+1, by definition of g_k. You do not get to change the function from f_n to f_k in the expression. It is only the argument that changes: in other words, f_n(n) becomes f_n(k). So you are talking nonsense.

No, I think that's OK.  It's a straight substitution n->k.  The trick is that g(n) is not some well defined specific function because n has infinite range.  So none of this works in a finite world.  But it's not surprising that there is incompleteness in an infinite theory.

Yes, I had misunderstood what g(n) was supposed to be -- it is simply a representation of the diagonal elements of the array, plus 1. But Bruno's attempt to use the diagonal argument here fails, because  he has to show that f_n(n)+1 is not contained in the infinite list. He has failed to do this.

Where?

I can't show where you are not doing something!

I repeat the argument. If f_n is an enumeration of *al*l computable functions from N to N, then we can define the function g by saying that on each n in N, g(n) = f_n(n) + 1. (Or if you prefer g = [n](f_n(n) + 1) if you dislike (like me) to use f(x) to describe a function. But most people fear the lambda notation ([n]), so I will use the secondary usual denotation of function, with the implicit meaning that x or n are the variable/input.

But now we have a contradiction if we assume f_n mechanically (recursively, computably) enumerable. 

In that case g is a computable defined on all n in N, and thus has to be in the enumeration of all computable function f_n. That means that it exists a number k such g is f_k, but then:

g(k) = f_k(k) (by g being a function computable from N to N)

But there is no need for this computable number to lie on the diagonal. It could well be f_k(x), for x!=k.

Bruce

[Bruno Marchal]

On Monday, June 1, 2020 at 12:13:32 PM UTC+2, Lawrence Crowell wrote:

It took a while for me to get to this. The formula f_k(k) = f(k)+1 and g such that you get 0 = 1 appears to be the contradiction one gets with the enumeration of all Gödel numbers. This illustrates in Gödel's first theorem that no axiomatic system can determine its consistency. 

It entails Gödel's first theorem (as I just showed): Any theory capable of defining the phi_i (the partial computable function, with the important remind that a Total function is a partial function, just that the subset of N on which it is defined is N itself), I say: any theory in which you can define the phi_i is essentially incomplete, if not it can decide if a code is for a total or or strictly partial, and you get can filter out the f_n from the phi_i and get back to 0 = 1.

Gödel's first theorem just say that all effective theories are incomplete. The second theorem illustrates indeed this by given a true but not provable by the machine/theory, and is proved by showing that the machine can prove the first theorem, but you need the machine to believe in the induction axiom, so this concerned what I call often the Gödel-Löbian machine, or shortly, the Löbian machine. You can characterise them by the universal machine/number which knows that they are universal (even in the weak technical sense that they can prove for all sigma_1 sentences p that p -> []p. (i.e. p -> bewesibar('p'), where 'p' encodes p in the language of arithmetic).

 

I am not sure about the claims with respect to the ontology of mathematics.

 

Gödel made statements about how his theorems indicated there is an objective reality to mathematics. I do not disagree with mathematics as objective, but on the other hand I am not sure this is a clear demonstration of that.

It demonstrates that this happens objectively from the points of view of all universal number/machine/effective-theory possible. They live in a reality which kick bad. The predicate TOT(n), which means that n codes a function from N to N, i.e. a total function, is already a sort of God, and that's part of the arithmetical reality where those digital machine lives in. TOT(n) can be shown to be a PI_2 Sort of God. (Here God, by default is simply the union of all SIGMA_i and PI_i true proposition, for all i). G and G* (the propositional theology of the machine) are decidable, but qG (quantified G) is PI_2, and qG* is PI_1 in GOD. (yes, Plato's Noùs is far more complex than God itself in the machine theology, but that is coherent with Plotinus wanting the ONE to be fundamentally simple).

Now all the is unavoidable that it applies to us once we make sense of the expression "universal digital machine", which is needed to have the idea of what technically happens when you bet or hope  that you will survive with a digital artificial brain. 

It exmaplins also why "saying yes" to the doctor requires faith, not just on Mechanism, but on the Doctor too, and his choice of the substitution level. 

 

The objective or epistemological nature of mathematics is something we endlessly debate. 

But with mechanism, and without, but with computer science and mathematical logic, we can compare the (objective) computer's science with the infinitely many computer's computer science. That is all what G¨is about: to compare the arithmetical reality with the way the numbers can understand it, perceive it, feel it, etc. 

Incompleteness and indefinability gives the "Lagragian" of arithmetic: it is Tarski minus Gödel, and the intensional variants, that is G1* minus G1, Z1* minus Z1, etc.

I recall that G is the logic of provability of the Gödel-Löbian machine, provable by the machine, and G* is the logic of provability which is true, and as we limit ourselves to sound machine G is included in G*. The axiom of G1 are are those of G plus p -> []p 

[](A -> B) -> ([]A -> []B)

and the Löb formula:

[]([]A -> A) -> []A

The axiome "1" is

p -> []p, for all atomic sentence (sentence letter).

For G1*, is is all axioms of G1, + []A -> A

The main difference is that G is closed for the rule of necessitation (from A, derive []A), and G* is not.

Note that <>t = ~[]f, so that []([]f -> f) -> []f), Löb formula with the boolean false constant instead of A, is Gödel's second theorem: <>t -> ~[]<>t, or ~[]f -> ~[](~[]f).

<>t (thou are consistent), <>[]f (shit can happen), are typical properly theological, that is true but not provable (belonging to G* minus G).

The theology of numbers is so "objective" that it can be found by any sound universal machine thinking a little bit, and it can tested, as it contains the logic of the observable (and the whole physics actually) which can be compared with our actual observations and theoretical anticipations.

Incompleteness enforced all possible nuances brought in the Theaetetus dialog, and refute Socrates refutation of Theaetetus. You get 3+2 = 8 modes:

1) p     (truth, god, the one) 

2) []p    Mind, Intellect, the Noùs (the universal machine)

3) []p & p (the soul, the first person, the knower, the owner of subjective private truth)

4) []p & <>t (the observable, the thing you can bet on, the measurable)

5) []p & <>t & p (the sensible, the thing you can evaluate, appreciate)

3+2 = 5, but it happens that 2), and 4) and 5) all splits in two, mathematically. 4) and 5) inherit the split of 2) which is the G/G* split, i.e.. incompleteness. The mystery and beauty here is that 3) does NOT split: the soul is really the fixed point of the doubt, like Descartes saw (I think).

The logic of 3) is intuitionist. And get quantum with "p -> []p". Like in Plotinus, the soul has a foot in Matter at the start!

The logic of 4) is quantum at the * level (along G*). Same for the logic of 5), but it is richer and provides more or less the qualia.

It's all in the head of the universal Turing machine!

Bruno

[Bruno Marchal]

On 1 Jun 2020, at 12:37, Bruce Kellett <bhkel...@gmail.com> wrote:

On Mon, Jun 1, 2020 at 7:08 PM Bruno Marchal <mar...@ulb.ac.be> wrote:

On 1 Jun 2020, at 00:23, Bruce Kellett <bhkel...@gmail.com> wrote:

On Mon, Jun 1, 2020 at 3:12 AM 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 5/30/2020 10:44 PM, Bruce Kellett wrote:

On Sun, May 31, 2020 at 2:26 AM Bruno Marchal <mar...@ulb.ac.be> wrote:

Let us write f_n for the function from N to N computed by nth expression. 

Now, the function g defined by g(n) = f_n(n) + 1 is computable, and is defined on all N. So it is a computable function from N to N. It is computable because it each f_n is computable, “+ 1” is computable, and, vy our hypothesis it get all and only all computable functions from N to N.

But then, g has have itself an expression in that universal language, of course. There there is a number k such that g = f_k. OK?

But then we get that g_k, applied to k has to give f_k(k), as g = f_k, and f_k(k) + 1, by definition of g. 

That is a fairly elementary blunder. g_k applied to k, g_k(k) = f_n(k)+1, by definition of g_k. You do not get to change the function from f_n to f_k in the expression. It is only the argument that changes: in other words, f_n(n) becomes f_n(k). So you are talking nonsense.

No, I think that's OK.  It's a straight substitution n->k.  The trick is that g(n) is not some well defined specific function because n has infinite range.  So none of this works in a finite world.  But it's not surprising that there is incompleteness in an infinite theory.

Yes, I had misunderstood what g(n) was supposed to be -- it is simply a representation of the diagonal elements of the array, plus 1. But Bruno's attempt to use the diagonal argument here fails, because  he has to show that f_n(n)+1 is not contained in the infinite list. He has failed to do this.

Where?

I can't show where you are not doing something!

I repeat the argument. If f_n is an enumeration of *al*l computable functions from N to N, then we can define the function g by saying that on each n in N, g(n) = f_n(n) + 1. (Or if you prefer g = [n](f_n(n) + 1) if you dislike (like me) to use f(x) to describe a function. But most people fear the lambda notation ([n]), so I will use the secondary usual denotation of function, with the implicit meaning that x or n are the variable/input.

But now we have a contradiction if we assume f_n mechanically (recursively, computably) enumerable. 

In that case g is a computable defined on all n in N, and thus has to be in the enumeration of all computable function f_n. That means that it exists a number k such g is f_k, but then:

g(k) = f_k(k) (by g being a function computable from N to N)

But there is no need for this computable number to lie on the diagonal. It could well be f_k(x), for x!=k.

It can be elsewhere, sure, the problem is that f_k(k) is necessarily also on the diagonal, like f_0(0), f_1(1), f_2(2), …, by the diagonal construction.

For all n, g(n) = f_k(n), thus in particular g(k) = f_k(k)

But by definition of g, g(k) = f_k(k) + 1.

f_k(k) and f_k(k) + 1 are both on the intersection of the row for f_k and the diagonal. No problem if they are both undefined, but big problem if they are both defined, as as f_k are all total, ex hypothesis, they are well defined.

Conclusion: either the f_k do not run through all total computable functions, and L is not universal, or we keep faith in the Church Turing thesis, we claim our L universal, but then the language get necessarily richer and describes strictly partial function, added "non algorithmically” to the f_n. We get the phi_i, or phi_n.

That shows that conventionalism does not make sense, because who would make the convention of the existence of something that we cannot compute?

Of course, you could define “computable” by a large subclass of the total computable functions, but you will have to make it non Turing universal if you want to secure the system. It is enough to send people to the goulag in case they would use an elementary operation not allowed, forbidden, in your language.

It shows that the universal machine is born with an hesitation between security and universality/liberty. 

Universality -> shit happens.

Löbianity -> you know that shit happens.

If arithmetic is conventional, I would like to meet the guy who decided to add shit, by convention!

Bruno

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[Brent Meeker]

On 6/1/2020 2:08 AM, Bruno Marchal wrote:
> Brent suggest that we might recover completeness by restricting N to a
> finite domain. That is correct, because all finite function are
> computable, but then, we have incompleteness directly with respect to
> the computable functions, even limited on finite but arbitrary domain.
> In fact, that moves makes the computer simply vanishing, and it makes
> Mechanism not even definable or expressible.

That's going to come as a big shock to IBM stockholders.

Brent

[Lawrence Crowell]

This can be thought of as Turing machine computation. The undecidability is a sort of epistemic horizon. We can make these functions a state and measurement, and the 0 = 1 result when an axiomatic system is set to enumerate all its Gödel numbers, or what happens with a universal Turing machine, can be modeled as a function for a state and measurement. The incompleteness result is equivalent to a form of uncertainty in measurements. The function is a pairing that is a bracket structure and this as a diagonalization leads to a form of g_n(n) = f(n) + 1 and 0 = 1. 

This is a part of my practical interest in this. This may have some impact on the nature of quantum computation as well. The modal logic aspects come from the Gödel second theorem that for a proposition P then NOT-Prove (gn(P) ↔ P is TRUE.  If B(x) is a probability predicate, here B = Bew, then for True(B(x) → x) we can let D(x) be B(x) → x and the diagonal result gives us True C↔ D(gn(C)) and True C ↔ (B(gn(x)) → x). From this we can get Löb’s theorem. The probability predicate serves as a meaning of “necessarily” and Löb’s theorem as a statement of modal logic can be found.

The Löb’s theorem has been said to be a from of sematic structure. With quantum mechanics and my work with the unprovability of any global solution system for all p-adic sets this may have some curious connection. In particular with gravitation and the measurement problem where nonlinearity enters in. This suggests the “white noise” structure of quantum mechanics gives way to sort of pink noise or a process that is not purely random in a Markovian sense. This does appear to happen with large quantized systems with molecules on up to biology etc. My work with quantum hair on black holes suggests a sort of dualism with the UV physics of quantum gravitation and the IR physics at low energy as

(UV physics of gravitation) = (IR physics of fields and matter),

which is a way of writing the Einstein field equation.

LC

[Bruno Marchal]

Why? On the contrary. IBM bets on universal machine and know well what is a computer: a finite arithmetical being in touch with the infinite, and indeed, always asking for more memory, which is the typical symptom of liberty/universality. IBM might be finitist, like Mechanism, but is not ultrafinist at all. Anyway, mathematically, Mechanism is consistent with ulrafinitsim, even if to prove this, you need to go beyond finitism, (but then that’s the case for all consistent theory: none can prove its own consistency once “rich enough” (= just Turing universal, not “Löbian”).

