Randomness of quantum processes and computability

[Alan Grayson]

I think Bruce's position is that quantum processes are inherently random and thus NOT computable. Doesn't this conclusion, if true, totally disconfirm Bruno's theory that the apparent physical universe comes into being by computations of arithmetic pre-existing principles or postulates? AG

[Stathis Papaioannou]

A deterministic branching algorithm can produce output that appears random for an observer in each branch.

On Fri, 21 Feb 2020 at 07:59, Alan Grayson <agrays...@gmail.com> wrote:

I think Bruce's position is that quantum processes are inherently random and thus NOT computable. Doesn't this conclusion, if true, totally disconfirm Bruno's theory that the apparent physical universe comes into being by computations of arithmetic pre-existing principles or postulates? AG

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Stathis Papaioannou

[Alan Grayson]

On Thursday, February 20, 2020 at 7:47:23 PM UTC-7, stathisp wrote:

A deterministic branching algorithm can produce output that appears random for an observer in each branch.

Can an appearance of randomness create a universe which is truly random at the quantum level? AG 

On Fri, 21 Feb 2020 at 07:59, Alan Grayson <agrays...@gmail.com> wrote:

I think Bruce's position is that quantum processes are inherently random and thus NOT computable. Doesn't this conclusion, if true, totally disconfirm Bruno's theory that the apparent physical universe comes into being by computations of arithmetic pre-existing principles or postulates? AG

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Stathis Papaioannou

[Brent Meeker]

Are you assuming an observer whose path is deterministic too, or are you relying on randomness of it his path thus creating self-locating uncertianty?

Brent

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[Philip Thrift]

On Thursday, February 20, 2020 at 2:59:05 PM UTC-6, Alan Grayson wrote:

I think Bruce's position is that quantum processes are inherently random and thus NOT computable. Doesn't this conclusion, if true, totally disconfirm Bruno's theory that the apparent physical universe comes into being by computations of arithmetic pre-existing principles or postulates? AG

William James thought belief in determinism is a form of religious bondage.

https://www.informationphilosopher.com/solutions/philosophers/james/

@philipthrift

[Stathis Papaioannou]

On Fri, 21 Feb 2020 at 14:25, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

Are you assuming an observer whose path is deterministic too, or are you relying on randomness of it his path thus creating self-locating uncertianty?

Self-locating uncertainty due to the observer branching.

On 2/20/2020 6:46 PM, Stathis Papaioannou wrote:

A deterministic branching algorithm can produce output that appears random for an observer in each branch.

On Fri, 21 Feb 2020 at 07:59, Alan Grayson <agrays...@gmail.com> wrote:

I think Bruce's position is that quantum processes are inherently random and thus NOT computable. Doesn't this conclusion, if true, totally disconfirm Bruno's theory that the apparent physical universe comes into being by computations of arithmetic pre-existing principles or postulates? AG

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Stathis Papaioannou

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Stathis Papaioannou

[Alan Grayson]

But true randomness, as the opposite of determinism, could be equated with UN-intelligibility. AG 

[Stathis Papaioannou]

On Fri, 21 Feb 2020 at 14:23, Alan Grayson <agrays...@gmail.com> wrote:

On Thursday, February 20, 2020 at 7:47:23 PM UTC-7, stathisp wrote:

A deterministic branching algorithm can produce output that appears random for an observer in each branch.

Can an appearance of randomness create a universe which is truly random at the quantum level? AG 

It’s truly random from the observer’s point of view. Not even an omniscient being can predict which branch the observer will see next.

On Fri, 21 Feb 2020 at 07:59, Alan Grayson <agrays...@gmail.com> wrote:

I think Bruce's position is that quantum processes are inherently random and thus NOT computable. Doesn't this conclusion, if true, totally disconfirm Bruno's theory that the apparent physical universe comes into being by computations of arithmetic pre-existing principles or postulates? AG

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Stathis Papaioannou

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Stathis Papaioannou

[Bruno Marchal]

On 20 Feb 2020, at 21:59, Alan Grayson <agrays...@gmail.com> wrote:

I think Bruce's position is that quantum processes are inherently random and thus NOT computable. Doesn't this conclusion, if true, totally disconfirm Bruno's theory that the apparent physical universe comes into being by computations of arithmetic pre-existing principles or postulates? AG

On the contrary, Mechanism reduces the apparent indeterminacy to the computable. With mechanism, things might be too much non computable, when you take the first person indeterminacy into account. Then the math shows that this refutation of mechanism does not work, as the computations are done with the exact redundancy making the physical reality enough computable to get stable histories. 

Mechanism entails that the physical reality cannot be entirely computed, like it predicted that no piece of matter can be cloned. Indeed, it emerges from a non computable statistics on infinitely many computation, which are not algorithmically recognisable in arithmetic. We can test mechanism by measuring its degree of non computability. Too much computable would be more problematic than too much non-computable.

Bruno

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[Bruno Marchal]

On 21 Feb 2020, at 04:23, Alan Grayson <agrays...@gmail.com> wrote:

On Thursday, February 20, 2020 at 7:47:23 PM UTC-7, stathisp wrote:

A deterministic branching algorithm can produce output that appears random for an observer in each branch.

Exactly.

Can an appearance of randomness create a universe which is truly random at the quantum level? AG 

How could an appearance create a universe?

It is much more simple. By emulating you in infinitely many occurrences in arithmetic, the simple laws of arithmetic explain consciousness (I don’t claim this is easy to understand) and the appearance of a partially random and partially computable universe. 

The arithmetical truth is highly not computable, and contains a lot of randomness, and even a lot of quite different sorts of randomness.

Bruno

On Fri, 21 Feb 2020 at 07:59, Alan Grayson <agrays...@gmail.com> wrote:

I think Bruce's position is that quantum processes are inherently random and thus NOT computable. Doesn't this conclusion, if true, totally disconfirm Bruno's theory that the apparent physical universe comes into being by computations of arithmetic pre-existing principles or postulates? AG

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Stathis Papaioannou

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[Bruno Marchal]

To postulate it is irrational. OK. But once the randomness admits a simple explanation, like with the self-duplicating procedure, it becomes intelligible. Everett saves physics from being un-intelligible, and indeed, leads to the explanation by arithmetic and its internal meta-arithmetic (à la Gödel).

Bruno

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[Philip Thrift]

"intelligibility" of course suggests a human-oriented desire to figure out an explanation - in a human language of course, as in terms of rules - for everything. 

"stochasticity" as an irreducible property of the universe is foreign to that.

cf. Albert Camus

@philipthift 

[Alan Grayson]

On Friday, February 21, 2020 at 3:40:51 AM UTC-7, Bruno Marchal wrote:

On 20 Feb 2020, at 21:59, Alan Grayson <agrays...@gmail.com> wrote:

I think Bruce's position is that quantum processes are inherently random and thus NOT computable. Doesn't this conclusion, if true, totally disconfirm Bruno's theory that the apparent physical universe comes into being by computations of arithmetic pre-existing principles or postulates? AG

On the contrary, Mechanism reduces the apparent indeterminacy to the computable. With mechanism, things might be too much non computable, when you take the first person indeterminacy into account. Then the math shows that this refutation of mechanism does not work, as the computations are done with the exact redundancy making the physical reality enough computable to get stable histories. 

Mechanism entails that the physical reality cannot be entirely computed, like it predicted that no piece of matter can be cloned. Indeed, it emerges from a non computable statistics on infinitely many computation, which are not algorithmically recognisable in arithmetic. We can test mechanism by measuring its degree of non computability. Too much computable would be more problematic than too much non-computable.

Bruno

I really can't follow the above. What I understand by mechanism (as you define it), is that the human nervous system, and presumably a human being, can be duplicated by computers. If that's what you mean, can you explain each sentence above. For example, how does mechanism reduce the apparent indeterminacy to the computable? And so forth. AG

On of the things I seriously dislike about MW, which makes it utterly REPELLENT (Steven Weinberg's word), is that there are too many damned worlds! For example, when one considers the worlds created when putting on left or right shoe first, this just scratches the surface of how many worlds are created -- not just two worlds, but uncountably many (if space is continuous) when one considers how many distinct worlds are created just for the case of the left shoe first. I am referring to the uncountably many ways to put on one's left shoe first. For me, this is hardly an ELEGANT Cosmos, but rather a downright SILLY one! AG

[Alan Grayson]

On Friday, February 21, 2020 at 3:48:56 AM UTC-7, Bruno Marchal wrote:

On 21 Feb 2020, at 09:47, Alan Grayson <agrays...@gmail.com> wrote:

On Friday, February 21, 2020 at 12:46:57 AM UTC-7, Philip Thrift wrote:

On Thursday, February 20, 2020 at 2:59:05 PM UTC-6, Alan Grayson wrote:

I think Bruce's position is that quantum processes are inherently random and thus NOT computable. Doesn't this conclusion, if true, totally disconfirm Bruno's theory that the apparent physical universe comes into being by computations of arithmetic pre-existing principles or postulates? AG

William James thought belief in determinism is a form of religious bondage.

https://www.informationphilosopher.com/solutions/philosophers/james/

@philipthrift

But true randomness, as the opposite of determinism, could be equated with UN-intelligibility. AG 

To postulate it is irrational. OK. But once the randomness admits a simple explanation, like with the self-duplicating procedure, it becomes intelligible. Everett saves physics from being un-intelligible, and indeed, leads to the explanation by arithmetic and its internal meta-arithmetic (à la Gödel).