Bruno



>
> Brent

>
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[Bruno Marchal]

On 2 Jun 2020, at 02:54, Lawrence Crowell <goldenfield...@gmail.com> wrote:

This can be thought of as Turing machine computation. The undecidability is a sort of epistemic horizon. We can make these functions a state and measurement, and the 0 = 1 result when an axiomatic system is set to enumerate all its Gödel numbers, or what happens with a universal Turing machine, can be modeled as a function for a state and measurement. The incompleteness result is equivalent to a form of uncertainty in measurements.

Incompleteness is about “[]p”, but ([]p & p) and other nuances inherit properties due to that incompleteness (like just existing to begin with). 

We have G* proves p <-> []p <-> ([]p & p) <-> ([]p & <>t) <-> ([]p & <>t & p).

This collapse all the epistemic nuances of the greeks, except that G does not prove those equivalence, which is what keep the physical hallucination going in the effective (“terrestrial”) plane.

It is not an epistemic horizon, it is more like a mathematical "surrational "corona, accessible by science, yet above the justifiable science, in between the rational (provable, justifiable) and the irrational (false, refutable or not).

The function is a pairing that is a bracket structure and this as a diagonalization leads to a form of g_n(n) = f(n) + 1 and 0 = 1. 

What is g_n, I guess you meant f_n. That is 

g(n) = f_n(n) + 1.

That equality has a very different meaning according to the presence or exclusion of the partial computable functions.

This is a part of my practical interest in this. This may have some impact on the nature of quantum computation as well.

The quantum appears in the observable/knowable levels:

[]p & p

[]p & <>t

[]p & <>t & p

With p sigma_1 arithmetical.

The modal logic aspects come from the Gödel second theorem that for a proposition P then NOT-Prove (gn(P) ↔ P is TRUE.  If B(x) is a probability predicate, here B = Bew, then for True(B(x) → x) we can let D(x) be B(x) → x and the diagonal result gives us True C↔ D(gn(C)) and True C ↔ (B(gn(x)) → x). From this we can get Löb’s theorem.

Yes. In arithmetic (not in the modal logic). I mean you cannot derive Löb’s formula from Gödel’s formula.

<>p -> ~[]<>p does not entail (modal logically) []([]p -> p) -> []p, It is easy to build a Kripke model satisfying Gödel’s formula, but not Löb’s formula.

The probability predicate serves as a meaning of “necessarily” and Löb’s theorem as a statement of modal logic can be found.

Yes, either as a rule:

Derive p from a proof of []p -> p,

Or as a formula:

[]([]p -> p) -> []p.  (Which implies Gödel’s second theorem: <>t -> <>[]f).

The Löb’s theorem has been said to be a from of sematic structure. With quantum mechanics and my work with the unprovability of any global solution system for all p-adic sets this may have some curious connection.

That relation with p-adic number is quite mysterious to me. If you have some link to some explanation, or if you can elaborate, perhaps...

In particular with gravitation and the measurement problem where nonlinearity enters in. This suggests the “white noise” structure of quantum mechanics gives way to sort of pink noise or a process that is not purely random in a Markovian sense. This does appear to happen with large quantized systems with molecules on up to biology etc. My work with quantum hair on black holes suggests a sort of dualism with the UV physics of quantum gravitation and the IR physics at low energy as

(UV physics of gravitation) = (IR physics of fields and matter),

which is a way of writing the Einstein field equation.

I can’t judge this. If true (and not contingent), it has to be derive from elementary arithmetic, or from Kxy = x and Sxyz = xz(yz). Note that we have already evidences (in the mechanist theory) that there is a ig physical core, and that K (which eliminates information) and S (which duplicates information) do not exist. The combinators S and K cannot exist (be implemented) in the physical reality core. They belong already to the number’s dream. If we use combinators in physics, it can only be true a (non Turing universal) BCI algebra (Bxyz = x(yz), Cxyz = xzy, Ix = x). 

Bruno

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[Brent Meeker]

On 6/2/2020 2:49 AM, Bruno Marchal wrote:
>> On 1 Jun 2020, at 22:43, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>
>>
>>
>> On 6/1/2020 2:08 AM, Bruno Marchal wrote:
>>> Brent suggest that we might recover completeness by restricting N to a finite domain. That is correct, because all finite function are computable, but then, we have incompleteness directly with respect to the computable functions, even limited on finite but arbitrary domain. In fact, that moves makes the computer simply vanishing, and it makes Mechanism not even definable or expressible.
>> That's going to come as a big shock to IBM stockholders.
>
> Why? On the contrary. IBM bets on universal machine

No, they bet only on finite machines, and they will be very surprised to
hear that they have vanished.

Brent

[Philip Thrift]

On Tuesday, June 2, 2020 at 12:34:37 PM UTC-5, Brent wrote:

On 6/2/2020 2:49 AM, Bruno Marchal wrote:
>> On 1 Jun 2020, at 22:43, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>
>>
>>
>> On 6/1/2020 2:08 AM, Bruno Marchal wrote:
>>> Brent suggest that we might recover completeness by restricting N to a finite domain. That is correct, because all finite function are computable, but then, we have incompleteness directly with respect to the computable functions, even limited on finite but arbitrary domain. In fact, that moves makes the computer simply vanishing, and it makes Mechanism not even definable or expressible.
>> That's going to come as a big shock to IBM stockholders.
>
> Why? On the contrary. IBM bets on universal machine

No, they bet only on finite machines, and they will be very surprised to
hear that they have vanished.

Brent

 

Next April 1, IBM should announce their IBM Zeno computer

       that allows a countably infinite number of algorithmic steps to be performed in finite time

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

@philipthrift 

[Lawrence Crowell]

On Tuesday, June 2, 2020 at 12:34:37 PM UTC-5, Brent wrote:

On 6/2/2020 2:49 AM, Bruno Marchal wrote:
>> On 1 Jun 2020, at 22:43, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>
>>
>>
>> On 6/1/2020 2:08 AM, Bruno Marchal wrote:
>>> Brent suggest that we might recover completeness by restricting N to a finite domain. That is correct, because all finite function are computable, but then, we have incompleteness directly with respect to the computable functions, even limited on finite but arbitrary domain. In fact, that moves makes the computer simply vanishing, and it makes Mechanism not even definable or expressible.
>> That's going to come as a big shock to IBM stockholders.
>
> Why? On the contrary. IBM bets on universal machine

No, they bet only on finite machines, and they will be very surprised to
hear that they have vanished.

Brent

For the most part computers are meant to run various algorithms that solve some restricted set of problems, say business applications. We use them largely as tools.

LC

[Philip Thrift]

On Wednesday, June 3, 2020 at 5:26:08 AM UTC-5, Lawrence Crowell wrote:

For the most part computers are meant to run various algorithms that solve some restricted set of problems, say business applications. We use them largely as tools.

LC

 

All of the (usable) theories of physics invented to date can be (and are) implemented on supercomputers (like those in the Dept. Of Energy national labs).

Some physicists though talk as if there is a Church they must go to -- where their minds are elevated into a Platonic realm where Physics is revealed to them. 

@philipthrift

[Lawrence Crowell]

I guess one might say that is what experiments do.

LC 

[Brent Meeker]

On 6/3/2020 3:26 AM, Lawrence Crowell wrote:

On Tuesday, June 2, 2020 at 12:34:37 PM UTC-5, Brent wrote:

On 6/2/2020 2:49 AM, Bruno Marchal wrote:
>> On 1 Jun 2020, at 22:43, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>
>>
>>
>> On 6/1/2020 2:08 AM, Bruno Marchal wrote:
>>> Brent suggest that we might recover completeness by restricting N to a finite domain. That is correct, because all finite function are computable, but then, we have incompleteness directly with respect to the computable functions, even limited on finite but arbitrary domain. In fact, that moves makes the computer simply vanishing, and it makes Mechanism not even definable or expressible.
>> That's going to come as a big shock to IBM stockholders.
>
> Why? On the contrary. IBM bets on universal machine

No, they bet only on finite machines, and they will be very surprised to
hear that they have vanished.

Brent

For the most part computers are meant to run various algorithms that solve some restricted set of problems, say business applications. We use them largely as tools.

Mathematics is largely a tool.  My pure mathematics friends over on math-fun seem to have most of their fun on Mathematica.

Brent

[Philip Thrift]

As I say,

      Physics = Math + Witchcraft.

(Computational Physics though is a programming domain.)

@philipthrift

[Philip Thrift]

I can't believe these mathematicians grovel themselves under the boot of a proprietary system. But I'm not surprised.

Mathematica is proprietary software restricted by both trade secret and copyright law. 

https://github.com/sympy/sympy/wiki/SymPy-vs.-Mathematica 

@philipthrift

[Bruno Marchal]

> On 2 Jun 2020, at 19:34, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>
>
>
> On 6/2/2020 2:49 AM, Bruno Marchal wrote:
>>> On 1 Jun 2020, at 22:43, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>>
>>>
>>>
>>> On 6/1/2020 2:08 AM, Bruno Marchal wrote:
>>>> Brent suggest that we might recover completeness by restricting N to a finite domain. That is correct, because all finite function are computable, but then, we have incompleteness directly with respect to the computable functions, even limited on finite but arbitrary domain. In fact, that moves makes the computer simply vanishing, and it makes Mechanism not even definable or expressible.
>>> That's going to come as a big shock to IBM stockholders.
>>
>> Why? On the contrary. IBM bets on universal machine
>
> No, they bet only on finite machines, and they will be very surprised to hear that they have vanished.

They bet on finite machines … including the universal machine, which I insist is a finite machine. That is even the reason why I called it from times to times universal number.

I recall that once we get the phi_i, which can be defined in elementary arithmetic, we get all the universal numbers, that is all u such that there phi_u(x, y) = phi_x(y), and such u can be used to define all the recursive enumeration of all digital machines.

The implementation of this fine but universal machines are called (physical) computer, and is the domain of expertise of IBM.

Bruno

>
> Brent
>
>> and know well what is a computer: a finite arithmetical being in touch with the infinite, and indeed, always asking for more memory, which is the typical symptom of liberty/universality. IBM might be finitist, like Mechanism, but is not ultrafinist at all. Anyway, mathematically, Mechanism is consistent with ulrafinitsim, even if to prove this, you need to go beyond finitism, (but then that’s the case for all consistent theory: none can prove its own consistency once “rich enough” (= just Turing universal, not “Löbian”).
>>
>> Bruno
>
>

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[Bruno Marchal]

That entry in wikipedia does not make much sense to me. Neither physically, nor mathematically. It needs to be elaborate … a lot.

Bruno

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[Bruno Marchal]

Of course, this is close to Aristotelian theology. It assumes that there is something which is not mathematical in some reality. A platonism or a pythegaorean think that he physical universe is but a tool, invented by the numbers to figure out what happens, and what is real.

But once you grasp that all computations exists in arithmetic (or more exactly, that they are enabled by the arithmetical true relations), even without Mechanism, the charge are reversed. It is those who claim (in metaphysics, not in physics) that there is a primitive universe who have the task to provide evidence. I have given the way to test this, and, thanks to QM, we can say that there are not yet any evidence found for a primitive physical universe. On the contrary, nature seems to obey exactly to what is needed for mechanism to be true.

Then, if we assume furthermore Mechanism, there is no more choice in this matter. Physics cannot be the fundamental science, it reduces to arithmetic (or any model of any Turing equivalent machinery) “seen-from inside”. 

Bruno

Brent

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[Bruno Marchal]

That is correct. But computational physics, like physics, cannot be the last word, and the physical reality, nor the psychological reality can be “entirely” computable. The universal machine is already not something totally computable, only partially. 

Any theory rich enough to define what is a computer has a non computable semantics.

Bruno

@philipthrift

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[Brent Meeker]

On 6/4/2020 4:07 AM, Bruno Marchal wrote:
>> On 2 Jun 2020, at 19:34, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>
>>
>>
>> On 6/2/2020 2:49 AM, Bruno Marchal wrote:
>>>> On 1 Jun 2020, at 22:43, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>>>
>>>>
>>>>
>>>> On 6/1/2020 2:08 AM, Bruno Marchal wrote:
>>>>> Brent suggest that we might recover completeness by restricting N to a finite domain. That is correct, because all finite function are computable, but then, we have incompleteness directly with respect to the computable functions, even limited on finite but arbitrary domain. In fact, that moves makes the computer simply vanishing, and it makes Mechanism not even definable or expressible.
>>>> That's going to come as a big shock to IBM stockholders.
>>> Why? On the contrary. IBM bets on universal machine
>> No, they bet only on finite machines, and they will be very surprised to hear that they have vanished.
> They bet on finite machines … including the universal machine, which I insist is a finite machine. That is even the reason why I called it from times to times universal number.
>
> I recall that once we get the phi_i,

i = 1 to inf.

[Brent Meeker]

On 6/4/2020 4:27 AM, Bruno Marchal wrote:

On 3 Jun 2020, at 21:47, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 6/3/2020 3:26 AM, Lawrence Crowell wrote:

On Tuesday, June 2, 2020 at 12:34:37 PM UTC-5, Brent wrote:

On 6/2/2020 2:49 AM, Bruno Marchal wrote:
>> On 1 Jun 2020, at 22:43, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>
>>
>>
>> On 6/1/2020 2:08 AM, Bruno Marchal wrote:
>>> Brent suggest that we might recover completeness by restricting N to a finite domain. That is correct, because all finite function are computable, but then, we have incompleteness directly with respect to the computable functions, even limited on finite but arbitrary domain. In fact, that moves makes the computer simply vanishing, and it makes Mechanism not even definable or expressible.
>> That's going to come as a big shock to IBM stockholders.
>
> Why? On the contrary. IBM bets on universal machine

No, they bet only on finite machines, and they will be very surprised to
hear that they have vanished.

Brent

For the most part computers are meant to run various algorithms that solve some restricted set of problems, say business applications. We use them largely as tools.

Mathematics is largely a tool.  My pure mathematics friends over on math-fun seem to have most of their fun on Mathematica.

Of course, this is close to Aristotelian theology. It assumes that there is something which is not mathematical in some reality. A platonism or a pythegaorean think that he physical universe is but a tool, invented by the numbers to figure out what happens, and what is real.

But once you grasp that all computations exists in arithmetic (or more exactly, that they are enabled by the arithmetical true relations), even without Mechanism, the charge are reversed. It is those who claim (in metaphysics, not in physics) that there is a primitive universe who have the task to provide evidence.

You have implicitly asserted that computation=reality.   With not proof, or even evidence.

Brent

I have given the way to test this, and, thanks to QM, we can say that there are not yet any evidence found for a primitive physical universe. On the contrary, nature seems to obey exactly to what is needed for mechanism to be true.

Then, if we assume furthermore Mechanism, there is no more choice in this matter. Physics cannot be the fundamental science, it reduces to arithmetic (or any model of any Turing equivalent machinery) “seen-from inside”. 

Bruno

Brent

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[Philip Thrift]

On Thursday, June 4, 2020 at 6:30:51 AM UTC-5, Bruno Marchal wrote:

On 4 Jun 2020, at 07:19, Philip Thrift <cloud...@gmail.com> wrote:

On Wednesday, June 3, 2020 at 1:03:58 PM UTC-5, Lawrence Crowell wrote:

On Wednesday, June 3, 2020 at 8:35:34 AM UTC-5, Philip Thrift wrote:

On Wednesday, June 3, 2020 at 5:26:08 AM UTC-5, Lawrence Crowell wrote:

For the most part computers are meant to run various algorithms that solve some restricted set of problems, say business applications. We use them largely as tools.

LC

 

All of the (usable) theories of physics invented to date can be (and are) implemented on supercomputers (like those in the Dept. Of Energy national labs).

Some physicists though talk as if there is a Church they must go to -- where their minds are elevated into a Platonic realm where Physics is revealed to them. 

@philipthrift

I guess one might say that is what experiments do.

LC 

As I say,

      Physics = Math + Witchcraft.

(Computational Physics though is a programming domain.)

That is correct. But computational physics, like physics, cannot be the last word, and the physical reality, nor the psychological reality can be “entirely” computable. The universal machine is already not something totally computable, only partially. 

Any theory rich enough to define what is a computer has a non computable semantics.

Bruno

Years ago I wrote about the Zetans

       http://poesophicalbits.blogspot.com/2012/04/persons-without-infinities.html

who never imagined infinities, nor found any reason to either think of them, or invent them.

They have a fine definition of computers and computing, and have found no need for anything more than finite mechanism in any of their theory of computing.

That we came to think "infinity" plays a role in computing (or in computing theory, or in mathematics in general) is just an aspect of our own peculiar psychology and history, but it is not needed.

@philipthrift 

[Lawrence Crowell]

On Thursday, June 4, 2020 at 6:07:45 AM UTC-5, Bruno Marchal wrote:

> On 2 Jun 2020, at 19:34, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>
>
>
> On 6/2/2020 2:49 AM, Bruno Marchal wrote:
>>> On 1 Jun 2020, at 22:43, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>>
>>>
>>>
>>> On 6/1/2020 2:08 AM, Bruno Marchal wrote:
>>>> Brent suggest that we might recover completeness by restricting N to a finite domain. That is correct, because all finite function are computable, but then, we have incompleteness directly with respect to the computable functions, even limited on finite but arbitrary domain. In fact, that moves makes the computer simply vanishing, and it makes Mechanism not even definable or expressible.
>>> That's going to come as a big shock to IBM stockholders.
>>
>> Why? On the contrary. IBM bets on universal machine
>
> No, they bet only on finite machines, and they will be very surprised to hear that they have vanished.

They bet on finite machines … including the universal machine, which I insist is a finite machine. That is even the reason why I called it from times to times universal number.

I recall that once we get the phi_i, which can be defined in elementary arithmetic, we get all the universal numbers, that is all u such that there phi_u(x, y) = phi_x(y), and such u can be used to define all the recursive enumeration of all digital machines.

The implementation of this fine but universal machines are called (physical) computer, and is the domain of expertise of IBM.

Bruno

Of course any computation is going to be finite or involve a finite number of bits. This happens as well with quantum computers, but there is one difference. Two states can be prepared and entangled so they have a continuum of probabilities depending upon measurement angle. This is what separates QM from classical mechanics. This separates entanglement of spins from the Bergman's socks, where knowing the left sock is in one box the right must be in the other. So while there is a finitude to the entanglement entropy or the quantity of quantum information, the possible ways an entanglement can register outcomes is infinite. This is what gives a violation of Bell's theorem in QM. With the measurement of a quantum system the pair of a state and measurement forms a type of Godel numbering. This connects QM foundations with the phi_u(x, y) = phi_x(y),you state above.

A classical computer will always be finite, and you can't have an infinite Cantor diagonalization. The computers that are manufactured are done so to solve certain problems, RSA encyrption, user interfaces for service personnel from travel agents to sales, word processors, games, cell phone signal shifters, data processors of medical measurements and on it goes. Even with quantum computers this will take off, and in fact I have thought quantum computing would be a way of managing a dynamics network defined by millions of drones over a city. Even if as I think the Godel-Turing result underlies obstructions between entanglement types quantum computers will in time become the province of engineering and business applications.

LC

 

>
> Brent
>
>> and know well what is a computer: a finite arithmetical being in touch with the infinite, and indeed, always asking for more memory, which is the typical symptom of liberty/universality. IBM might be finitist, like Mechanism, but is not ultrafinist at all. Anyway, mathematically, Mechanism is consistent with ulrafinitsim, even if to prove this, you need to go beyond finitism, (but then that’s the case for all consistent theory: none can prove its own consistency once “rich enough” (= just Turing universal, not “Löbian”).
>>
>> Bruno
>
>
> --
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[Brent Meeker]

On 6/4/2020 3:39 PM, Lawrence Crowell wrote:

On Thursday, June 4, 2020 at 6:07:45 AM UTC-5, Bruno Marchal wrote:

> On 2 Jun 2020, at 19:34, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>
>
>
> On 6/2/2020 2:49 AM, Bruno Marchal wrote:
>>> On 1 Jun 2020, at 22:43, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>>
>>>
>>>
>>> On 6/1/2020 2:08 AM, Bruno Marchal wrote:
>>>> Brent suggest that we might recover completeness by restricting N to a finite domain. That is correct, because all finite function are computable, but then, we have incompleteness directly with respect to the computable functions, even limited on finite but arbitrary domain. In fact, that moves makes the computer simply vanishing, and it makes Mechanism not even definable or expressible.
>>> That's going to come as a big shock to IBM stockholders.
>>
>> Why? On the contrary. IBM bets on universal machine
>
> No, they bet only on finite machines, and they will be very surprised to hear that they have vanished.

They bet on finite machines … including the universal machine, which I insist is a finite machine. That is even the reason why I called it from times to times universal number.

I recall that once we get the phi_i, which can be defined in elementary arithmetic, we get all the universal numbers, that is all u such that there phi_u(x, y) = phi_x(y), and such u can be used to define all the recursive enumeration of all digital machines.

The implementation of this fine but universal machines are called (physical) computer, and is the domain of expertise of IBM.

Bruno

Of course any computation is going to be finite or involve a finite number of bits. This happens as well with quantum computers, but there is one difference. Two states can be prepared and entangled so they have a continuum of probabilities depending upon measurement angle. This is what separates QM from classical mechanics.

I've wondered about this.  Of course a lot variables in the theory are continua; not just angle but also position.  Yet none of those can be measured to arbitrary precision.  And the more precisely one is, the less precisely it's conjugate can be...which is what separate QM from classical mechanics.  Holevo's theorem limits what we can know about a state.

Brent

This separates entanglement of spins from the Bergman's socks, where knowing the left sock is in one box the right must be in the other. So while there is a finitude to the entanglement entropy or the quantity of quantum information, the possible ways an entanglement can register outcomes is infinite. This is what gives a violation of Bell's theorem in QM. With the measurement of a quantum system the pair of a state and measurement forms a type of Godel numbering. This connects QM foundations with the phi_u(x, y) = phi_x(y),you state above.

A classical computer will always be finite, and you can't have an infinite Cantor diagonalization. The computers that are manufactured are done so to solve certain problems, RSA encyrption, user interfaces for service personnel from travel agents to sales, word processors, games, cell phone signal shifters, data processors of medical measurements and on it goes. Even with quantum computers this will take off, and in fact I have thought quantum computing would be a way of managing a dynamics network defined by millions of drones over a city. Even if as I think the Godel-Turing result underlies obstructions between entanglement types quantum computers will in time become the province of engineering and business applications.

LC

 

>
> Brent
>
>> and know well what is a computer: a finite arithmetical being in touch with the infinite, and indeed, always asking for more memory, which is the typical symptom of liberty/universality. IBM might be finitist, like Mechanism, but is not ultrafinist at all. Anyway, mathematically, Mechanism is consistent with ulrafinitsim, even if to prove this, you need to go beyond finitism, (but then that’s the case for all consistent theory: none can prove its own consistency once “rich enough” (= just Turing universal, not “Löbian”).
>>
>> Bruno
>
>
> --
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[Philip Thrift]

On Thursday, June 4, 2020 at 11:01:20 PM UTC-5, Brent wrote:

On 6/4/2020 3:39 PM, Lawrence Crowell wrote:

Of course any computation is going to be finite or involve a finite number of bits. This happens as well with quantum computers, but there is one difference. Two states can be prepared and entangled so they have a continuum of probabilities depending upon measurement angle. This is what separates QM from classical mechanics.

I've wondered about this.  Of course a lot variables in the theory are continua; not just angle but also position.  Yet none of those can be measured to arbitrary precision.  And the more precisely one is, the less precisely it's conjugate can be...which is what separate QM from classical mechanics.  Holevo's theorem limits what we can know about a state.

Brent

I was going to comment here but just to re-quote Max Tegmark's dictum:

Our challenge as physicists is to discover the infinity-free equations describing it—the true laws of physics.

 

It seems to always be needed to restate to physicists *as Vic Stenger did): 

Just because a mathematical theory someone came up with to model physical stuff has property X doesn't mean that the physical stuff has property X.

@philipthrift

[Bruno Marchal]

> On 4 Jun 2020, at 20:28, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>
>
>
> On 6/4/2020 4:07 AM, Bruno Marchal wrote:
>>> On 2 Jun 2020, at 19:34, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>>
>>>
>>>
>>> On 6/2/2020 2:49 AM, Bruno Marchal wrote:
>>>>> On 1 Jun 2020, at 22:43, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>>>>
>>>>>
>>>>>
>>>>> On 6/1/2020 2:08 AM, Bruno Marchal wrote:
>>>>>> Brent suggest that we might recover completeness by restricting N to a finite domain. That is correct, because all finite function are computable, but then, we have incompleteness directly with respect to the computable functions, even limited on finite but arbitrary domain. In fact, that moves makes the computer simply vanishing, and it makes Mechanism not even definable or expressible.
>>>>> That's going to come as a big shock to IBM stockholders.
>>>> Why? On the contrary. IBM bets on universal machine
>>> No, they bet only on finite machines, and they will be very surprised to hear that they have vanished.
>> They bet on finite machines … including the universal machine, which I insist is a finite machine. That is even the reason why I called it from times to times universal number.
>>
>> I recall that once we get the phi_i,
>
> i = 1 to inf.

That is the potential infinite, that you already need for a concept like the square root of 2, used all the time in elementary quantum mechanics.
Without it, neither CT, nor YD makes any sense. We could aswell stop doing any math, if not stop thinking.

The axioms that I use are just Kxy = x, and Sxyz = xz(yz).

There is no axiom of infinity, nor even induction axiom. That belongs only to the observers, and the proof of their existence requires only the two axiom above, or the arithmetic one, or anything Turing equivalent. With less than that, there is no computer, nor laptop … The universal machinery is potentially infinite. The universal machine is finite.

Bruno

>
>> which can be defined in elementary arithmetic, we get all the universal numbers, that is all u such that there phi_u(x, y) = phi_x(y), and such u can be used to define all the recursive enumeration of all digital machines.
>>
>> The implementation of this fine but universal machines are called (physical) computer, and is the domain of expertise of IBM.
>>
>> Bruno
>>
>>
>>
>>> Brent
>>>
>>>> and know well what is a computer: a finite arithmetical being in touch with the infinite, and indeed, always asking for more memory, which is the typical symptom of liberty/universality. IBM might be finitist, like Mechanism, but is not ultrafinist at all. Anyway, mathematically, Mechanism is consistent with ulrafinitsim, even if to prove this, you need to go beyond finitism, (but then that’s the case for all consistent theory: none can prove its own consistency once “rich enough” (= just Turing universal, not “Löbian”).
>>>>
>>>> Bruno
>>>
>>> --
>>> You received this message because you are subscribed to the Google Groups "Everything List" group.
>>> To unsubscribe from this group and stop receiving emails from it, send an email to everything-li...@googlegroups.com.
>>> To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/3068f558-7f61-56cb-61fe-44832ec28a91%40verizon.net.
>
>
> --
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[Bruno Marchal]

On 4 Jun 2020, at 20:35, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 6/4/2020 4:27 AM, Bruno Marchal wrote:

On 3 Jun 2020, at 21:47, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 6/3/2020 3:26 AM, Lawrence Crowell wrote:

On Tuesday, June 2, 2020 at 12:34:37 PM UTC-5, Brent wrote:

On 6/2/2020 2:49 AM, Bruno Marchal wrote:
>> On 1 Jun 2020, at 22:43, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>
>>
>>
>> On 6/1/2020 2:08 AM, Bruno Marchal wrote:
>>> Brent suggest that we might recover completeness by restricting N to a finite domain. That is correct, because all finite function are computable, but then, we have incompleteness directly with respect to the computable functions, even limited on finite but arbitrary domain. In fact, that moves makes the computer simply vanishing, and it makes Mechanism not even definable or expressible.
>> That's going to come as a big shock to IBM stockholders.
>
> Why? On the contrary. IBM bets on universal machine

No, they bet only on finite machines, and they will be very surprised to
hear that they have vanished.