Bruno

But, as I just pointed out in my previous message, the price paid is way too high to avoid randomness; that is, self-duplication is too silly to be believable. I prefer a possible middle ground; that the universe isn't really stochastic  (an inference from QM), but pseudo random. AG

[John Clark]

On Fri, Feb 21, 2020 at 8:25 AM Alan Grayson <agrays...@gmail.com> wrote:

> how does mechanism reduce the apparent indeterminacy to the computable?

Determinism doesn't mean every fact is computable. We know for certain the first 5 Busy Beaver numbers are 0, 1, 4, 6 and 13, but after that things get dicey. Someday we *might* be able to prove the 6th one is 4098 (it can't be smaller) and we know the 7th Busy Beaver number can't be smaller than 1.29*10^865. And we can prove that even with infinite computing power nobody will ever be able to know what the 1919'th Busy Beaver number is, it hasn't been proven but I wouldn't be surprised if the same thing was true for the 6th.

 

> On of the things I seriously dislike about MW, which makes it utterly REPELLENT (Steven Weinberg's word), is that there are too many damned worlds! 

Repellent is a very emotional word, and I think that's the primary reason MW didn't become the standard quantum interpretation 50 years ago, it was rejected for emotional reasons not intelectual ones. But nature is what it is and doesn't take our delicate sensibilities into account before deciding what to be.

John K Clark

[Philip Thrift]

On Friday, February 21, 2020 at 7:40:24 AM UTC-6, Alan Grayson wrote:

 I prefer a possible middle ground; that the universe isn't really stochastic  (an inference from QM), but pseudo random. AG

Typically, "a pseudorandom variable is a variable which is created by a deterministic algorithm, often a computer program or subroutine". 

So if actual randomness is removed from nature, and one supposes this is replaced by pseudorandomness in nature, then it would part of science to discover What is this algorithm?

This would be a fool's errand if actual randomness is in the reality of nature.

@philipthrift

[Brent Meeker]

You should read Ruth Kastner's book on "The Transactional Interpretation".

Brent

[Alan Grayson]

On Friday, February 21, 2020 at 9:38:18 AM UTC-7, Philip Thrift wrote:

On Friday, February 21, 2020 at 7:40:24 AM UTC-6, Alan Grayson wrote:

 I prefer a possible middle ground; that the universe isn't really stochastic  (an inference from QM), but pseudo random. AG

Typically, "a pseudorandom variable is a variable which is created by a deterministic algorithm, often a computer program or subroutine". 

So if actual randomness is removed from nature, and one supposes this is replaced by pseudorandomness in nature, then it would part of science to discover What is this algorithm?

If it exists, wouldn't that be a hidden variable? AG 

[Philip Thrift]

Pseudorandom hidden variables would be a superdeterministic hidden variable theory.

Stochastic hidden variable theories have been considered for a long time, like one's of Price and Wharton (Feynman Integral Symmetry Hypothesis). 

@philipthrift

[Brent Meeker]

On 2/21/2020 12:28 PM, Alan Grayson wrote:

On Friday, February 21, 2020 at 9:38:18 AM UTC-7, Philip Thrift wrote:

On Friday, February 21, 2020 at 7:40:24 AM UTC-6, Alan Grayson wrote:

 I prefer a possible middle ground; that the universe isn't really stochastic  (an inference from QM), but pseudo random. AG

Typically, "a pseudorandom variable is a variable which is created by a deterministic algorithm, often a computer program or subroutine". 

So if actual randomness is removed from nature, and one supposes this is replaced by pseudorandomness in nature, then it would part of science to discover What is this algorithm?

If it exists, wouldn't that be a hidden variable? AG

Which would be non-local and therefore consistent with Bell experiments.

Brent

This would be a fool's errand if actual randomness is in the reality of nature.

@philipthrift

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[Alan Grayson]

On Friday, February 21, 2020 at 4:01:04 PM UTC-7, Brent wrote:

On 2/21/2020 12:28 PM, Alan Grayson wrote:

On Friday, February 21, 2020 at 9:38:18 AM UTC-7, Philip Thrift wrote:

On Friday, February 21, 2020 at 7:40:24 AM UTC-6, Alan Grayson wrote:

 I prefer a possible middle ground; that the universe isn't really stochastic  (an inference from QM), but pseudo random. AG

Typically, "a pseudorandom variable is a variable which is created by a deterministic algorithm, often a computer program or subroutine". 

So if actual randomness is removed from nature, and one supposes this is replaced by pseudorandomness in nature, then it would part of science to discover What is this algorithm?

If it exists, wouldn't that be a hidden variable? AG

Which would be non-local and therefore consistent with Bell experiments.

Brent

Why non-local? AG 

This would be a fool's errand if actual randomness is in the reality of nature.

@philipthrift

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[Brent Meeker]

On 2/21/2020 4:59 PM, Alan Grayson wrote:

On Friday, February 21, 2020 at 4:01:04 PM UTC-7, Brent wrote:

On 2/21/2020 12:28 PM, Alan Grayson wrote:

On Friday, February 21, 2020 at 9:38:18 AM UTC-7, Philip Thrift wrote:

On Friday, February 21, 2020 at 7:40:24 AM UTC-6, Alan Grayson wrote:

 I prefer a possible middle ground; that the universe isn't really stochastic  (an inference from QM), but pseudo random. AG

Typically, "a pseudorandom variable is a variable which is created by a deterministic algorithm, often a computer program or subroutine". 

So if actual randomness is removed from nature, and one supposes this is replaced by pseudorandomness in nature, then it would part of science to discover What is this algorithm?

If it exists, wouldn't that be a hidden variable? AG

Which would be non-local and therefore consistent with Bell experiments.

Brent

Why non-local? AG

Otherwise it couldn't account for the entanglement that violates Bell's inequality.

Brent

This would be a fool's errand if actual randomness is in the reality of nature.

@philipthrift

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[Alan Grayson]

On Friday, February 21, 2020 at 7:26:26 AM UTC-7, John Clark wrote:

On Fri, Feb 21, 2020 at 8:25 AM Alan Grayson <agrays...@gmail.com> wrote:

> how does mechanism reduce the apparent indeterminacy to the computable?

Determinism doesn't mean every fact is computable.

I was referring to "mechanism" as Bruno defines it; namely, that a human being can be replicated by a computer. AG

[Alan Grayson]

Thanks, but I am not an enthusiast of the TI, since it requires pro-active processes for each particles going backward in time. I've asked this before, but haven't gotten a reply, or at least one I can recall. What's wrong with just assuming that in a superposition of states, the amplitudes give us the probability of each state in the sum, and NOT that the system is in all states simultaneously? Doesn't this interpretation resolves most, or all of the alleged paradoxes of QM? TIA, AG 

[Brent Meeker]

On 2/21/2020 7:02 PM, Alan Grayson wrote:

On Friday, February 21, 2020 at 12:42:20 PM UTC-7, Brent wrote:

On 2/21/2020 5:40 AM, Alan Grayson wrote:

On Friday, February 21, 2020 at 3:48:56 AM UTC-7, Bruno Marchal wrote:

On 21 Feb 2020, at 09:47, Alan Grayson <agrays...@gmail.com> wrote:

On Friday, February 21, 2020 at 12:46:57 AM UTC-7, Philip Thrift wrote:

On Thursday, February 20, 2020 at 2:59:05 PM UTC-6, Alan Grayson wrote:

I think Bruce's position is that quantum processes are inherently random and thus NOT computable. Doesn't this conclusion, if true, totally disconfirm Bruno's theory that the apparent physical universe comes into being by computations of arithmetic pre-existing principles or postulates? AG

William James thought belief in determinism is a form of religious bondage.

https://www.informationphilosopher.com/solutions/philosophers/james/

@philipthrift

But true randomness, as the opposite of determinism, could be equated with UN-intelligibility. AG 

To postulate it is irrational. OK. But once the randomness admits a simple explanation, like with the self-duplicating procedure, it becomes intelligible. Everett saves physics from being un-intelligible, and indeed, leads to the explanation by arithmetic and its internal meta-arithmetic (à la Gödel).

Bruno

But, as I just pointed out in my previous message, the price paid is way too high to avoid randomness; that is, self-duplication is too silly to be believable. I prefer a possible middle ground; that the universe isn't really stochastic  (an inference from QM), but pseudo random. AG

You should read Ruth Kastner's book on "The Transactional Interpretation".

Brent

Thanks, but I am not an enthusiast of the TI, since it requires pro-active processes for each particles going backward in time.

Kastner modifies that by hypothesizing a "possibility space" in which the "hand-shake" takes place.  But it still involves a "confirmation wave" which extends back in time from the absorber (and forward in time from the emitter). 

I've asked this before, but haven't gotten a reply, or at least one I can recall. What's wrong with just assuming that in a superposition of states, the amplitudes give us the probability of each state in the sum, and NOT that the system is in all states simultaneously?

Think of applying that to a silver atom in an SG experiment.  It is in an UP spin state (with probability 1.0) but it's also in LEFT spin state with probability 0.5 and a RIGHT spin state with probability 0.5.  So it's total probability is 2.0.

Doesn't this interpretation resolves most, or all of the alleged paradoxes of QM? TIA, AG

No.  The problem arises when there's a measurement and the problem has three parts:

1. What basis will the result be in, i.e. why is the cat |alive> or |dead> and never 0.7|alive>+0.3|dead> ?

2. When is the measurement process complete?  The problem of Wigner's friend.

3. Why does the Born rule hold?

I think Zurek's envariance based quantum Darwinism is closest to have a complete solution; but it still seems to have multiple worlds.

Brent

[Alan Grayson]

On Friday, February 21, 2020 at 8:28:32 PM UTC-7, Brent wrote:

On 2/21/2020 7:02 PM, Alan Grayson wrote:

On Friday, February 21, 2020 at 12:42:20 PM UTC-7, Brent wrote:

On 2/21/2020 5:40 AM, Alan Grayson wrote:

On Friday, February 21, 2020 at 3:48:56 AM UTC-7, Bruno Marchal wrote:

On 21 Feb 2020, at 09:47, Alan Grayson <agrays...@gmail.com> wrote:

On Friday, February 21, 2020 at 12:46:57 AM UTC-7, Philip Thrift wrote:

On Thursday, February 20, 2020 at 2:59:05 PM UTC-6, Alan Grayson wrote:

I think Bruce's position is that quantum processes are inherently random and thus NOT computable. Doesn't this conclusion, if true, totally disconfirm Bruno's theory that the apparent physical universe comes into being by computations of arithmetic pre-existing principles or postulates? AG

William James thought belief in determinism is a form of religious bondage.

https://www.informationphilosopher.com/solutions/philosophers/james/

@philipthrift

But true randomness, as the opposite of determinism, could be equated with UN-intelligibility. AG 

To postulate it is irrational. OK. But once the randomness admits a simple explanation, like with the self-duplicating procedure, it becomes intelligible. Everett saves physics from being un-intelligible, and indeed, leads to the explanation by arithmetic and its internal meta-arithmetic (à la Gödel).

Bruno

But, as I just pointed out in my previous message, the price paid is way too high to avoid randomness; that is, self-duplication is too silly to be believable. I prefer a possible middle ground; that the universe isn't really stochastic  (an inference from QM), but pseudo random. AG

You should read Ruth Kastner's book on "The Transactional Interpretation".

Brent

Thanks, but I am not an enthusiast of the TI, since it requires pro-active processes for each particles going backward in time.

Kastner modifies that by hypothesizing a "possibility space" in which the "hand-shake" takes place.  But it still involves a "confirmation wave" which extends back in time from the absorber (and forward in time from the emitter). 

I've asked this before, but haven't gotten a reply, or at least one I can recall. What's wrong with just assuming that in a superposition of states, the amplitudes give us the probability of each state in the sum, and NOT that the system is in all states simultaneously?

Think of applying that to a silver atom in an SG experiment.  It is in an UP spin state (with probability 1.0) but it's also in LEFT spin state with probability 0.5 and a RIGHT spin state with probability 0.5.  So it's total probability is 2.0.

I was taught that the sum of probabilities in any basis must be 1.0. I never heard of adding up probabilities in more than one basis. AG 

Doesn't this interpretation resolves most, or all of the alleged paradoxes of QM? TIA, AG

No.  The problem arises when there's a measurement and the problem has three parts:

1. What basis will the result be in, i.e. why is the cat |alive> or |dead> and never 0.7|alive>+0.3|dead> ?

The cat inherits the probabilities of the radioactive source, which I suppose is .5 for decayed and undecayed. AG 

2. When is the measurement process complete?  The problem of Wigner's friend.

3. Why does the Born rule hold?

Why does any rule hold? Why, for example, are conjugate observables anti-commutative? AG 

[Brent Meeker]

On 2/21/2020 8:19 PM, Alan Grayson wrote:

On Friday, February 21, 2020 at 8:28:32 PM UTC-7, Brent wrote:

On 2/21/2020 7:02 PM, Alan Grayson wrote:

On Friday, February 21, 2020 at 12:42:20 PM UTC-7, Brent wrote:

On 2/21/2020 5:40 AM, Alan Grayson wrote:

On Friday, February 21, 2020 at 3:48:56 AM UTC-7, Bruno Marchal wrote:

On 21 Feb 2020, at 09:47, Alan Grayson <agrays...@gmail.com> wrote:

On Friday, February 21, 2020 at 12:46:57 AM UTC-7, Philip Thrift wrote:

On Thursday, February 20, 2020 at 2:59:05 PM UTC-6, Alan Grayson wrote:

I think Bruce's position is that quantum processes are inherently random and thus NOT computable. Doesn't this conclusion, if true, totally disconfirm Bruno's theory that the apparent physical universe comes into being by computations of arithmetic pre-existing principles or postulates? AG

William James thought belief in determinism is a form of religious bondage.

https://www.informationphilosopher.com/solutions/philosophers/james/

@philipthrift

But true randomness, as the opposite of determinism, could be equated with UN-intelligibility. AG 

To postulate it is irrational. OK. But once the randomness admits a simple explanation, like with the self-duplicating procedure, it becomes intelligible. Everett saves physics from being un-intelligible, and indeed, leads to the explanation by arithmetic and its internal meta-arithmetic (à la Gödel).

Bruno

But, as I just pointed out in my previous message, the price paid is way too high to avoid randomness; that is, self-duplication is too silly to be believable. I prefer a possible middle ground; that the universe isn't really stochastic  (an inference from QM), but pseudo random. AG

You should read Ruth Kastner's book on "The Transactional Interpretation".

Brent

Thanks, but I am not an enthusiast of the TI, since it requires pro-active processes for each particles going backward in time.

Kastner modifies that by hypothesizing a "possibility space" in which the "hand-shake" takes place.  But it still involves a "confirmation wave" which extends back in time from the absorber (and forward in time from the emitter). 

I've asked this before, but haven't gotten a reply, or at least one I can recall. What's wrong with just assuming that in a superposition of states, the amplitudes give us the probability of each state in the sum, and NOT that the system is in all states simultaneously?

Think of applying that to a silver atom in an SG experiment.  It is in an UP spin state (with probability 1.0) but it's also in LEFT spin state with probability 0.5 and a RIGHT spin state with probability 0.5.  So it's total probability is 2.0.

I was taught that the sum of probabilities in any basis must be 1.0. I never heard of adding up probabilities in more than one basis. AG

That's the point.  P=2.0 makes no sense.  Yet those two states are mathematically the same in QM. How are you going to get P(UP)=1.0 by summing over states of LEFT and RIGHT?

Doesn't this interpretation resolves most, or all of the alleged paradoxes of QM? TIA, AG

No.  The problem arises when there's a measurement and the problem has three parts:

1. What basis will the result be in, i.e. why is the cat |alive> or |dead> and never 0.7|alive>+0.3|dead> ?

The cat inherits the probabilities of the radioactive source, which I suppose is .5 for decayed and undecayed. AG

Then why isn't it in a state 0.707|alive>+0.707|dead>?

2. When is the measurement process complete?  The problem of Wigner's friend.

3. Why does the Born rule hold?

Why does any rule hold? Why, for example, are conjugate observables anti-commutative? AG

See Vic's "Comprehensible Cosmos" appendix on QM.

Brent

I think Zurek's envariance based quantum Darwinism is closest to have a complete solution; but it still seems to have multiple worlds.

Brent

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[Alan Grayson]

If you sum over either representation, you get 1.0. I don't see any problem. AG

Doesn't this interpretation resolves most, or all of the alleged paradoxes of QM? TIA, AG

No.  The problem arises when there's a measurement and the problem has three parts:

1. What basis will the result be in, i.e. why is the cat |alive> or |dead> and never 0.7|alive>+0.3|dead> ?

The cat inherits the probabilities of the radioactive source, which I suppose is .5 for decayed and undecayed. AG 

Then why isn't it in a state 0.707|alive>+0.707|dead>?

Where did .5 for |alive> and |dead> states come from? AG 

2. When is the measurement process complete?  The problem of Wigner's friend.

3. Why does the Born rule hold?

Why does any rule hold? Why, for example, are conjugate observables anti-commutative? AG 

See Vic's "Comprehensible Cosmos" appendix on QM.

I have it. I'll look. AG 

[Alan Grayson]

On Friday, February 21, 2020 at 6:44:12 PM UTC-7, Brent wrote:

On 2/21/2020 4:59 PM, Alan Grayson wrote:

On Friday, February 21, 2020 at 4:01:04 PM UTC-7, Brent wrote:

On 2/21/2020 12:28 PM, Alan Grayson wrote:

On Friday, February 21, 2020 at 9:38:18 AM UTC-7, Philip Thrift wrote:

On Friday, February 21, 2020 at 7:40:24 AM UTC-6, Alan Grayson wrote:

 I prefer a possible middle ground; that the universe isn't really stochastic  (an inference from QM), but pseudo random. AG

Typically, "a pseudorandom variable is a variable which is created by a deterministic algorithm, often a computer program or subroutine". 

So if actual randomness is removed from nature, and one supposes this is replaced by pseudorandomness in nature, then it would part of science to discover What is this algorithm?

If it exists, wouldn't that be a hidden variable? AG

Which would be non-local and therefore consistent with Bell experiments.

Brent

Why non-local? AG

Otherwise it couldn't account for the entanglement that violates Bell's inequality.