Brent

For the most part computers are meant to run various algorithms that solve some restricted set of problems, say business applications. We use them largely as tools.

Mathematics is largely a tool.  My pure mathematics friends over on math-fun seem to have most of their fun on Mathematica.

Of course, this is close to Aristotelian theology. It assumes that there is something which is not mathematical in some reality. A platonism or a pythegaorean think that he physical universe is but a tool, invented by the numbers to figure out what happens, and what is real.

But once you grasp that all computations exists in arithmetic (or more exactly, that they are enabled by the arithmetical true relations), even without Mechanism, the charge are reversed. It is those who claim (in metaphysics, not in physics) that there is a primitive universe who have the task to provide evidence.

You have implicitly asserted that computation=reality.   With not proof, or even evidence.

?

The UDA *proves* that the fundamental reality = arithmetic. And AUDA (arithmetical Dovetailer Argument) makes the proof constructive, and it makes Mechanism testable, and the evidences for mechanism are striking, at a place where we know since 1500 years that Materialism is already refuted. Oh, yes, that is well hidden since 1500 years, by all gnostic (atheist or non atheists).

You are the one who seems to claim the existence of an ontological physical universe, where there is no proof nor any evidence. Evidences for a physical reality is not evidences for an ontological or primitive physical reality. The confusion between both of those is know as Aristotle theology. The belief in primary matter or physicalism (mathematicalist or not).

Bruno

Brent

I have given the way to test this, and, thanks to QM, we can say that there are not yet any evidence found for a primitive physical universe. On the contrary, nature seems to obey exactly to what is needed for mechanism to be true.

Then, if we assume furthermore Mechanism, there is no more choice in this matter. Physics cannot be the fundamental science, it reduces to arithmetic (or any model of any Turing equivalent machinery) “seen-from inside”. 

Bruno

Brent

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[Bruno Marchal]

That makes some sense. You can compute without axiom of infinity, and indeed you can define what is a computer just by using the two axioms Kxy = x, and Sxyz = xz(yz), as I have explicitly shown on this list. Similarly you can define a computer using only elementary arithmetic, like Gödel did implicitly and Kleene did explicitly. But to prove anything non trivial, you need induction, and to get semantics treated mathematically, you need actual infinity axioms, like with the notion of real numbers, etc.

To *understand* Kxy = x …, you need an axiom of infinity at the meta-level, and this is required by all scientist-numbers in arithmetic, so “infinity” is more than welcome to define the notion of observer, and for the notion of physical *laws*.

Bruno

@philipthrift 

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[Lawrence Crowell]

On Thursday, June 4, 2020 at 11:01:20 PM UTC-5, Brent wrote:

On 6/4/2020 3:39 PM, Lawrence Crowell wrote:

On Thursday, June 4, 2020 at 6:07:45 AM UTC-5, Bruno Marchal wrote:

> On 2 Jun 2020, at 19:34, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>
>
>
> On 6/2/2020 2:49 AM, Bruno Marchal wrote:
>>> On 1 Jun 2020, at 22:43, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>>
>>>
>>>
>>> On 6/1/2020 2:08 AM, Bruno Marchal wrote:
>>>> Brent suggest that we might recover completeness by restricting N to a finite domain. That is correct, because all finite function are computable, but then, we have incompleteness directly with respect to the computable functions, even limited on finite but arbitrary domain. In fact, that moves makes the computer simply vanishing, and it makes Mechanism not even definable or expressible.
>>> That's going to come as a big shock to IBM stockholders.
>>
>> Why? On the contrary. IBM bets on universal machine
>
> No, they bet only on finite machines, and they will be very surprised to hear that they have vanished.

They bet on finite machines … including the universal machine, which I insist is a finite machine. That is even the reason why I called it from times to times universal number.

I recall that once we get the phi_i, which can be defined in elementary arithmetic, we get all the universal numbers, that is all u such that there phi_u(x, y) = phi_x(y), and such u can be used to define all the recursive enumeration of all digital machines.

The implementation of this fine but universal machines are called (physical) computer, and is the domain of expertise of IBM.

Bruno

Of course any computation is going to be finite or involve a finite number of bits. This happens as well with quantum computers, but there is one difference. Two states can be prepared and entangled so they have a continuum of probabilities depending upon measurement angle. This is what separates QM from classical mechanics.

I've wondered about this.  Of course a lot variables in the theory are continua; not just angle but also position.  Yet none of those can be measured to arbitrary precision.  And the more precisely one is, the less precisely it's conjugate can be...which is what separate QM from classical mechanics.  Holevo's theorem limits what we can know about a state.

Brent

The main thing is that QM in violating the Bell inequalities has this vast degree of possible outcomes that would not happen with a purely classical system. We might say this large number of possible outcomes are a sort of epistemological horizon that prevent complete knowledge of a system. A classical system has both configuration and momentum accessible, while a QM system in configuration variables is half of that, purely Lagrangian. and this is what Holevo's theorem tells us. The pairing of a system state and measurement is a form of Gödel numbering, and this is not an aspect of classical mechanics. 

The difficult aspect to physics is not really complicated mathematics or calculations. It is really in coming up with simple fundamental statements. I think, of course I might be in some sort of delusion, that I am finding ways in this direction of late. I suppose if you persist with thinking about this long enough this can happen.

LC

 

This separates entanglement of spins from the Bergman's socks, where knowing the left sock is in one box the right must be in the other. So while there is a finitude to the entanglement entropy or the quantity of quantum information, the possible ways an entanglement can register outcomes is infinite. This is what gives a violation of Bell's theorem in QM. With the measurement of a quantum system the pair of a state and measurement forms a type of Godel numbering. This connects QM foundations with the phi_u(x, y) = phi_x(y),you state above.

A classical computer will always be finite, and you can't have an infinite Cantor diagonalization. The computers that are manufactured are done so to solve certain problems, RSA encyrption, user interfaces for service personnel from travel agents to sales, word processors, games, cell phone signal shifters, data processors of medical measurements and on it goes. Even with quantum computers this will take off, and in fact I have thought quantum computing would be a way of managing a dynamics network defined by millions of drones over a city. Even if as I think the Godel-Turing result underlies obstructions between entanglement types quantum computers will in time become the province of engineering and business applications.

LC

 

>
> Brent
>
>> and know well what is a computer: a finite arithmetical being in touch with the infinite, and indeed, always asking for more memory, which is the typical symptom of liberty/universality. IBM might be finitist, like Mechanism, but is not ultrafinist at all. Anyway, mathematically, Mechanism is consistent with ulrafinitsim, even if to prove this, you need to go beyond finitism, (but then that’s the case for all consistent theory: none can prove its own consistency once “rich enough” (= just Turing universal, not “Löbian”).
>>
>> Bruno
>
>
> --
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[Bruno Marchal]

On 5 Jun 2020, at 00:39, Lawrence Crowell <goldenfield...@gmail.com> wrote:

On Thursday, June 4, 2020 at 6:07:45 AM UTC-5, Bruno Marchal wrote:

> On 2 Jun 2020, at 19:34, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>
>
>
> On 6/2/2020 2:49 AM, Bruno Marchal wrote:
>>> On 1 Jun 2020, at 22:43, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>>
>>>
>>>
>>> On 6/1/2020 2:08 AM, Bruno Marchal wrote:
>>>> Brent suggest that we might recover completeness by restricting N to a finite domain. That is correct, because all finite function are computable, but then, we have incompleteness directly with respect to the computable functions, even limited on finite but arbitrary domain. In fact, that moves makes the computer simply vanishing, and it makes Mechanism not even definable or expressible.
>>> That's going to come as a big shock to IBM stockholders.
>>
>> Why? On the contrary. IBM bets on universal machine
>
> No, they bet only on finite machines, and they will be very surprised to hear that they have vanished.

They bet on finite machines … including the universal machine, which I insist is a finite machine. That is even the reason why I called it from times to times universal number.

I recall that once we get the phi_i, which can be defined in elementary arithmetic, we get all the universal numbers, that is all u such that there phi_u(x, y) = phi_x(y), and such u can be used to define all the recursive enumeration of all digital machines.

The implementation of this fine but universal machines are called (physical) computer, and is the domain of expertise of IBM.

Bruno

Of course any computation is going to be finite or involve a finite number of bits.

Any halting computation.

Some non halting computation requires infinite time and space, virtual, arithmetical or physical.

This happens as well with quantum computers, but there is one difference. Two states can be prepared and entangled so they have a continuum of probabilities depending upon measurement angle. This is what separates QM from classical mechanics. This separates entanglement of spins from the Bergman's socks, where knowing the left sock is in one box the right must be in the other. So while there is a finitude to the entanglement entropy or the quantity of quantum information, the possible ways an entanglement can register outcomes is infinite. This is what gives a violation of Bell's theorem in QM. With the measurement of a quantum system the pair of a state and measurement forms a type of Godel numbering. This connects QM foundations with the phi_u(x, y) = phi_x(y),you state above.

OK. But you assume some quantum universe, where the UDA explains why you have to derive the quantum from arithmetic or (Turing) equivalent.

A classical computer will always be finite, and you can't have an infinite Cantor diagonalization.

The Kleene diagonalisation is constructive. It shows the inexistence of some finite machine having some “arithmetical omniscience”. It requires potential infinities, not the cantorian infinities.

The computers that are manufactured are done so to solve certain problems, RSA encyrption, user interfaces for service personnel from travel agents to sales, word processors, games, cell phone signal shifters, data processors of medical measurements and on it goes.

All computers exists in arithmetic, and all computations exist in an internal limit of arithmetic (by step 2, actually!).

With mechanism, the physical reality is not the fundamental reality. The physical reality emerges from the computation executed in virtue of the number relations, like the prime number distribution, for example. 

Even with quantum computers this will take off, and in fact I have thought quantum computing would be a way of managing a dynamics network defined by millions of drones over a city. Even if as I think the Godel-Turing result underlies obstructions between entanglement types quantum computers will in time become the province of engineering and business applications.

No doubt on this. It is just that with mechanism, the physical universe is not ontological, but more like a collective hallucination made by the relative universal number relations which are as true, and independent of the physical laws, than 117 is composite and 317 is prime.

Bruno

<<
Of this reality, as I explained […], I take a 'realistic" view. At any rate (and this is my main point) this realistic view is much more plausible of mathematical than of physical reality, because mathematical objects are so much more what they seem. A chair or a star is not in the least like what it seems to be ; the more we think of it, the fuzzier its outlines become in the haze of sensations which surrounds it; but '2' and '317' has nothing to do with sensations, and its properties stand out the more clearly the more closely we scrutinize it. It may be that modern physics fits best in the framework of idealistic philosophy---I do not believe it, but there are eminent physicist who say so. Pure Mathematics, on the other hand, seems to me a rock on which all idealism founders: 317 is prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical is built that way.
>>
G. H. Hardy, "A Mathematician's Apology", Cambridge University Press, 1940 (1998).

LC

 

>
> Brent
>
>> and know well what is a computer: a finite arithmetical being in touch with the infinite, and indeed, always asking for more memory, which is the typical symptom of liberty/universality. IBM might be finitist, like Mechanism, but is not ultrafinist at all. Anyway, mathematically, Mechanism is consistent with ulrafinitsim, even if to prove this, you need to go beyond finitism, (but then that’s the case for all consistent theory: none can prove its own consistency once “rich enough” (= just Turing universal, not “Löbian”).
>>
>> Bruno
>
>
> --
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[Bruno Marchal]

On 5 Jun 2020, at 08:09, Philip Thrift <cloud...@gmail.com> wrote:

On Thursday, June 4, 2020 at 11:01:20 PM UTC-5, Brent wrote:

On 6/4/2020 3:39 PM, Lawrence Crowell wrote:

Of course any computation is going to be finite or involve a finite number of bits. This happens as well with quantum computers, but there is one difference. Two states can be prepared and entangled so they have a continuum of probabilities depending upon measurement angle. This is what separates QM from classical mechanics.

I've wondered about this.  Of course a lot variables in the theory are continua; not just angle but also position.  Yet none of those can be measured to arbitrary precision.  And the more precisely one is, the less precisely it's conjugate can be...which is what separate QM from classical mechanics.  Holevo's theorem limits what we can know about a state.

Brent

I was going to comment here but just to re-quote Max Tegmark's dictum:

Our challenge as physicists is to discover the infinity-free equations describing it—the true laws of physics.

Kxy = x

Sxyz = xz(yz).

But a degree 4 diophantine polynomial works as well. That is infinity-free. 