Brent

Not sure what you mean. For me "local" means the variable exists and is set when an entangled pair is created. If the variable exists in the form of an algorithm, and predicts the result of a measurement, is this sufficient to call it "non local"? AG 

[John Clark]

On Fri, Feb 21, 2020 at 9:02 PM Alan Grayson <agrays...@gmail.com> wrote:

>>> how does mechanism reduce the apparent indeterminacy to the computable?

>> Determinism doesn't mean every fact is computable.

> I was referring to "mechanism" as Bruno defines it; namely, that a human being can be replicated by a computer. AG

I was never sure what Bruno meant by the word but if that's what you mean then the answer to your original question above is obvious, Chaos Theory and the fact that calculations take time and you can't know what the result of a calculation you're making will be until you've finished making the calculation.

 John K Clark

[Brent Meeker]

If you say the state is |LEFT>+|RIGHT>  and "the amplitudes give us the probability of each state in the sum" what is the probability of |UP>?

Brent

[Brent Meeker]

In order to agree with experiment it would have to be an algorithm that depended on what measurement was made on each particle of the entangled pair, and that's what would make it non-local.

Brent

[Alan Grayson]

If the probability of UP is 1.0, I assume the system has been measured and is in the UP state; or possibly UP is pointed along the x-axis with no other component. How is this different from having a unit vector along the x-axis, and the same vector represented by a linear combination of two other vectors, say pointing in the same general direction, but rotated up and down from the x-axis? AG

Brent

[Alan Grayson]

How does this differ from a local hidden variable? AG 

[Alan Grayson]

This doesn't work since, presumably, LEFT and RIGHT must be anti-parallel.  If UP has unit probability, it must imply that a measurement has already taken place. This could effect LEFT and RIGHT, but I'm not sure how. AG

Brent

[Bruno Marchal]

On 21 Feb 2020, at 14:25, Alan Grayson <agrays...@gmail.com> wrote:

On Friday, February 21, 2020 at 3:40:51 AM UTC-7, Bruno Marchal wrote:

On 20 Feb 2020, at 21:59, Alan Grayson <agrays...@gmail.com> wrote:

I think Bruce's position is that quantum processes are inherently random and thus NOT computable. Doesn't this conclusion, if true, totally disconfirm Bruno's theory that the apparent physical universe comes into being by computations of arithmetic pre-existing principles or postulates? AG

On the contrary, Mechanism reduces the apparent indeterminacy to the computable. With mechanism, things might be too much non computable, when you take the first person indeterminacy into account. Then the math shows that this refutation of mechanism does not work, as the computations are done with the exact redundancy making the physical reality enough computable to get stable histories. 

Mechanism entails that the physical reality cannot be entirely computed, like it predicted that no piece of matter can be cloned. Indeed, it emerges from a non computable statistics on infinitely many computation, which are not algorithmically recognisable in arithmetic. We can test mechanism by measuring its degree of non computability. Too much computable would be more problematic than too much non-computable.

Bruno

I really can't follow the above. What I understand by mechanism (as you define it), is that the human nervous system, and presumably a human being, can be duplicated by computers.

Yes. If you want, we could define a mechanist practitioners as someone accepting an artificial prosthesis, like an artificial heart (a pump), or an artificial kidney ( filter machine), etc. A mechanist is then someone accepting this whatever organ is concerned, and in particular an artificial and digital brain.

If that's what you mean, can you explain each sentence above. For example, how does mechanism reduce the apparent indeterminacy to the computable? And so forth. AG

Self-duplication is made possible by the Digital Mechanism. If you agree that with self-duplication, like in the Washington-Moscoow thought experiment, the person is unable to predict his immediate particular personal future feeling, despite being certain (assuming Mechanism and all default hyoptheses) that it will be either like feeling to be in Moscow or like feeling to be in Washington, then you understand that mechanism entails the existence of a personal, subjective (first person) indeterminacy.

Then you will need to understand that the elementary arithmetical truth implement all computations (including all quantum computations, notably), in the original sense of Turing, Church, Kleene (cf the 1930s papers), which makes us, in any “here and now” indexical state, indeterminate on which computations continue us, making eventually the physical science as a statistics on all (relative) computations, among an infinity (which are indeed realised in that elementary part of arithmetic).

On of the things I seriously dislike about MW, which makes it utterly REPELLENT (Steven Weinberg's word), is that there are too many damned worlds! For example, when one considers the worlds created when putting on left or right shoe first, this just scratches the surface of how many worlds are created -- not just two worlds, but uncountably many (if space is continuous) when one considers how many distinct worlds are created just for the case of the left shoe first. I am referring to the uncountably many ways to put on one's left shoe first. For me, this is hardly an ELEGANT Cosmos, but rather a downright SILLY one! AG

When you decide to put the left shoes, in bot “quantum Mechanism” à-la Everett, or in arithmetic, only the consistent extensions must be taken into account, and if you decide to put the left shoe, you will put in all continuation that left shoe. Both the quantum and the purely Mechanist frame does not make everything happening, only the consistent (with you local belief) extension will occur, in the vast majority of extensions.

Bruno

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[Bruno Marchal]

On 21 Feb 2020, at 15:25, John Clark <johnk...@gmail.com> wrote:

On Fri, Feb 21, 2020 at 8:25 AM Alan Grayson <agrays...@gmail.com> wrote:

> how does mechanism reduce the apparent indeterminacy to the computable?

Determinism doesn't mean every fact is computable.

Determinism is indeed very different from predictability.

We know for certain the first 5 Busy Beaver numbers are 0, 1, 4, 6 and 13, but after that things get dicey. Someday we *might* be able to prove the 6th one is 4098 (it can't be smaller) and we know the 7th Busy Beaver number can't be smaller than 1.29*10^865. And we can prove that even with infinite computing power nobody will ever be able to know what the 1919'th Busy Beaver number is, it hasn't been proven but I wouldn't be surprised if the same thing was true for the 6th.

The BB is computable already with the Halting oracle, and thus with the totality oracle, etc. So you need less infinite power for computing the BB than computing if a machine compute a total or a partial computable function.

 

> On of the things I seriously dislike about MW, which makes it utterly REPELLENT (Steven Weinberg's word), is that there are too many damned worlds! 

Repellent is a very emotional word, and I think that's the primary reason MW didn't become the standard quantum interpretation 50 years ago, it was rejected for emotional reasons not intelectual ones. But nature is what it is and doesn't take our delicate sensibilities into account before deciding what to be.

Absolutely.

Bruno

John K Clark

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[Bruno Marchal]

… especially when you understand that randomness occurs everywhere in the arithmetical reality, and even a lot of different sort of randomness emerge from it, from the long term non predictability of determinist machines, and from the existence of non deterministic machines, and the existence of quantum computer executed in arithmetic, … to the first person indeterminacy which is unavoidable from the machine personal views, etc.

Bruno

@philipthrift

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[John Clark]

On Sun, Feb 23, 2020 at 7:12 AM Bruno Marchal <mar...@ulb.ac.be> wrote:

> The BB is computable already with the Halting oracle

But a Halting oracle produces paradoxes, and I don't just mean weird situations I mean genuine logical contradictions.

John K Clark 

[Alan Grayson]

IMO, there's nothing mysterious about Bruno's definition of mechanism. It's what's generally believed by most physicists; namely, that everything in the universe can be explained by the interaction of particles (and waves), and its possible extension; namely, that a human being can be copied by computers and appropriate algorithms.  I would add that Bruno doesn't necessary affirm mechanism, which is a prudent decision since it's obviously false. Although one must assume the existence of space and time when using a computer, but one cannot create them using a computer. AG

[Alan Grayson]

On Sunday, February 23, 2020 at 5:08:16 AM UTC-7, Bruno Marchal wrote:

On 21 Feb 2020, at 14:25, Alan Grayson <agrays...@gmail.com> wrote:

On Friday, February 21, 2020 at 3:40:51 AM UTC-7, Bruno Marchal wrote:

On 20 Feb 2020, at 21:59, Alan Grayson <agrays...@gmail.com> wrote:

I think Bruce's position is that quantum processes are inherently random and thus NOT computable. Doesn't this conclusion, if true, totally disconfirm Bruno's theory that the apparent physical universe comes into being by computations of arithmetic pre-existing principles or postulates? AG

On the contrary, Mechanism reduces the apparent indeterminacy to the computable. With mechanism, things might be too much non computable, when you take the first person indeterminacy into account. Then the math shows that this refutation of mechanism does not work, as the computations are done with the exact redundancy making the physical reality enough computable to get stable histories. 

Mechanism entails that the physical reality cannot be entirely computed, like it predicted that no piece of matter can be cloned. Indeed, it emerges from a non computable statistics on infinitely many computation, which are not algorithmically recognisable in arithmetic. We can test mechanism by measuring its degree of non computability. Too much computable would be more problematic than too much non-computable.

Bruno

I really can't follow the above. What I understand by mechanism (as you define it), is that the human nervous system, and presumably a human being, can be duplicated by computers.

Yes. If you want, we could define a mechanist practitioners as someone accepting an artificial prosthesis, like an artificial heart (a pump), or an artificial kidney ( filter machine), etc. A mechanist is then someone accepting this whatever organ is concerned, and in particular an artificial and digital brain.

If that's what you mean, can you explain each sentence above. For example, how does mechanism reduce the apparent indeterminacy to the computable? And so forth. AG

Self-duplication is made possible by the Digital Mechanism. If you agree that with self-duplication,

I don't. Although it has some plausibility, there's no way that arithmetic alone can CREATE space and time. A computer can create "points" in a hypothetical grid, and various types of distance formulas, but it cannot create space or time. BTW, what's your definition of physicalism? AG 

like in the Washington-Moscoow thought experiment, the person is unable to predict his immediate particular personal future feeling, despite being certain (assuming Mechanism and all default hyoptheses) that it will be either like feeling to be in Moscow or like feeling to be in Washington, then you understand that mechanism entails the existence of a personal, subjective (first person) indeterminacy.

Then you will need to understand that the elementary arithmetical truth implement all computations (including all quantum computations, notably), in the original sense of Turing, Church, Kleene (cf the 1930s papers), which makes us, in any “here and now” indexical state, indeterminate on which computations continue us, making eventually the physical science as a statistics on all (relative) computations, among an infinity (which are indeed realised in that elementary part of arithmetic).

On of the things I seriously dislike about MW, which makes it utterly REPELLENT (Steven Weinberg's word), is that there are too many damned worlds! For example, when one considers the worlds created when putting on left or right shoe first, this just scratches the surface of how many worlds are created -- not just two worlds, but uncountably many (if space is continuous) when one considers how many distinct worlds are created just for the case of the left shoe first. I am referring to the uncountably many ways to put on one's left shoe first. For me, this is hardly an ELEGANT Cosmos, but rather a downright SILLY one! AG

When you decide to put the left shoes, in bot “quantum Mechanism” à-la Everett, or in arithmetic, only the consistent extensions must be taken into account, and if you decide to put the left shoe, you will put in all continuation that left shoe. Both the quantum and the purely Mechanist frame does not make everything happening, only the consistent (with you local belief) extension will occur, in the vast majority of extensions.

Bruno

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[Bruno Marchal]

On 23 Feb 2020, at 14:34, John Clark <johnk...@gmail.com> wrote:

On Sun, Feb 23, 2020 at 7:12 AM Bruno Marchal <mar...@ulb.ac.be> wrote:

> The BB is computable already with the Halting oracle

But a Halting oracle produces paradoxes,

I don’t see why you say this. The quantum oracle is just a (non computable) set of numbers, and it has been used by Turing and others to show that even with such oracle, we cannot solve some problem in arithmetic, like the totality problem TOT (deciding if a number code for a total or strictly partial computable functions). 

Of course, with an oracle for TOT you can solve the halting problem, but the pont is that with the halting oracle, you still cannot decide TOT. This shows that TOT is more unsolvable than HALT, or that HALT is simpler than TOT. This is the starting result of the study of the degrees of unsolvability, which is mainly Recursion Theory.

All this means that adding “divine abilities” to machine, does NOT overcome incompleteness. The universal machine is Sigma_1 complete. HALT is Pi_1 complete, TOT is Pi_2 complete (like quantified G, btw). To be the code of a finite set is Sigma_2 complete, to be the code of a recursive set is sigma_3 complete, etc.

and I don't just mean weird situations I mean genuine logical contradictions.

There is no contradiction coming from the existence of an oracle for the halting problem.

See my older post to see how to modify a Turing machine so that the can consult an oracle.

(I can explain again if you want).

Bruno

John K Clark 

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[Bruno Marchal]

On 24 Feb 2020, at 05:44, Alan Grayson <agrays...@gmail.com> wrote:

On Sunday, February 23, 2020 at 5:08:16 AM UTC-7, Bruno Marchal wrote:

On 21 Feb 2020, at 14:25, Alan Grayson <agrays...@gmail.com> wrote:

On Friday, February 21, 2020 at 3:40:51 AM UTC-7, Bruno Marchal wrote:

On 20 Feb 2020, at 21:59, Alan Grayson <agrays...@gmail.com> wrote:

I think Bruce's position is that quantum processes are inherently random and thus NOT computable. Doesn't this conclusion, if true, totally disconfirm Bruno's theory that the apparent physical universe comes into being by computations of arithmetic pre-existing principles or postulates? AG

On the contrary, Mechanism reduces the apparent indeterminacy to the computable. With mechanism, things might be too much non computable, when you take the first person indeterminacy into account. Then the math shows that this refutation of mechanism does not work, as the computations are done with the exact redundancy making the physical reality enough computable to get stable histories. 

Mechanism entails that the physical reality cannot be entirely computed, like it predicted that no piece of matter can be cloned. Indeed, it emerges from a non computable statistics on infinitely many computation, which are not algorithmically recognisable in arithmetic. We can test mechanism by measuring its degree of non computability. Too much computable would be more problematic than too much non-computable.

Bruno

I really can't follow the above. What I understand by mechanism (as you define it), is that the human nervous system, and presumably a human being, can be duplicated by computers.

Yes. If you want, we could define a mechanist practitioners as someone accepting an artificial prosthesis, like an artificial heart (a pump), or an artificial kidney ( filter machine), etc. A mechanist is then someone accepting this whatever organ is concerned, and in particular an artificial and digital brain.

If that's what you mean, can you explain each sentence above. For example, how does mechanism reduce the apparent indeterminacy to the computable? And so forth. AG

Self-duplication is made possible by the Digital Mechanism. If you agree that with self-duplication,

I don't. Although it has some plausibility, there's no way that arithmetic alone can CREATE space and time. A computer can create "points" in a hypothetical grid, and various types of distance formulas, but it cannot create space or time.

Arithmetic cannot create space and time. I agree with you. That would be rather magical indeed.

But once you assume digital mechanism, and once you understand that the simple arithmetical truth emulates all computations, you can understand that arithmetic create the experience of space-time, and indeed, with, apparently until now, the right redundancy which explain the “many-world” aspect of the physical reality, and that is confirmed mathematically, in the sense that we recover quantum logic for the logic of experimental s-certainty, relatively to the computational states accessible from arithmetic, by universal numbers/machines “living” in there.

The quantum is explained by the digital “seen from inside”. We get the qubits for the bits when seen from the bits, roughly speaking.

BTW, what's your definition of physicalism? AG 

The idea that physics is the fundamental science. 

With mechanism, this does not work, (I can show this), and we have to explain the “illusion of a physical world” by a statistic on all machine “dreams", where a dream is just a computation rich enough to support a “self-aware observer” (which can be defined in arithmetic by a consistent machine having enough rich cognitive abilities (precisely: believing in enough induction axioms, if you have heard about theories like Peano arithmetic, or Zermelo-Fraenkel set theory, those are typical examples of such digital (immaterial) machine.

With mechanism, physics is reduced to computer science or to arithmetic, in the same sense that for most educated people today think that biology can be reduced to physics, in principle of course. 

My first older result is simply that mechanism (the idea that we could survive with a digital brain/body) is incompatible with weak-materialism or with physicalism (the idea that there is a physical universe having a fundamental ontology, not reducible to any other science).

Bruno

like in the Washington-Moscoow thought experiment, the person is unable to predict his immediate particular personal future feeling, despite being certain (assuming Mechanism and all default hyoptheses) that it will be either like feeling to be in Moscow or like feeling to be in Washington, then you understand that mechanism entails the existence of a personal, subjective (first person) indeterminacy.

Then you will need to understand that the elementary arithmetical truth implement all computations (including all quantum computations, notably), in the original sense of Turing, Church, Kleene (cf the 1930s papers), which makes us, in any “here and now” indexical state, indeterminate on which computations continue us, making eventually the physical science as a statistics on all (relative) computations, among an infinity (which are indeed realised in that elementary part of arithmetic).

On of the things I seriously dislike about MW, which makes it utterly REPELLENT (Steven Weinberg's word), is that there are too many damned worlds! For example, when one considers the worlds created when putting on left or right shoe first, this just scratches the surface of how many worlds are created -- not just two worlds, but uncountably many (if space is continuous) when one considers how many distinct worlds are created just for the case of the left shoe first. I am referring to the uncountably many ways to put on one's left shoe first. For me, this is hardly an ELEGANT Cosmos, but rather a downright SILLY one! AG

When you decide to put the left shoes, in bot “quantum Mechanism” à-la Everett, or in arithmetic, only the consistent extensions must be taken into account, and if you decide to put the left shoe, you will put in all continuation that left shoe. Both the quantum and the purely Mechanist frame does not make everything happening, only the consistent (with you local belief) extension will occur, in the vast majority of extensions.

Bruno

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[John Clark]

On Mon, Feb 24, 2020 at 8:40 AM Bruno Marchal <mar...@ulb.ac.be> wrote:

>> a Halting oracle produces paradoxes,

> I don’t see why you say this.

I have a halting oracle machine and it has 2 input slots, one slot for the logical blueprints of a computer in digital form and the other slot for a program, also in digital form, to run on that computer. The oracle machine will then output either the words "Halt" or "Not Halt" depending on what a program running on that computer will do. I decide to use the oracle machine as one part of a new 3 part machine I will call machine X. The first part of machine X is just a photocopier that makes two copies of its input and then feeds them into the 2 input slots of the oracle machine. The last part of machine X is the negator, if it receives a "Not Halt" input from the oracle machine then the negator will output "Halt" and then stop, if the negator receives a "Halt" input from the oracle the negator will go into an infinite loop and never stop. The entire X machine as constructed has one input slot and one output slot.