 

It seems to always be needed to restate to physicists *as Vic Stenger did): 

Just because a mathematical theory someone came up with to model physical stuff has property X doesn't mean that the physical stuff has property X.

That is valid. But once you bet on YD + CT, the charge are reversed, and it is the believer in a material or physical ontology who has to provide some evidences. I never found even one evidence for it. Each time people give me evidence, hey are easy to refute with the dream argument, especially its post-Turing mathematical version.

Bruno 

@philipthrift

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[Philip Thrift]

On Friday, June 5, 2020 at 4:47:14 AM UTC-5, Bruno Marchal wrote:

On 4 Jun 2020, at 22:33, Philip Thrift <cloud...@gmail.com> wrote:

Years ago I wrote about the Zetans

       http://poesophicalbits.blogspot.com/2012/04/persons-without-infinities.html

who never imagined infinities, nor found any reason to either think of them, or invent them.

They have a fine definition of computers and computing, and have found no need for anything more than finite mechanism in any of their theory of computing.

That we came to think "infinity" plays a role in computing (or in computing theory, or in mathematics in general) is just an aspect of our own peculiar psychology and history, but it is not needed.

That makes some sense. You can compute without axiom of infinity, and indeed you can define what is a computer just by using the two axioms Kxy = x, and Sxyz = xz(yz), as I have explicitly shown on this list. Similarly you can define a computer using only elementary arithmetic, like Gödel did implicitly and Kleene did explicitly. But to prove anything non trivial, you need induction, and to get semantics treated mathematically, you need actual infinity axioms, like with the notion of real numbers, etc.

To *understand* Kxy = x …, you need an axiom of infinity at the meta-level, and this is required by all scientist-numbers in arithmetic, so “infinity” is more than welcome to define the notion of observer, and for the notion of physical *laws*.

Bruno

As my old post shows, the Zetans do have their Axiom of Infinity - called the Axiom of Zillions.

They are happy with their finite set of numbers, but sometimes they construct bigger numbers to add to the set. But they do not think there are numbers between the bigger numbers. There are gaps. 

There is no potential infinity since they don't think their galaxy will last forever.

( All this follows from https://www.jstor.org/stable/2273942 )

@philipthrift 

[Brent Meeker]

On 6/5/2020 2:32 AM, Bruno Marchal wrote:
>> On 4 Jun 2020, at 20:28, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>
>>
>>
>> On 6/4/2020 4:07 AM, Bruno Marchal wrote:
>>>> On 2 Jun 2020, at 19:34, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>>>
>>>>
>>>>
>>>> On 6/2/2020 2:49 AM, Bruno Marchal wrote:
>>>>>> On 1 Jun 2020, at 22:43, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>>>>>
>>>>>>
>>>>>>
>>>>>> On 6/1/2020 2:08 AM, Bruno Marchal wrote:
>>>>>>> Brent suggest that we might recover completeness by restricting N to a finite domain. That is correct, because all finite function are computable, but then, we have incompleteness directly with respect to the computable functions, even limited on finite but arbitrary domain. In fact, that moves makes the computer simply vanishing, and it makes Mechanism not even definable or expressible.
>>>>>> That's going to come as a big shock to IBM stockholders.
>>>>> Why? On the contrary. IBM bets on universal machine
>>>> No, they bet only on finite machines, and they will be very surprised to hear that they have vanished.
>>> They bet on finite machines … including the universal machine, which I insist is a finite machine. That is even the reason why I called it from times to times universal number.
>>>
>>> I recall that once we get the phi_i,
>> i = 1 to inf.
> That is the potential infinite,

No, you can't diagonalize on an infinity that is only potential.

> that you already need for a concept like the square root of 2, used all the time in elementary quantum mechanics.

And in every computer...which uses on finitely many bits.

> Without it, neither CT, nor YD makes any sense.

CT doesn't.  So much the worse for CT.   YD makes better sense since the
doctor can now be sure he only needs to reproduce finitely many functions.

> We could aswell stop doing any math, if not stop thinking.

At least stop imagining the supernatural.

>
> The axioms that I use are just Kxy = x, and Sxyz = xz(yz).

But you allow rules of inference that permit inferences about the
enumerated array of all functions.

Brent

[Brent Meeker]

On 6/5/2020 2:39 AM, Bruno Marchal wrote:

On 4 Jun 2020, at 20:35, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 6/4/2020 4:27 AM, Bruno Marchal wrote:

On 3 Jun 2020, at 21:47, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 6/3/2020 3:26 AM, Lawrence Crowell wrote:

On Tuesday, June 2, 2020 at 12:34:37 PM UTC-5, Brent wrote:

On 6/2/2020 2:49 AM, Bruno Marchal wrote:
>> On 1 Jun 2020, at 22:43, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>
>>
>>
>> On 6/1/2020 2:08 AM, Bruno Marchal wrote:
>>> Brent suggest that we might recover completeness by restricting N to a finite domain. That is correct, because all finite function are computable, but then, we have incompleteness directly with respect to the computable functions, even limited on finite but arbitrary domain. In fact, that moves makes the computer simply vanishing, and it makes Mechanism not even definable or expressible.
>> That's going to come as a big shock to IBM stockholders.
>
> Why? On the contrary. IBM bets on universal machine

No, they bet only on finite machines, and they will be very surprised to
hear that they have vanished.

Brent

For the most part computers are meant to run various algorithms that solve some restricted set of problems, say business applications. We use them largely as tools.

Mathematics is largely a tool.  My pure mathematics friends over on math-fun seem to have most of their fun on Mathematica.

Of course, this is close to Aristotelian theology. It assumes that there is something which is not mathematical in some reality. A platonism or a pythegaorean think that he physical universe is but a tool, invented by the numbers to figure out what happens, and what is real.

But once you grasp that all computations exists in arithmetic (or more exactly, that they are enabled by the arithmetical true relations), even without Mechanism, the charge are reversed. It is those who claim (in metaphysics, not in physics) that there is a primitive universe who have the task to provide evidence.

You have implicitly asserted that computation=reality.   With not proof, or even evidence.

?

The UDA *proves* that the fundamental reality = arithmetic.

All proofs are relative to their premises.  You just assume arithmetic is real.

Brent

And AUDA (arithmetical Dovetailer Argument) makes the proof constructive, and it makes Mechanism testable, and the evidences for mechanism are striking, at a place where we know since 1500 years that Materialism is already refuted. Oh, yes, that is well hidden since 1500 years, by all gnostic (atheist or non atheists).

You are the one who seems to claim the existence of an ontological physical universe, where there is no proof nor any evidence.

When I kick it, it kicks back.

Evidences for a physical reality is not evidences for an ontological or primitive physical reality.

Nobody said it was.  But it is evidence for physical reality.  The thirst for an absolute primitive is a sickness of philosophy.

Brent

The confusion between both of those is know as Aristotle theology. The belief in primary matter or physicalism (mathematicalist or not).

Bruno

Brent

I have given the way to test this, and, thanks to QM, we can say that there are not yet any evidence found for a primitive physical universe. On the contrary, nature seems to obey exactly to what is needed for mechanism to be true.

Then, if we assume furthermore Mechanism, there is no more choice in this matter. Physics cannot be the fundamental science, it reduces to arithmetic (or any model of any Turing equivalent machinery) “seen-from inside”. 

Bruno

Brent

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[Bruno Marchal]

On 5 Jun 2020, at 13:34, Philip Thrift <cloud...@gmail.com> wrote:

On Friday, June 5, 2020 at 4:47:14 AM UTC-5, Bruno Marchal wrote:

On 4 Jun 2020, at 22:33, Philip Thrift <cloud...@gmail.com> wrote:

Years ago I wrote about the Zetans

       http://poesophicalbits.blogspot.com/2012/04/persons-without-infinities.html

who never imagined infinities, nor found any reason to either think of them, or invent them.

They have a fine definition of computers and computing, and have found no need for anything more than finite mechanism in any of their theory of computing.

That we came to think "infinity" plays a role in computing (or in computing theory, or in mathematics in general) is just an aspect of our own peculiar psychology and history, but it is not needed.

That makes some sense. You can compute without axiom of infinity, and indeed you can define what is a computer just by using the two axioms Kxy = x, and Sxyz = xz(yz), as I have explicitly shown on this list. Similarly you can define a computer using only elementary arithmetic, like Gödel did implicitly and Kleene did explicitly. But to prove anything non trivial, you need induction, and to get semantics treated mathematically, you need actual infinity axioms, like with the notion of real numbers, etc.

To *understand* Kxy = x …, you need an axiom of infinity at the meta-level, and this is required by all scientist-numbers in arithmetic, so “infinity” is more than welcome to define the notion of observer, and for the notion of physical *laws*.

Bruno

As my old post shows, the Zetans do have their Axiom of Infinity - called the Axiom of Zillions.

?

They are happy with their finite set of numbers, but sometimes they construct bigger numbers to add to the set. But they do not think there are numbers between the bigger numbers. There are gaps. 

There is no potential infinity since they don't think their galaxy will last forever.

The notion of “forever” and/ or “not forever” needs potential infinity. Like the notion of (natural) law. 

Anyway, you are free to interpreted Kxy = x, or arithmetic, … ultrafinitaitsically. That does not change the Mechanist phenomenological predictions.

Bruno

( All this follows from https://www.jstor.org/stable/2273942 )

@philipthrift 

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[Lawrence Crowell]

On Friday, June 5, 2020 at 4:57:06 AM UTC-5, Bruno Marchal wrote:

On 5 Jun 2020, at 00:39, Lawrence Crowell <goldenfield...@gmail.com> wrote:

On Thursday, June 4, 2020 at 6:07:45 AM UTC-5, Bruno Marchal wrote:

> On 2 Jun 2020, at 19:34, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>
>
>
> On 6/2/2020 2:49 AM, Bruno Marchal wrote:
>>> On 1 Jun 2020, at 22:43, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>>
>>>
>>>
>>> On 6/1/2020 2:08 AM, Bruno Marchal wrote:
>>>> Brent suggest that we might recover completeness by restricting N to a finite domain. That is correct, because all finite function are computable, but then, we have incompleteness directly with respect to the computable functions, even limited on finite but arbitrary domain. In fact, that moves makes the computer simply vanishing, and it makes Mechanism not even definable or expressible.
>>> That's going to come as a big shock to IBM stockholders.
>>
>> Why? On the contrary. IBM bets on universal machine
>
> No, they bet only on finite machines, and they will be very surprised to hear that they have vanished.

They bet on finite machines … including the universal machine, which I insist is a finite machine. That is even the reason why I called it from times to times universal number.

I recall that once we get the phi_i, which can be defined in elementary arithmetic, we get all the universal numbers, that is all u such that there phi_u(x, y) = phi_x(y), and such u can be used to define all the recursive enumeration of all digital machines.

The implementation of this fine but universal machines are called (physical) computer, and is the domain of expertise of IBM.

Bruno

Of course any computation is going to be finite or involve a finite number of bits.

Any halting computation.

Some non halting computation requires infinite time and space, virtual, arithmetical or physical.

It is easy to make a nonhalting computation that is finite. A few lines of code with a line 

4 GOTO 3

will be nonhalting. When your desk or lap computer freezes up it has entered into a state where something like this is happening. With codes that have a bounded length it is in principle possible for a UTM that can catch all nonhalting codes. Though for programs with thousands of lines the number of possibilities is ~ n^{1000}, which is a tad large. The finite UTM is very difficult. Just ponder Microsoft the next time your machine freezes up.

 

This happens as well with quantum computers, but there is one difference. Two states can be prepared and entangled so they have a continuum of probabilities depending upon measurement angle. This is what separates QM from classical mechanics. This separates entanglement of spins from the Bergman's socks, where knowing the left sock is in one box the right must be in the other. So while there is a finitude to the entanglement entropy or the quantity of quantum information, the possible ways an entanglement can register outcomes is infinite. This is what gives a violation of Bell's theorem in QM. With the measurement of a quantum system the pair of a state and measurement forms a type of Godel numbering. This connects QM foundations with the phi_u(x, y) = phi_x(y),you state above.

OK. But you assume some quantum universe, where the UDA explains why you have to derive the quantum from arithmetic or (Turing) equivalent.

As John Wheeler asked, "Why the quantum?" The quantum horizon and a general undecidability of all possible propositions are related and maybe ultimately equivalent. Quantum mechanics may entirely be due to this limitation. I wrote a paper, under adjudication right now, that entanglement types are topologically distinct removed from any unitary equivalence by topological obstruction due to mathematical undecidability.

 

A classical computer will always be finite, and you can't have an infinite Cantor diagonalization.

The Kleene diagonalisation is constructive. It shows the inexistence of some finite machine having some “arithmetical omniscience”. It requires potential infinities, not the cantorian infinities.

Yes, I have heard of this. It is similar to my approach to Hilbert space as finite and unbounded. By making it unbounded it just means that even if the HIlbert space is infinite the most extreme UV states are unoccupied or inaccessible. With quantum gravitation this is where the Planck scale comes in. Quantum states with a wavelength smaller than the Planck length ℓ_p = √(Għ/c^3) are unentangled with states in the rest of the universe. The accelerated expansion of the universe increases their wavelength and in effect pulls them out of this tiny scale. These then become entangled with the rest of the observable universe. Quantum states that have very low energy are similarly stretched out these eventually exceed the cosmological horizon length √(3/Λ) for the cosmological constant Λ = 8πGρ/c^2 and are then no longer entangled with any local region bounded by the cosmological horizon. This keeps the number of quantum states finite.