I will now input machine X with machine X's own blueprints, so after the photocopier has done its work the oracle machine will receive identical inputs in both slots and the oracle machine will have to figure out what will happen to the X machine when the X machine is fed it's own blueprint as input. If the oracle says under those circumstances the X machine will halt then the X machine will never halt, and if the oracle says the X machine will not halt then the X machine will print "Halt" and stop. So the halting oracle machine always makes predictions that are wrong. So there is no such thing as a halting oracle machine. QED.

John K Clark

[Alan Grayson]

On Monday, February 24, 2020 at 6:55:59 AM UTC-7, Bruno Marchal wrote:

On 24 Feb 2020, at 05:44, Alan Grayson <agrays...@gmail.com> wrote:

On Sunday, February 23, 2020 at 5:08:16 AM UTC-7, Bruno Marchal wrote:

On 21 Feb 2020, at 14:25, Alan Grayson <agrays...@gmail.com> wrote:

On Friday, February 21, 2020 at 3:40:51 AM UTC-7, Bruno Marchal wrote:

On 20 Feb 2020, at 21:59, Alan Grayson <agrays...@gmail.com> wrote:

I think Bruce's position is that quantum processes are inherently random and thus NOT computable. Doesn't this conclusion, if true, totally disconfirm Bruno's theory that the apparent physical universe comes into being by computations of arithmetic pre-existing principles or postulates? AG

On the contrary, Mechanism reduces the apparent indeterminacy to the computable. With mechanism, things might be too much non computable, when you take the first person indeterminacy into account. Then the math shows that this refutation of mechanism does not work, as the computations are done with the exact redundancy making the physical reality enough computable to get stable histories. 

Mechanism entails that the physical reality cannot be entirely computed, like it predicted that no piece of matter can be cloned. Indeed, it emerges from a non computable statistics on infinitely many computation, which are not algorithmically recognisable in arithmetic. We can test mechanism by measuring its degree of non computability. Too much computable would be more problematic than too much non-computable.

Bruno

I really can't follow the above. What I understand by mechanism (as you define it), is that the human nervous system, and presumably a human being, can be duplicated by computers.

Yes. If you want, we could define a mechanist practitioners as someone accepting an artificial prosthesis, like an artificial heart (a pump), or an artificial kidney ( filter machine), etc. A mechanist is then someone accepting this whatever organ is concerned, and in particular an artificial and digital brain.

If that's what you mean, can you explain each sentence above. For example, how does mechanism reduce the apparent indeterminacy to the computable? And so forth. AG

Self-duplication is made possible by the Digital Mechanism. If you agree that with self-duplication,

I don't. Although it has some plausibility, there's no way that arithmetic alone can CREATE space and time. A computer can create "points" in a hypothetical grid, and various types of distance formulas, but it cannot create space or time.

Arithmetic cannot create space and time. I agree with you. That would be rather magical indeed.

Then arithmetic cannot create other worlds! End of story. Case closed. AG 

But once you assume digital mechanism,

If arithmetic can't create other worlds, it throws grave doubt that digital mechanism is true. AG

 

and once you understand that the simple arithmetical truth emulates all computations, you can understand that arithmetic create the experience of space-time, and indeed, with, apparently until now, the right redundancy which explain the “many-world” aspect of the physical reality, and that is confirmed mathematically, in the sense that we recover quantum logic for the logic of experimental s-certainty, relatively to the computational states accessible from arithmetic, by universal numbers/machines “living” in there.

The quantum is explained by the digital “seen from inside”. We get the qubits for the bits when seen from the bits, roughly speaking.

BTW, what's your definition of physicalism? AG 

The idea that physics is the fundamental science. 

With mechanism, this does not work, (I can show this), and we have to explain the “illusion of a physical world” by a statistic on all machine “dreams", where a dream is just a computation rich enough to support a “self-aware observer”

So, with a false theory of mechanism, or digital mechanism, and arithmetic which you admit can't create space (and probably time as well), you claim to derive a self-aware observer? AG

 

(which can be defined in arithmetic by a consistent machine having enough rich cognitive abilities (precisely: believing in enough induction axioms, if you have heard about theories like Peano arithmetic, or Zermelo-Fraenkel set theory, those are typical examples of such digital (immaterial) machine.

Yes, I am aware of those axioms, and, as I have written, Peano's axioms imply arithmetic, but not IMO, of other worlds. AG 

With mechanism, physics is reduced to computer science or to arithmetic, in the same sense that for most educated people today think that biology can be reduced to physics, in principle of course. 

My first older result is simply that mechanism (the idea that we could survive with a digital brain/body) is incompatible with weak-materialism or with physicalism (the idea that there is a physical universe having a fundamental ontology, not reducible to any other science).

Bruno

like in the Washington-Moscoow thought experiment, the person is unable to predict his immediate particular personal future feeling, despite being certain (assuming Mechanism and all default hyoptheses) that it will be either like feeling to be in Moscow or like feeling to be in Washington, then you understand that mechanism entails the existence of a personal, subjective (first person) indeterminacy.

Then you will need to understand that the elementary arithmetical truth implement all computations (including all quantum computations, notably), in the original sense of Turing, Church, Kleene (cf the 1930s papers), which makes us, in any “here and now” indexical state, indeterminate on which computations continue us, making eventually the physical science as a statistics on all (relative) computations, among an infinity (which are indeed realised in that elementary part of arithmetic).

On of the things I seriously dislike about MW, which makes it utterly REPELLENT (Steven Weinberg's word), is that there are too many damned worlds! For example, when one considers the worlds created when putting on left or right shoe first, this just scratches the surface of how many worlds are created -- not just two worlds, but uncountably many (if space is continuous) when one considers how many distinct worlds are created just for the case of the left shoe first. I am referring to the uncountably many ways to put on one's left shoe first. For me, this is hardly an ELEGANT Cosmos, but rather a downright SILLY one! AG

When you decide to put the left shoes, in bot “quantum Mechanism” à-la Everett, or in arithmetic, only the consistent extensions must be taken into account, and if you decide to put the left shoe, you will put in all continuation that left shoe. Both the quantum and the purely Mechanist frame does not make everything happening, only the consistent (with you local belief) extension will occur, in the vast majority of extensions.

Bruno

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[Alan Grayson]

Test of memory; is Zermelo-Fraenkel set theory just Peano's Axioms (PA) plus the Axiom of Choice? TIA, AG 

[Bruno Marchal]

This shows that the machine+halting-oracle cannot solve the halting (machine+halting-oracle) problem, but the halting-oracle is supposed fro be concerned only for the machine halting problem, i.e. the machine without oracle. 

Bruno

John K Clark

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[John Clark]

On Tue, Feb 25, 2020 at 8:34 AM Bruno Marchal <mar...@ulb.ac.be> wrote:

>> I have a halting oracle machine and it has 2 input slots, one slot for the logical blueprints of a computer in digital form and the other slot for a program, also in digital form, to run on that computer. The oracle machine will then output either the words "Halt" or "Not Halt" depending on what a program running on that computer will do. I decide to use the oracle machine as one part of a new 3 part machine I will call machine X. The first part of machine X is just a photocopier that makes two copies of its input and then feeds them into the 2 input slots of the oracle machine. The last part of machine X is the negator, if it receives a "Not Halt" input from the oracle machine then the negator will output "Halt" and then stop, if the negator receives a "Halt" input from the oracle the negator will go into an infinite loop and never stop. The entire X machine as constructed has one input slot and one output slot.
I will now input machine X with machine X's own blueprints, so after the photocopier has done its work the oracle machine will receive identical inputs in both slots and the oracle machine will have to figure out what will happen to the X machine when the X machine is fed it's own blueprint as input. If the oracle says under those circumstances the X machine will halt then the X machine will never halt, and if the oracle says the X machine will not halt then the X machine will print "Halt" and stop. So the halting oracle machine always makes predictions that are wrong. So there is no such thing as a halting oracle machine. QED.

> This shows that the machine+halting-oracle cannot solve the halting (machine+halting-oracle) problem,

Yes, it can't solve the halting problem.

> but the halting-oracle is supposed fro be concerned only for the machine halting problem, i.e. the machine without oracle. 

Supposed? That's a rather silly thing to say, if the oracle machine is well defined and the machine is well defined then the new combined machine is also well defined and the so called halting oracle fails miserably when it tries to predict if this new well defined machine will halt or not.

John K Clark

[Bruno Marchal]

On 25 Feb 2020, at 05:06, Alan Grayson <agrays...@gmail.com> wrote:

On Monday, February 24, 2020 at 6:55:59 AM UTC-7, Bruno Marchal wrote:

On 24 Feb 2020, at 05:44, Alan Grayson <agrays...@gmail.com> wrote:

On Sunday, February 23, 2020 at 5:08:16 AM UTC-7, Bruno Marchal wrote:

On 21 Feb 2020, at 14:25, Alan Grayson <agrays...@gmail.com> wrote:

On Friday, February 21, 2020 at 3:40:51 AM UTC-7, Bruno Marchal wrote:

On 20 Feb 2020, at 21:59, Alan Grayson <agrays...@gmail.com> wrote:

I think Bruce's position is that quantum processes are inherently random and thus NOT computable. Doesn't this conclusion, if true, totally disconfirm Bruno's theory that the apparent physical universe comes into being by computations of arithmetic pre-existing principles or postulates? AG

On the contrary, Mechanism reduces the apparent indeterminacy to the computable. With mechanism, things might be too much non computable, when you take the first person indeterminacy into account. Then the math shows that this refutation of mechanism does not work, as the computations are done with the exact redundancy making the physical reality enough computable to get stable histories. 