 

The computers that are manufactured are done so to solve certain problems, RSA encyrption, user interfaces for service personnel from travel agents to sales, word processors, games, cell phone signal shifters, data processors of medical measurements and on it goes.

All computers exists in arithmetic, and all computations exist in an internal limit of arithmetic (by step 2, actually!).

With mechanism, the physical reality is not the fundamental reality. The physical reality emerges from the computation executed in virtue of the number relations, like the prime number distribution, for example. 

I guess I have not seen any reason to be concerned with the distinction between mechanism and physicalism. This strikes me as sort of philosophical hair splitting that is of no utility to doing physics. Again, the ultimate relationship between the physical world and that of mathematics is not something that may be knowable. At some point here we are in an existential nullity, where our understanding of what it means to exist is not longer definite. With QM we have less certitude over what is meant by an event or objective meaning to an object and with mathematics there is no clear definition of what is meant by mathematical truth. Hardy tries to raise this point below, but it appears we have today a lack of clarity with respect to what is meant by a proof and clearly Gödel showed that within an axiomatic system there are holes of undecidability and these things have a dualism of being objective and almost Platonic, but also on the other hand little more than model systems.

LC

 

Even with quantum computers this will take off, and in fact I have thought quantum computing would be a way of managing a dynamics network defined by millions of drones over a city. Even if as I think the Godel-Turing result underlies obstructions between entanglement types quantum computers will in time become the province of engineering and business applications.

No doubt on this. It is just that with mechanism, the physical universe is not ontological, but more like a collective hallucination made by the relative universal number relations which are as true, and independent of the physical laws, than 117 is composite and 317 is prime.

Bruno

<<
Of this reality, as I explained […], I take a 'realistic" view. At any rate (and this is my main point) this realistic view is much more plausible of mathematical than of physical reality, because mathematical objects are so much more what they seem. A chair or a star is not in the least like what it seems to be ; the more we think of it, the fuzzier its outlines become in the haze of sensations which surrounds it; but '2' and '317' has nothing to do with sensations, and its properties stand out the more clearly the more closely we scrutinize it. It may be that modern physics fits best in the framework of idealistic philosophy---I do not believe it, but there are eminent physicist who say so. Pure Mathematics, on the other hand, seems to me a rock on which all idealism founders: 317 is prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical is built that way.
>>
G. H. Hardy, "A Mathematician's Apology", Cambridge University Press, 1940 (1998).

LC

 

>
> Brent
>
>> and know well what is a computer: a finite arithmetical being in touch with the infinite, and indeed, always asking for more memory, which is the typical symptom of liberty/universality. IBM might be finitist, like Mechanism, but is not ultrafinist at all. Anyway, mathematically, Mechanism is consistent with ulrafinitsim, even if to prove this, you need to go beyond finitism, (but then that’s the case for all consistent theory: none can prove its own consistency once “rich enough” (= just Turing universal, not “Löbian”).
>>
>> Bruno
>
>
> --
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[Bruno Marchal]

> On 5 Jun 2020, at 21:10, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>
>
>
> On 6/5/2020 2:32 AM, Bruno Marchal wrote:
>>> On 4 Jun 2020, at 20:28, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>>
>>>
>>>
>>> On 6/4/2020 4:07 AM, Bruno Marchal wrote:
>>>>> On 2 Jun 2020, at 19:34, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>>>>
>>>>>
>>>>>
>>>>> On 6/2/2020 2:49 AM, Bruno Marchal wrote:
>>>>>>> On 1 Jun 2020, at 22:43, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> On 6/1/2020 2:08 AM, Bruno Marchal wrote:
>>>>>>>> Brent suggest that we might recover completeness by restricting N to a finite domain. That is correct, because all finite function are computable, but then, we have incompleteness directly with respect to the computable functions, even limited on finite but arbitrary domain. In fact, that moves makes the computer simply vanishing, and it makes Mechanism not even definable or expressible.
>>>>>>> That's going to come as a big shock to IBM stockholders.
>>>>>> Why? On the contrary. IBM bets on universal machine
>>>>> No, they bet only on finite machines, and they will be very surprised to hear that they have vanished.
>>>> They bet on finite machines … including the universal machine, which I insist is a finite machine. That is even the reason why I called it from times to times universal number.
>>>>
>>>> I recall that once we get the phi_i,
>>> i = 1 to inf.
>> That is the potential infinite,
>
> No, you can't diagonalize on an infinity that is only potential.

That is not true. Cantor’s diagonal cannot be done, but Kleene’s diagonal (the one I have explained) does not require any actual infinities. You might reread it.

>
>> that you already need for a concept like the square root of 2, used all the time in elementary quantum mechanics.
>
> And in every computer...which uses on finitely many bits.

At each moment, but with question like will a machine stop or not, you need potential infinity.

>
>> Without it, neither CT, nor YD makes any sense.
>
> CT doesn't. So much the worse for CT.

OK. But that’s says something. CT is the most solid non trivial facts in modern science, I would say.

> YD makes better sense since the doctor can now be sure he only needs to reproduce finitely many functions.

YD is a far more stronger principle that CT. I have never met someone doubting CT, actually. It is not so for YD, which requires some sort of faith.

>
>> We could aswell stop doing any math, if not stop thinking.
>
> At least stop imagining the supernatural.

Better to stop imagining the “natural”, to begin with.

>
>>
>> The axioms that I use are just Kxy = x, and Sxyz = xz(yz).
>
> But you allow rules of inference that permit inferences about the enumerated array of all functions.

Right. Here is the complete set of ontological assumptions:

AXIOMS

KAB = A
SABC = AC(BC)

RULES:

If A = B and A = C, then B = C
If A = B then AC = BC
If A = B then CA = CB

For the phenomenology, we can use as much as we want, like in physics. It is provably never enough.

Bruno

>
> Brent
>>
>> There is no axiom of infinity, nor even induction axiom. That belongs only to the observers, and the proof of their existence requires only the two axiom above, or the arithmetic one, or anything Turing equivalent. With less than that, there is no computer, nor laptop … The universal machinery is potentially infinite. The universal machine is finite.
>>
>> Bruno
>>
>>
>>
>>>> which can be defined in elementary arithmetic, we get all the universal numbers, that is all u such that there phi_u(x, y) = phi_x(y), and such u can be used to define all the recursive enumeration of all digital machines.
>>>>
>>>> The implementation of this fine but universal machines are called (physical) computer, and is the domain of expertise of IBM.
>>>>
>>>> Bruno
>>>>
>>>>
>>>>
>>>>> Brent
>>>>>
>>>>>> and know well what is a computer: a finite arithmetical being in touch with the infinite, and indeed, always asking for more memory, which is the typical symptom of liberty/universality. IBM might be finitist, like Mechanism, but is not ultrafinist at all. Anyway, mathematically, Mechanism is consistent with ulrafinitsim, even if to prove this, you need to go beyond finitism, (but then that’s the case for all consistent theory: none can prove its own consistency once “rich enough” (= just Turing universal, not “Löbian”).
>>>>>>
>>>>>> Bruno
>>>>> --
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>>>>> To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/3068f558-7f61-56cb-61fe-44832ec28a91%40verizon.net.
>>>
>>> --
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>
>
> --
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[Bruno Marchal]

On 5 Jun 2020, at 21:13, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 6/5/2020 2:39 AM, Bruno Marchal wrote:

On 4 Jun 2020, at 20:35, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 6/4/2020 4:27 AM, Bruno Marchal wrote:

On 3 Jun 2020, at 21:47, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 6/3/2020 3:26 AM, Lawrence Crowell wrote:

On Tuesday, June 2, 2020 at 12:34:37 PM UTC-5, Brent wrote:

On 6/2/2020 2:49 AM, Bruno Marchal wrote:
>> On 1 Jun 2020, at 22:43, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>
>>
>>
>> On 6/1/2020 2:08 AM, Bruno Marchal wrote:
>>> Brent suggest that we might recover completeness by restricting N to a finite domain. That is correct, because all finite function are computable, but then, we have incompleteness directly with respect to the computable functions, even limited on finite but arbitrary domain. In fact, that moves makes the computer simply vanishing, and it makes Mechanism not even definable or expressible.
>> That's going to come as a big shock to IBM stockholders.
>
> Why? On the contrary. IBM bets on universal machine

No, they bet only on finite machines, and they will be very surprised to
hear that they have vanished.

Brent

For the most part computers are meant to run various algorithms that solve some restricted set of problems, say business applications. We use them largely as tools.

Mathematics is largely a tool.  My pure mathematics friends over on math-fun seem to have most of their fun on Mathematica.

Of course, this is close to Aristotelian theology. It assumes that there is something which is not mathematical in some reality. A platonism or a pythegaorean think that he physical universe is but a tool, invented by the numbers to figure out what happens, and what is real.

But once you grasp that all computations exists in arithmetic (or more exactly, that they are enabled by the arithmetical true relations), even without Mechanism, the charge are reversed. It is those who claim (in metaphysics, not in physics) that there is a primitive universe who have the task to provide evidence.

You have implicitly asserted that computation=reality.   With not proof, or even evidence.

?

The UDA *proves* that the fundamental reality = arithmetic.

All proofs are relative to their premises.  You just assume arithmetic is real.

To assume arithmetic is real is ambiguous, if not non sensical.  

All we have to assume is that a number added to 0 gives that number, that 0 is not a successor, etc. We have just to believe that 817 is prime or not prime, or that (x + 4 = 9) admits a solution, or not. I have never met a physicist who does not believe in those truth.

Now, given that all computations are run in arithmetic, a believer in Matter is invited to provide evidences, but as Plato understood already, there are no evidences at all. Not one. All the evidences we have today points on the immaterial consequence of Digital mechanism, from molecular biology to QM without collapse.

Bruno

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[Brent Meeker]

On 6/6/2020 3:47 AM, Bruno Marchal wrote:

The axioms that I use are just Kxy = x, and Sxyz = xz(yz).

But you allow rules of inference that permit inferences about the enumerated array of all functions.

Right. Here is the complete set of ontological assumptions:

AXIOMS

KAB = A
SABC = AC(BC)

RULES:

If A = B and A = C, then B = C
If A = B then AC = BC
If A = B then CA = CB

Those are not the rules you have used to infer incompleteness.  Incompleteness and provabilty aren't even expressible within that ontology.

Brent

[Brent Meeker]

A proposition cannot be ambiguous or nonsensical and also proven: "The UDA *proves* that the fundamental reality = arithmetic."

All we have to assume is that a number added to 0 gives that number, that 0 is not a successor, etc. We have just to believe that 817 is prime or not prime, or that (x + 4 = 9) admits a solution, or not. I have never met a physicist who does not believe in those truth.

I've never met a physicist who confused true with real.  But I've met such mathematicians.

Now, given that all computations are run in arithmetic, a believer in Matter is invited to provide evidences, but as Plato understood already, there are no evidences at all.

No evidence for what? 

Not one. All the evidences we have today points on the immaterial consequence of Digital mechanism, from molecular biology to QM without collapse.

A theory that depends on the collapse interpretation of QM as proof that mathematics is real; is self contradictory since there are mathematically consistent theories of QM in which the wave function does collapse.

Brent

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[Bruno Marchal]

On 6 Jun 2020, at 12:44, Lawrence Crowell <goldenfield...@gmail.com> wrote:

On Friday, June 5, 2020 at 4:57:06 AM UTC-5, Bruno Marchal wrote:

On 5 Jun 2020, at 00:39, Lawrence Crowell <goldenfield...@gmail.com> wrote:

On Thursday, June 4, 2020 at 6:07:45 AM UTC-5, Bruno Marchal wrote:

> On 2 Jun 2020, at 19:34, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>
>
>
> On 6/2/2020 2:49 AM, Bruno Marchal wrote:
>>> On 1 Jun 2020, at 22:43, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>>
>>>
>>>
>>> On 6/1/2020 2:08 AM, Bruno Marchal wrote:
>>>> Brent suggest that we might recover completeness by restricting N to a finite domain. That is correct, because all finite function are computable, but then, we have incompleteness directly with respect to the computable functions, even limited on finite but arbitrary domain. In fact, that moves makes the computer simply vanishing, and it makes Mechanism not even definable or expressible.
>>> That's going to come as a big shock to IBM stockholders.
>>
>> Why? On the contrary. IBM bets on universal machine
>
> No, they bet only on finite machines, and they will be very surprised to hear that they have vanished.

They bet on finite machines … including the universal machine, which I insist is a finite machine. That is even the reason why I called it from times to times universal number.

I recall that once we get the phi_i, which can be defined in elementary arithmetic, we get all the universal numbers, that is all u such that there phi_u(x, y) = phi_x(y), and such u can be used to define all the recursive enumeration of all digital machines.

The implementation of this fine but universal machines are called (physical) computer, and is the domain of expertise of IBM.

Bruno

Of course any computation is going to be finite or involve a finite number of bits.

Any halting computation.

Some non halting computation requires infinite time and space, virtual, arithmetical or physical.

It is easy to make a nonhalting computation that is finite. A few lines of code with a line 

4 GOTO 3

The one line "4 goto 4” would be enough. 

Or just the combinators MM (SII(SII), or S(SKK)(SKK). Mx = xx, and MM does not stop.

But my point was that not only to get all total computable functions, we need to have all partial one too, and that no effective theories at all can distinguish all total function codes from codes of the strictly partial functions. 

will be nonhalting. When your desk or lap computer freezes up it has entered into a state where something like this is happening. With codes that have a bounded length

All codes are fine words on a finite alphabet. Of course some have generalised this, using non standard models, where some finite object are actually infinite, but not seen as such from the pow of some machines.

it is in principle possible for a UTM that can catch all nonhalting codes.