Mechanism entails that the physical reality cannot be entirely computed, like it predicted that no piece of matter can be cloned. Indeed, it emerges from a non computable statistics on infinitely many computation, which are not algorithmically recognisable in arithmetic. We can test mechanism by measuring its degree of non computability. Too much computable would be more problematic than too much non-computable.

Bruno

I really can't follow the above. What I understand by mechanism (as you define it), is that the human nervous system, and presumably a human being, can be duplicated by computers.

Yes. If you want, we could define a mechanist practitioners as someone accepting an artificial prosthesis, like an artificial heart (a pump), or an artificial kidney ( filter machine), etc. A mechanist is then someone accepting this whatever organ is concerned, and in particular an artificial and digital brain.

If that's what you mean, can you explain each sentence above. For example, how does mechanism reduce the apparent indeterminacy to the computable? And so forth. AG

Self-duplication is made possible by the Digital Mechanism. If you agree that with self-duplication,

I don't. Although it has some plausibility, there's no way that arithmetic alone can CREATE space and time. A computer can create "points" in a hypothetical grid, and various types of distance formulas, but it cannot create space or time.

Arithmetic cannot create space and time. I agree with you. That would be rather magical indeed.

Then arithmetic cannot create other worlds! End of story. Case closed. AG 

It creates all the individual and collective stories. The case would be close if you could prove the existence of a world, but if you are a consistent machine (to be short), then incompleteness precludes this. 

Even intuitively, you can understand that no one can prove the existence of anything from scratch. We need to assume some theories, and with mechanism, eventually we have to abandon the idea of an ontological physical universe. Still, we get an explanation of where the illusion comes from, and why it is lawful, etc.

But once you assume digital mechanism,

If arithmetic can't create other worlds, it throws grave doubt that digital mechanism is true. AG

It “creates” all computations, which includes (as we assume mechanism here) all the computations simulating your brain right now. We don’t need an ontological physical universe, once we can explain the physical laws in terms of histories which are emulated in arithmetic. We need only to test if the statistics obtained match what we see. (And thanks to QM-without-collapse), ir fits rather well up to now.

 

and once you understand that the simple arithmetical truth emulates all computations, you can understand that arithmetic create the experience of space-time, and indeed, with, apparently until now, the right redundancy which explain the “many-world” aspect of the physical reality, and that is confirmed mathematically, in the sense that we recover quantum logic for the logic of experimental s-certainty, relatively to the computational states accessible from arithmetic, by universal numbers/machines “living” in there.

The quantum is explained by the digital “seen from inside”. We get the qubits for the bits when seen from the bits, roughly speaking.

BTW, what's your definition of physicalism? AG 

The idea that physics is the fundamental science. 

With mechanism, this does not work, (I can show this), and we have to explain the “illusion of a physical world” by a statistic on all machine “dreams", where a dream is just a computation rich enough to support a “self-aware observer”

So, with a false theory of mechanism, or digital mechanism, and arithmetic which you admit can't create space (and probably time as well), you claim to derive a self-aware observer? AG

Yes, because that is an attribute of the person attached to the machine, and I got them by using the oldest definition of the knower, given by Theaetetus to socrates. Socrates demolish that theory, but the incompleteness theorem refutes Socrates’ refutation. I model rational belief by Gödel’s provability predicate, which is a normal thing to do, because incompleteness prevents it to be a knowledge predicate, and so it is a sort of rational belief. 

"I know p” is modelled by “(I prove) p and p’. That obeys the axiom of knowledge, but also make the “knower” non nameable, which confirms many theories already existing, and is a startling feature of knowledge, which appears to be non definable by the machine, and private.

 

(which can be defined in arithmetic by a consistent machine having enough rich cognitive abilities (precisely: believing in enough induction axioms, if you have heard about theories like Peano arithmetic, or Zermelo-Fraenkel set theory, those are typical examples of such digital (immaterial) machine.

Yes, I am aware of those axioms, and, as I have written, Peano's axioms imply arithmetic, but not IMO, of other worlds. AG 

I write PA for Peano’s axiom (or theorems).

Concerning the (standard) model of arithmetic, PA can only scratch the surface, and ZF, despite proving much more, can also only scratch the surface of the arithmetical reality/standard-model.

But  PA you can prove the existence of an infinity of computations going through your state right now, and that includes your feeling that there is a physical universe out there, but in this case, we know that it is not the case, and so it could be “not the case” for you right here and now too.

You miss the work of the logicians in the 1930s, that is the discovery in mathematics, and then in arithmetic, of the universal machines and all its activities.  

Just to give a little example, you can encode the truth that “the machine register ( a, b , c) contains b, by the arithmetical proposition saying that the number (3^b divides A, and (3^b+1) does not) with:

A = (2^”a”)*(3^”b”)*(5^”c”), with 2, 3, 5, … being the prime factors, so that we can exploit the prime factorisation theorem (truth) to be sure this is not ambiguous. The expression “a”, “b” , “c” denotes some preliminary encoding of the letters a, b and c with numbers.

A universal machine cannot see or feel the difference between being run by this or that universal machinery, and the theory eventually predict that below its substitution level, she has to see the symptoms of infinitely many histories, coming from all computations, run in arithmetic, going through your state and having refined description of the computations. We can come back on this.

Bruno

With mechanism, physics is reduced to computer science or to arithmetic, in the same sense that for most educated people today think that biology can be reduced to physics, in principle of course. 

My first older result is simply that mechanism (the idea that we could survive with a digital brain/body) is incompatible with weak-materialism or with physicalism (the idea that there is a physical universe having a fundamental ontology, not reducible to any other science).

Bruno

like in the Washington-Moscoow thought experiment, the person is unable to predict his immediate particular personal future feeling, despite being certain (assuming Mechanism and all default hyoptheses) that it will be either like feeling to be in Moscow or like feeling to be in Washington, then you understand that mechanism entails the existence of a personal, subjective (first person) indeterminacy.

Then you will need to understand that the elementary arithmetical truth implement all computations (including all quantum computations, notably), in the original sense of Turing, Church, Kleene (cf the 1930s papers), which makes us, in any “here and now” indexical state, indeterminate on which computations continue us, making eventually the physical science as a statistics on all (relative) computations, among an infinity (which are indeed realised in that elementary part of arithmetic).

On of the things I seriously dislike about MW, which makes it utterly REPELLENT (Steven Weinberg's word), is that there are too many damned worlds! For example, when one considers the worlds created when putting on left or right shoe first, this just scratches the surface of how many worlds are created -- not just two worlds, but uncountably many (if space is continuous) when one considers how many distinct worlds are created just for the case of the left shoe first. I am referring to the uncountably many ways to put on one's left shoe first. For me, this is hardly an ELEGANT Cosmos, but rather a downright SILLY one! AG

When you decide to put the left shoes, in bot “quantum Mechanism” à-la Everett, or in arithmetic, only the consistent extensions must be taken into account, and if you decide to put the left shoe, you will put in all continuation that left shoe. Both the quantum and the purely Mechanist frame does not make everything happening, only the consistent (with you local belief) extension will occur, in the vast majority of extensions.

Bruno

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[Bruno Marchal]

ZF is set theory. It contains a faithful realisation of arithmetic. Actually it contains many such representations. The usual one use von Neuman representation where 0 is represented by the empty set {}, and n is represented by n united to {n}.

That gives 3 = {0, 1, 2} = { {}, {{}}, {{}{{}}} } for example. Each number becomes the set of its predecessors.

This makes ZF-minus, or ZFC-minus (that is ZF, or ZF + the axiom of choice, minus the axiom of infinity) equivalent to Peano arithmetic.

Now ZF, or ZFC is just ZF-minus, or ZFC-minus to which you add the axiom of infinity; which says that there is a set omega such that 

 1) it contains 0 (the empty set), and 

 2) it is such that if it contains x, it contains x union {x}.

That makes it close for the set theoretical successor relation s(x) = x union {x}. It contains (intuitively) all natural numbers.

And then you can prove in ZF that there are an infinity of higher and higher infinite cardinals, etc.

So, by abusing language, up to the faithful representation, it is correct to say that ZF is mainly PA + an axiom of infinity.

Now, interestingly, the axiom of choice is not needed by ZF to prove anything in arithmetic. So ZF and ZFC proves the same theorems of arithmetic. You can use the axiom of choice freely, as it can be proved that we can eliminate its use after. This requires a bit more mathematics to be shown. So, with respect to the arithmetical reality, ZF and ZFC are the same.

Bruno

With mechanism, physics is reduced to computer science or to arithmetic, in the same sense that for most educated people today think that biology can be reduced to physics, in principle of course. 

My first older result is simply that mechanism (the idea that we could survive with a digital brain/body) is incompatible with weak-materialism or with physicalism (the idea that there is a physical universe having a fundamental ontology, not reducible to any other science).

Bruno

like in the Washington-Moscoow thought experiment, the person is unable to predict his immediate particular personal future feeling, despite being certain (assuming Mechanism and all default hyoptheses) that it will be either like feeling to be in Moscow or like feeling to be in Washington, then you understand that mechanism entails the existence of a personal, subjective (first person) indeterminacy.