?

No. That is impossible. If such machine exists, you can use it to build a recursive enumeration of all codes of all total computable functions, and Kleene’s diagonal will bring again 0 = 1.

Though for programs with thousands of lines the number of possibilities is ~ n^{1000}, which is a tad large. The finite UTM is very difficult. Just ponder Microsoft the next time your machine freezes up.

All UTM are finite (unless working in a non standard model, but that has to be made precise, and it is something entirely different).

Ana dall finite UTM leads to undecidability. For all UTM M and U, there is some x for which no UTM M cannot say if Ux stops or not.

 

This happens as well with quantum computers, but there is one difference. Two states can be prepared and entangled so they have a continuum of probabilities depending upon measurement angle. This is what separates QM from classical mechanics. This separates entanglement of spins from the Bergman's socks, where knowing the left sock is in one box the right must be in the other. So while there is a finitude to the entanglement entropy or the quantity of quantum information, the possible ways an entanglement can register outcomes is infinite. This is what gives a violation of Bell's theorem in QM. With the measurement of a quantum system the pair of a state and measurement forms a type of Godel numbering. This connects QM foundations with the phi_u(x, y) = phi_x(y),you state above.

OK. But you assume some quantum universe, where the UDA explains why you have to derive the quantum from arithmetic or (Turing) equivalent.

As John Wheeler asked, "Why the quantum?”

And Wheeler gave the correct answer: It for bit. Although a more precise one is: qubits from the theology of bits.

The quantum horizon and a general undecidability of all possible propositions are related and maybe ultimately equivalent. Quantum mechanics may entirely be due to this limitation.

It has to. This is what I have proven. (Assuming the digital mechanist theory, to be sure).

I wrote a paper, under adjudication right now, that entanglement types are topologically distinct removed from any unitary equivalence by topological obstruction due to mathematical undecidability.

If you can give a link toward your text, I would be pleased.

 

A classical computer will always be finite, and you can't have an infinite Cantor diagonalization.

The Kleene diagonalisation is constructive. It shows the inexistence of some finite machine having some “arithmetical omniscience”. It requires potential infinities, not the cantorian infinities.

Yes, I have heard of this. It is similar to my approach to Hilbert space as finite and unbounded. By making it unbounded it just means that even if the HIlbert space is infinite the most extreme UV states are unoccupied or inaccessible.

OK. But here, you are presupposing the quantum, which cannot be done if we want to derive it from the theology of universal numbers.

With quantum gravitation this is where the Planck scale comes in. Quantum states with a wavelength smaller than the Planck length ℓ_p = √(Għ/c^3) are unentangled with states in the rest of the universe. The accelerated expansion of the universe increases their wavelength and in effect pulls them out of this tiny scale. These then become entangled with the rest of the observable universe. Quantum states that have very low energy are similarly stretched out these eventually exceed the cosmological horizon length √(3/Λ) for the cosmological constant Λ = 8πGρ/c^2 and are then no longer entangled with any local region bounded by the cosmological horizon. This keeps the number of quantum states finite.

This might help, or not. Today, even the role of PI in physics is still an open problem, … (it is important to understand than mechanism is not supposed to replace physics, that would be like using superstring theory to make a cup of coffee). The goal of mechanism is to give a context where the mind-body becomes a problem in pure mathematics, yet with testable experimental consequences.

 

The computers that are manufactured are done so to solve certain problems, RSA encyrption, user interfaces for service personnel from travel agents to sales, word processors, games, cell phone signal shifters, data processors of medical measurements and on it goes.

All computers exists in arithmetic, and all computations exist in an internal limit of arithmetic (by step 2, actually!).

With mechanism, the physical reality is not the fundamental reality. The physical reality emerges from the computation executed in virtue of the number relations, like the prime number distribution, for example. 

I guess I have not seen any reason to be concerned with the distinction between mechanism and physicalism. This strikes me as sort of philosophical hair splitting that is of no utility to doing physics.

Sure it does, if the goal is doing physics. But the goal is o solve the mind body problem, and if we assume digital mechanism (YD + CT), we have no choice: we have to derive QM and GR from Kxy = x and Sxyz = xz(yz), without assuming anything else (beyond some equality rules). Precisely: the theory of everything is given by:

RULES:

1) If A = B and A = C, then B = C
2) If A = B then AC = BC
3) If A = B then CA = CB

AXIOMS:

4) KAB = A
5) SABC = AC(BC)

Again, the ultimate relationship between the physical world and that of mathematics is not something that may be knowable.

It is knowable modulo the mechanist hypothesis. The physical world is a collective universal number hallucination.

At some point here we are in an existential nullity, where our understanding of what it means to exist is not longer definite.

It depends on your assumption in the theory of mind (aka cognitive science).

With QM we have less certitude over what is meant by an event or objective meaning to an object and with mathematics there is no clear definition of what is meant by mathematical truth.

Mechanism allows to limit ourselves to the arithmetical truth, which is well defined. I mean that it is as well defined than real numbers, Hilbert space, etc.

Hardy tries to raise this point below, but it appears we have today a lack of clarity with respect to what is meant by a proof

We have a total clarity fro first order logic proof, like in PA or ZF, etc. This is enough to make the derivation of physics from arithmetic, as asked by the mechanist hypothesis, so that we can test Mechanism. If confirmed, we learn nothing. If refuted, we learn that either mechanism is false, or we are in some second order type of simulation, by malevolent descendent amor ancestors. That is a bit conspiratorial, and most people would conclude simply that mechanism is false. Form a logic point of view, some conspiracy theories are hard to evacuate. 

and clearly Gödel showed that within an axiomatic system

Or any mind of any machine (finite by definition of (standard) machine).

there are holes of undecidability and these things have a dualism of being objective and almost Platonic, but also on the other hand little more than model systems.

There are just theories or machines. Logician, like painters, use “models” for “realities”. A group is a model of the theory of group. A set universe is a model of “set theory” (which should be called set-universe theory, if we use “of” in the same set as in “theory of groups”. 

Bruno

LC

 

Even with quantum computers this will take off, and in fact I have thought quantum computing would be a way of managing a dynamics network defined by millions of drones over a city. Even if as I think the Godel-Turing result underlies obstructions between entanglement types quantum computers will in time become the province of engineering and business applications.

No doubt on this. It is just that with mechanism, the physical universe is not ontological, but more like a collective hallucination made by the relative universal number relations which are as true, and independent of the physical laws, than 117 is composite and 317 is prime.

Bruno

<<
Of this reality, as I explained […], I take a 'realistic" view. At any rate (and this is my main point) this realistic view is much more plausible of mathematical than of physical reality, because mathematical objects are so much more what they seem. A chair or a star is not in the least like what it seems to be ; the more we think of it, the fuzzier its outlines become in the haze of sensations which surrounds it; but '2' and '317' has nothing to do with sensations, and its properties stand out the more clearly the more closely we scrutinize it. It may be that modern physics fits best in the framework of idealistic philosophy---I do not believe it, but there are eminent physicist who say so. Pure Mathematics, on the other hand, seems to me a rock on which all idealism founders: 317 is prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical is built that way.
>>
G. H. Hardy, "A Mathematician's Apology", Cambridge University Press, 1940 (1998).

LC

 

>
> Brent
>
>> and know well what is a computer: a finite arithmetical being in touch with the infinite, and indeed, always asking for more memory, which is the typical symptom of liberty/universality. IBM might be finitist, like Mechanism, but is not ultrafinist at all. Anyway, mathematically, Mechanism is consistent with ulrafinitsim, even if to prove this, you need to go beyond finitism, (but then that’s the case for all consistent theory: none can prove its own consistency once “rich enough” (= just Turing universal, not “Löbian”).
>>
>> Bruno
>
>
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[Bruno Marchal]

They are, once you add the induction axioms. That is Gödel’s main contribution: to show that meta-arithmetic, which talks about completeness, incompleteness are entirely expressible in arithmetic (aka combinators), and *provable* once you add enough axioms to get what I called Gödel-Löbian entity. In that case, G formalise what the machine can prove, and G¨formaises what is true (the machine being able, or not, to prove it). As we limit ourself to sound machines, G is (properly) included in G*

The theory above (KAB = A, etc.) is equivalent with RA. In such theories you can prove the existence of RA+INDUCTION, or of S and K and of machine believing in S and K and the theory above + INDUCTION. To get the phenomenology, you need to interrogate such machine, in the ontology given by RA.

So you are right, I derive incompleteness in PA, but I derive the existence of PA, and ZF, and you and me, in RA.

This is a subtle, but very important point.

Bruno

Brent

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[Bruno Marchal]

A proposition cannot be ambiguous or nonsensical and also proven: "The UDA *proves* that the fundamental reality = arithmetic.”

But the “UDA proves that …” is not derived from “arithmetic is real”. It is derived from x + 0 = x, etc.

You seem to confuse the theory/machine (and what its says) with the arithmetical reality. Those do not belong to the same level of explanation. The arithmetical reality proves nothing: it is not a theory.

All we have to assume is that a number added to 0 gives that number, that 0 is not a successor, etc. We have just to believe that 817 is prime or not prime, or that (x + 4 = 9) admits a solution, or not. I have never met a physicist who does not believe in those truth.

I've never met a physicist who confused true with real.  But I've met such mathematicians.

Because they interpret “real” by physically real. They are studying the physical reality, not the ultimate truth, which is the object of philosophy/metaphysics/theology/whatever-you-call-this.

“Real” without an adjective making it precise, is the same as truth. The existence of the moon is a physical truth, and that is what makes the moon physically real, but that does not mean that the moon is a primitively real thing in the ontology.

Now, given that all computations are run in arithmetic, a believer in Matter is invited to provide evidences, but as Plato understood already, there are no evidences at all.

No evidence for what? 

For a PRIMITIVELY ontological physical reality. For the ideas that we have to assume that a physical universe exist, and cannot be explained by something conceptually simpler, like natural numbers.

Not one. All the evidences we have today points on the immaterial consequence of Digital mechanism, from molecular biology to QM without collapse.

A theory that depends on the collapse interpretation of QM

From what follows, I guess you meant “non-collapse” here.

as proof that mathematics is real;

Of course I do not do that. I do not assume collapse or non-collapse, given that I assume only elementary arithmetic. I assume only that 2+3 = 5, and alike, with classical logic, or much less if I use KAB = A, etc. Then I use Mechanism, of course, but no more in the theory of everything extracted from mechanism, and that is the one in which the physical laws are mathematically derivable, and I have derived the propositional physics, and it is not yet refuted, unlike physics when used by physicalist, which is refuted by the existence of consciousness and first person, already, since long.

is self contradictory since there are mathematically consistent theories of QM in which the wave function does collapse.

Plausibly.

But not relevant. It just means that if you find an evidence for a collapse, you refute Mechanism.With QM we have either “many-world” or “non mechanism”. But with mechanism, even without QM, we have already the many-world/histories, so why invent a Universe, collapse, … except for wishful thinking?

When we do metaphysics with the scientific method, we cannot invoke god or universe. Any ontological commitment is problematic, except for the terms needed to define what we are taking about when postulating mechanism in the cognitive science.

Bruno

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[Brent Meeker]

On 6/7/2020 5:08 AM, Bruno Marchal wrote:
>>>>> The UDA *proves* that the fundamental reality = arithmetic.
>>>>
>>>> All proofs are relative to their premises.  You just assume
>>>> arithmetic is real.
>>>
>>> To assume arithmetic is real is ambiguous, if not non sensical.
>>
>> A proposition cannot be ambiguous or nonsensical and also proven:
>> "The UDA *proves* that the fundamental reality = arithmetic.”
>
> But the “UDA proves that …” is not derived from “arithmetic is real”.
> It is derived from x + 0 = x, etc.
>
> You seem to confuse the theory/machine (and what its says) with the
> arithmetical reality. Those do not belong to the same level of
> explanation. The arithmetical reality proves nothing: it is not a theory.

I'm not confused.  You made two statements that are implicitly
contradictory:

(1) To assume arithmetic is real is ambiguous, if not non sensical.
(2) The UDA *proves* that the fundamental reality = arithmetic.

I just made the contradiction explicit by pointing out that any
proposition that can be proven, cannot be ambiguous or nonsensical and
hence can be unambiguously assumed.

Brent

[Lawrence Crowell]

I mis-wrote that. I should have set a finite set of algorithms. 

 

 

 

Though for programs with thousands of lines the number of possibilities is ~ n^{1000}, which is a tad large. The finite UTM is very difficult. Just ponder Microsoft the next time your machine freezes up.

All UTM are finite (unless working in a non standard model, but that has to be made precise, and it is something entirely different).

Ana dall finite UTM leads to undecidability. For all UTM M and U, there is some x for which no UTM M cannot say if Ux stops or not.

I meant a finite list.

 

 

This happens as well with quantum computers, but there is one difference. Two states can be prepared and entangled so they have a continuum of probabilities depending upon measurement angle. This is what separates QM from classical mechanics. This separates entanglement of spins from the Bergman's socks, where knowing the left sock is in one box the right must be in the other. So while there is a finitude to the entanglement entropy or the quantity of quantum information, the possible ways an entanglement can register outcomes is infinite. This is what gives a violation of Bell's theorem in QM. With the measurement of a quantum system the pair of a state and measurement forms a type of Godel numbering. This connects QM foundations with the phi_u(x, y) = phi_x(y),you state above.

OK. But you assume some quantum universe, where the UDA explains why you have to derive the quantum from arithmetic or (Turing) equivalent.

As John Wheeler asked, "Why the quantum?”

And Wheeler gave the correct answer: It for bit. Although a more precise one is: qubits from the theology of bits.

It from bits that are in self-referential loops.

 

The quantum horizon and a general undecidability of all possible propositions are related and maybe ultimately equivalent. Quantum mechanics may entirely be due to this limitation.

It has to. This is what I have proven. (Assuming the digital mechanist theory, to be sure).

I wrote a paper, under adjudication right now, that entanglement types are topologically distinct removed from any unitary equivalence by topological obstruction due to mathematical undecidability.

If you can give a link toward your text, I would be pleased.

I entered this into the FQXi essay contest. I read the announcement on this topic of undecidability and uncertainty, and was inspired to work like crazy on this. I had been kicking this idea around for a while. I furiously worked this theory and hammered out this paper.

https://fqxi.org/data/essay-contest-files/Crowell_fqxi_2020.pdf

My essay did fair to decent in the voting. If you click on community rating in the page linked below you can see my paper made nearly 90 percentile rating.

https://fqxi.org/community/forum/category/31427?sort=community

This essay contest is not the highest ranked place to host a paper. but on the other hand this is such a "way out there" sort of idea that it is hard to get this sort of thing published in more standard peer review. The FQXi poobahs tend to give the prizes to their own members and members of the Perimeter Institute, so my prospects are not entirely great.

I will try to get to the rest later

LC

 

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[Bruno Marchal]

I am not sure this changes something. I mean that Kleene’s diagonal just explains why anything finite (or recursively enumarbale) can distinguish all total function code from strictly partial function code. That is a result of absolute unsolvability, accepting the Church Turing thesis (which make computability into an absolute notion).

 

 

 

Though for programs with thousands of lines the number of possibilities is ~ n^{1000}, which is a tad large. The finite UTM is very difficult. Just ponder Microsoft the next time your machine freezes up.

All UTM are finite (unless working in a non standard model, but that has to be made precise, and it is something entirely different).

Ana dall finite UTM leads to undecidability. For all UTM M and U, there is some x for which no UTM M cannot say if Ux stops or not.

I meant a finite list.

That is, at best, ambiguous. If a finite list ofd algorithm can do something, a UTM can do it, except in learning theory, but learning” is quite different from computing (despite diagonalisation is very useful too, as shown by Joihn Case, notably).

 

 

This happens as well with quantum computers, but there is one difference. Two states can be prepared and entangled so they have a continuum of probabilities depending upon measurement angle. This is what separates QM from classical mechanics. This separates entanglement of spins from the Bergman's socks, where knowing the left sock is in one box the right must be in the other. So while there is a finitude to the entanglement entropy or the quantity of quantum information, the possible ways an entanglement can register outcomes is infinite. This is what gives a violation of Bell's theorem in QM. With the measurement of a quantum system the pair of a state and measurement forms a type of Godel numbering. This connects QM foundations with the phi_u(x, y) = phi_x(y),you state above.

OK. But you assume some quantum universe, where the UDA explains why you have to derive the quantum from arithmetic or (Turing) equivalent.

As John Wheeler asked, "Why the quantum?”

And Wheeler gave the correct answer: It for bit. Although a more precise one is: qubits from the theology of bits.

It from bits that are in self-referential loops.

Yes, making Wheeler very close to my (much older) work. 

 

The quantum horizon and a general undecidability of all possible propositions are related and maybe ultimately equivalent. Quantum mechanics may entirely be due to this limitation.

It has to. This is what I have proven. (Assuming the digital mechanist theory, to be sure).

I wrote a paper, under adjudication right now, that entanglement types are topologically distinct removed from any unitary equivalence by topological obstruction due to mathematical undecidability.

If you can give a link toward your text, I would be pleased.

I entered this into the FQXi essay contest. I read the announcement on this topic of undecidability and uncertainty, and was inspired to work like crazy on this. I had been kicking this idea around for a while. I furiously worked this theory and hammered out this paper.

https://fqxi.org/data/essay-contest-files/Crowell_fqxi_2020.pdf

I will take a look. You might start at the point I am arriving, in which case your work would be very useful. But with mechanism, we cannot start from quantum mechanics. It has to be derived from Kab = a + Sabc = ac(bc), and nothing more!

My essay did fair to decent in the voting. If you click on community rating in the page linked below you can see my paper made nearly 90 percentile rating.

https://fqxi.org/community/forum/category/31427?sort=community

This essay contest is not the highest ranked place to host a paper. but on the other hand this is such a "way out there" sort of idea that it is hard to get this sort of thing published in more standard peer review. The FQXi poobahs tend to give the prizes to their own members and members of the Perimeter Institute, so my prospects are not entirely great.

With people like Chalmers it is a bit hopeless …

I will try to get to the rest later

Take your time. This and new week it is the oral exam, at a distance, which is much more work, also for the students, which are (understandably) quite stressed, by exams + confinement… So I will have not much time, and don’t worry if I am late in my comments.

Bruno

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[Bruno Marchal]

> On 7 Jun 2020, at 23:03, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>
>
>
> On 6/7/2020 5:08 AM, Bruno Marchal wrote:
>>>>>> The UDA *proves* that the fundamental reality = arithmetic.
>>>>>
>>>>> All proofs are relative to their premises. You just assume arithmetic is real.
>>>>
>>>> To assume arithmetic is real is ambiguous, if not non sensical.
>>>
>>> A proposition cannot be ambiguous or nonsensical and also proven: "The UDA *proves* that the fundamental reality = arithmetic.”
>>
>> But the “UDA proves that …” is not derived from “arithmetic is real”. It is derived from x + 0 = x, etc.
>>
>> You seem to confuse the theory/machine (and what its says) with the arithmetical reality. Those do not belong to the same level of explanation. The arithmetical reality proves nothing: it is not a theory.
>
> I'm not confused. You made two statements that are implicitly contradictory:
>
> (1) To assume arithmetic is real is ambiguous, if not non sensical.

It is unclear if by “assuming arithmetic” you are are assuming 0 + 0 = 0, 1 + 0 = 1, etc. or if you are assuming that the theory exists and is consistent (that is: assuming that a model of arithmetic exists, which when formalised assumes much more, like infinite sets, etc.).

> (2) The UDA *proves* that the fundamental reality = arithmetic.

UDA shows that we cannot use the assumption that there is a universe to explain why we see a universe. It shows rigorously that this idea does not work. Of course, the neoplatonician udesrood this since long, but without the Church thesis, their argument (mainly the dream argument) is not constructive, and does not provide the means of verification.

>
> I just made the contradiction explicit by pointing out that any proposition that can be proven, cannot be ambiguous or nonsensical and hence can be unambiguously assumed.

The expression “assuming arithmetic” is unclear. With mechanism (which is an heavy assumption) we isolate by meta-reasoning a theory of everything which has very few assumptions: just 0 + 0 = 0, 1 + 0 = 1, etc. That is quite different than assuming that arithmetic is consistent, or make sense, etc.

There is a subtlety here, no doubt. As we assume as much math as we needed at the meta-level, and for the internal phenomenology as well, but all this is done without assuming more than elementary arithmetic at the fundamental ontological level. Mechanism justifies such an approach. All the machine interviews in the context of RA, believes far more proposition than RA. Arithmetic explains why numbers believe (even “richly”) in much more than arithmetic, indeed, they believe in most of the objects that they are dreaming…

Bruno




>
> Brent
>
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[Brent Meeker]

On 6/8/2020 4:36 AM, Bruno Marchal wrote:
>> On 7 Jun 2020, at 23:03, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>
>>
>>
>> On 6/7/2020 5:08 AM, Bruno Marchal wrote:
>>>>>>> The UDA *proves* that the fundamental reality = arithmetic.
>>>>>> All proofs are relative to their premises. You just assume arithmetic is real.
>>>>> To assume arithmetic is real is ambiguous, if not non sensical.
>>>> A proposition cannot be ambiguous or nonsensical and also proven: "The UDA *proves* that the fundamental reality = arithmetic.”
>>> But the “UDA proves that …” is not derived from “arithmetic is real”. It is derived from x + 0 = x, etc.
>>>
>>> You seem to confuse the theory/machine (and what its says) with the arithmetical reality. Those do not belong to the same level of explanation. The arithmetical reality proves nothing: it is not a theory.
>> I'm not confused. You made two statements that are implicitly contradictory:
>>
>> (1) To assume arithmetic is real is ambiguous, if not non sensical.
> It is unclear if by “assuming arithmetic” you are are assuming 0 + 0 = 0, 1 + 0 = 1, etc.

Ask yourself what you were assuming.  It's your statement.

> or if you are assuming that the theory exists and is consistent (that is: assuming that a model of arithmetic exists, which when formalised assumes much more, like infinite sets, etc.).
>
>
>
>> (2) The UDA *proves* that the fundamental reality = arithmetic.
>
> UDA shows that we cannot use the assumption that there is a universe to explain why we see a universe. It shows rigorously that this idea does not work.

But you can assume the UDA.  Proofs are relative to their premises.

But that is beside my point: It is contradictory to say that to assume
proposition X is nonsense and also that proposition X can be proven. 
Any proposition that can be proven (in any logical system) is a
proposition that can be consistently added to the axioms.

Brent

[Bruno Marchal]

Oops, I discover this mail now.

On 8 Jun 2020, at 22:51, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 6/8/2020 4:36 AM, Bruno Marchal wrote:

On 7 Jun 2020, at 23:03, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:



On 6/7/2020 5:08 AM, Bruno Marchal wrote:

The UDA *proves* that the fundamental reality = arithmetic.

All proofs are relative to their premises.  You just assume arithmetic is real.

To assume arithmetic is real is ambiguous, if not non sensical.

A proposition cannot be ambiguous or nonsensical and also proven: "The UDA *proves* that the fundamental reality = arithmetic.”

But the “UDA proves that …” is not derived from “arithmetic is real”. It is derived from x + 0 = x, etc.

You seem to confuse the theory/machine (and what its says) with the arithmetical reality. Those do not belong to the same level of explanation. The arithmetical reality proves nothing: it is not a theory.

I'm not confused.  You made two statements that are implicitly contradictory:

(1) To assume arithmetic is real is ambiguous, if not non sensical.

It is unclear if by “assuming arithmetic” you are are assuming 0 + 0 = 0, 1 + 0 = 1, etc.

Ask yourself what you were assuming.  It's your statement.

So to be clear, I assume Mechanism (YD + CT), and from Mechanism, I derive that we cannot assume more than the axioms of classical logic + the non logical arithmetical axioms:

1) 0 ≠ s(x)

2) x ≠ y -> s(x) ≠ s(y)

3) x ≠ 0 -> Ey(x = s(y)) 

4) x+0 = x

5) x+s(y) = s(x+y)

6) x*0=0

7) x*s(y)=(x*y)+x

I can assume much less, but this is OK.

or if you are assuming that the theory exists and is consistent (that is: assuming that a model of arithmetic exists, which when formalised assumes much more, like infinite sets, etc.).

(2) The UDA *proves* that the fundamental reality = arithmetic.

UDA shows that we cannot use the assumption that there is a universe to explain why we see a universe. It shows rigorously that this idea does not work.

But you can assume the UDA. 

You cannot assume a proof? You assume only a theory. The theory is shown above. The notion of computations is defined in the language of that theory, without adding any other assumption (except YD + CT at the meta-level, to help making sense of the relation between consciousness as used in the thought experience and its metamathematical definition).

Proofs are relative to their premises.

Indeed.

But that is beside my point: It is contradictory to say that to assume proposition X is nonsense and also that proposition X can be proven. 

Absolutely, unless X is an axiom, in which case you can prove it in two/three words, like “see the axioms”. 

Any proposition that can be proven (in any logical system) is a proposition that can be consistently added to the axioms.

That gives a redundant theory, but that is not a problem. 

I got the feeling that you seem to believe that I have both prove something and assume it, but I don’t see what you allude too.

I just said that assuming arithmetic is fuzzy. I just assume Mechanism, and all what is needed to give sense to mechanism, which is just elementary arithmetic, without induction (RA). Then I define an observer as being anything, provably existing in very elmemrentary arithmetic, believing in much more than that: the same + the induction axioms (PA).

RA can prove the existence of PA, and can mimic PA, but of course, RA remains much more cognitively weaker than PA.

Bruno

Brent

Of course, the neoplatonician udesrood this since long, but without the Church thesis, their argument (mainly the dream argument) is not constructive, and does not provide the means of verification.

I just made the contradiction explicit by pointing out that any proposition that can be proven, cannot be ambiguous or nonsensical and hence can be unambiguously assumed.

The expression “assuming arithmetic” is unclear. With mechanism (which is an heavy assumption) we isolate by meta-reasoning a theory of everything which has very few assumptions: just 0 + 0 = 0, 1 + 0 = 1, etc. That is quite different than assuming that arithmetic is consistent, or make sense, etc.

There is a subtlety here, no doubt. As we assume as much math as we needed at the meta-level, and for the internal phenomenology as well, but all this is done without assuming more than elementary arithmetic at the fundamental ontological level. Mechanism justifies such an approach. All the machine interviews in the context of RA, believes far more proposition than RA. Arithmetic explains why numbers believe (even “richly”) in much more than arithmetic, indeed, they believe in most of the objects that they are dreaming…

Bruno

Brent

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