Then you will need to understand that the elementary arithmetical truth implement all computations (including all quantum computations, notably), in the original sense of Turing, Church, Kleene (cf the 1930s papers), which makes us, in any “here and now” indexical state, indeterminate on which computations continue us, making eventually the physical science as a statistics on all (relative) computations, among an infinity (which are indeed realised in that elementary part of arithmetic).

On of the things I seriously dislike about MW, which makes it utterly REPELLENT (Steven Weinberg's word), is that there are too many damned worlds! For example, when one considers the worlds created when putting on left or right shoe first, this just scratches the surface of how many worlds are created -- not just two worlds, but uncountably many (if space is continuous) when one considers how many distinct worlds are created just for the case of the left shoe first. I am referring to the uncountably many ways to put on one's left shoe first. For me, this is hardly an ELEGANT Cosmos, but rather a downright SILLY one! AG

When you decide to put the left shoes, in bot “quantum Mechanism” à-la Everett, or in arithmetic, only the consistent extensions must be taken into account, and if you decide to put the left shoe, you will put in all continuation that left shoe. Both the quantum and the purely Mechanist frame does not make everything happening, only the consistent (with you local belief) extension will occur, in the vast majority of extensions.

Bruno

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[Brent Meeker]

On 2/25/2020 6:01 AM, Bruno Marchal wrote:

Then arithmetic cannot create other worlds! End of story. Case closed. AG 

It creates all the individual and collective stories. The case would be close if you could prove the existence of a world, but if you are a consistent machine (to be short), then incompleteness precludes this. 

Even intuitively, you can understand that no one can prove the existence of anything from scratch.

Dr. Marchal, meet Dr. Johnson.

Brent

[Bruno Marchal]

On 25 Feb 2020, at 14:49, John Clark <johnk...@gmail.com> wrote:

On Tue, Feb 25, 2020 at 8:34 AM Bruno Marchal <mar...@ulb.ac.be> wrote:

>> I have a halting oracle machine and it has 2 input slots, one slot for the logical blueprints of a computer in digital form and the other slot for a program, also in digital form, to run on that computer. The oracle machine will then output either the words "Halt" or "Not Halt" depending on what a program running on that computer will do. I decide to use the oracle machine as one part of a new 3 part machine I will call machine X. The first part of machine X is just a photocopier that makes two copies of its input and then feeds them into the 2 input slots of the oracle machine. The last part of machine X is the negator, if it receives a "Not Halt" input from the oracle machine then the negator will output "Halt" and then stop, if the negator receives a "Halt" input from the oracle the negator will go into an infinite loop and never stop. The entire X machine as constructed has one input slot and one output slot.
I will now input machine X with machine X's own blueprints, so after the photocopier has done its work the oracle machine will receive identical inputs in both slots and the oracle machine will have to figure out what will happen to the X machine when the X machine is fed it's own blueprint as input. If the oracle says under those circumstances the X machine will halt then the X machine will never halt, and if the oracle says the X machine will not halt then the X machine will print "Halt" and stop. So the halting oracle machine always makes predictions that are wrong. So there is no such thing as a halting oracle machine. QED.

> This shows that the machine+halting-oracle cannot solve the halting (machine+halting-oracle) problem,

Yes, it can't solve the halting problem.

Let us be clear. The machine+halting-oracle can obviously solve the halting problem.

And what you proved above, is that the machine+halting-oracle cannot solve the machine+halting-oracle halting problem.

> but the halting-oracle is supposed fro be concerned only for the machine halting problem, i.e. the machine without oracle. 

Supposed? That's a rather silly thing to say, if the oracle machine is well defined and the machine is well defined then the new combined machine is also well defined

Yes, but it has a sort of divine abilities (to use Boolos term). Turing used the term oracle to make clear that it does not have to be computable, and indeed the oracle notion makes his office when being not computable. 

and the so called halting oracle fails miserably when it tries to predict if this new well defined machine will halt or not.

Yes, that’s correct.

Bruno

John K Clark

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[Alan Grayson]

What are the elements in the large brackets on the right? AG 

This makes ZF-minus, or ZFC-minus (that is ZF, or ZF + the axiom of choice, minus the axiom of infinity) equivalent to Peano arithmetic.

Now ZF, or ZFC is just ZF-minus, or ZFC-minus to which you add the axiom of infinity; which says that there is a set omega such that 

 1) it contains 0 (the empty set), and 

 2) it is such that if it contains x, it contains x union {x}.

How do you distinguish x from {x}? AG

[Bruno Marchal]

I write again the whole set representing 3

0 is represented by the empty set { }.

1 is the set which has a unique element which is the empty set:  { { } }

2 is represented  by the set { 0, 1} which is equal to { { } { { } }  }

3 is represented by the set {0, 1, 2 }, which is {  { }, { { } }, { { } { { } } }.  }.

This makes ZF-minus, or ZFC-minus (that is ZF, or ZF + the axiom of choice, minus the axiom of infinity) equivalent to Peano arithmetic.

Now ZF, or ZFC is just ZF-minus, or ZFC-minus to which you add the axiom of infinity; which says that there is a set omega such that 

 1) it contains 0 (the empty set), and 

 2) it is such that if it contains x, it contains x union {x}.

How do you distinguish x from {x}? AG

x is some object, usually some set in “pure set theory”, but it could be anything.

{x} is the set with a unique object which is x.

OK?

Bruno

 

That makes it close for the set theoretical successor relation s(x) = x union {x}. It contains (intuitively) all natural numbers.

And then you can prove in ZF that there are an infinity of higher and higher infinite cardinals, etc.

So, by abusing language, up to the faithful representation, it is correct to say that ZF is mainly PA + an axiom of infinity.

Now, interestingly, the axiom of choice is not needed by ZF to prove anything in arithmetic. So ZF and ZFC proves the same theorems of arithmetic. You can use the axiom of choice freely, as it can be proved that we can eliminate its use after. This requires a bit more mathematics to be shown. So, with respect to the arithmetical reality, ZF and ZFC are the same.

Bruno

With mechanism, physics is reduced to computer science or to arithmetic, in the same sense that for most educated people today think that biology can be reduced to physics, in principle of course. 

My first older result is simply that mechanism (the idea that we could survive with a digital brain/body) is incompatible with weak-materialism or with physicalism (the idea that there is a physical universe having a fundamental ontology, not reducible to any other science).

Bruno

like in the Washington-Moscoow thought experiment, the person is unable to predict his immediate particular personal future feeling, despite being certain (assuming Mechanism and all default hyoptheses) that it will be either like feeling to be in Moscow or like feeling to be in Washington, then you understand that mechanism entails the existence of a personal, subjective (first person) indeterminacy.

Then you will need to understand that the elementary arithmetical truth implement all computations (including all quantum computations, notably), in the original sense of Turing, Church, Kleene (cf the 1930s papers), which makes us, in any “here and now” indexical state, indeterminate on which computations continue us, making eventually the physical science as a statistics on all (relative) computations, among an infinity (which are indeed realised in that elementary part of arithmetic).

On of the things I seriously dislike about MW, which makes it utterly REPELLENT (Steven Weinberg's word), is that there are too many damned worlds! For example, when one considers the worlds created when putting on left or right shoe first, this just scratches the surface of how many worlds are created -- not just two worlds, but uncountably many (if space is continuous) when one considers how many distinct worlds are created just for the case of the left shoe first. I am referring to the uncountably many ways to put on one's left shoe first. For me, this is hardly an ELEGANT Cosmos, but rather a downright SILLY one! AG

When you decide to put the left shoes, in bot “quantum Mechanism” à-la Everett, or in arithmetic, only the consistent extensions must be taken into account, and if you decide to put the left shoe, you will put in all continuation that left shoe. Both the quantum and the purely Mechanist frame does not make everything happening, only the consistent (with you local belief) extension will occur, in the vast majority of extensions.

Bruno

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[Alan Grayson]

It's extraordinarily subtle. Not sure it's OK. After all, the concept of "set" is primitive and more or less undefined, as is the empty set. You name the empty set, zero, or 0. Then we have an element or some object called x, and the set containing only x, as well as the union of the two. Is the union also a set if x isn't? AG

 

That makes it close for the set theoretical successor relation s(x) = x union {x}. It contains (intuitively) all natural numbers.

And then you can prove in ZF that there are an infinity of higher and higher infinite cardinals, etc.

So, by abusing language, up to the faithful representation, it is correct to say that ZF is mainly PA + an axiom of infinity.

Now, interestingly, the axiom of choice is not needed by ZF to prove anything in arithmetic. So ZF and ZFC proves the same theorems of arithmetic. You can use the axiom of choice freely, as it can be proved that we can eliminate its use after. This requires a bit more mathematics to be shown. So, with respect to the arithmetical reality, ZF and ZFC are the same.

Bruno

With mechanism, physics is reduced to computer science or to arithmetic, in the same sense that for most educated people today think that biology can be reduced to physics, in principle of course. 

My first older result is simply that mechanism (the idea that we could survive with a digital brain/body) is incompatible with weak-materialism or with physicalism (the idea that there is a physical universe having a fundamental ontology, not reducible to any other science).

Bruno

like in the Washington-Moscoow thought experiment, the person is unable to predict his immediate particular personal future feeling, despite being certain (assuming Mechanism and all default hyoptheses) that it will be either like feeling to be in Moscow or like feeling to be in Washington, then you understand that mechanism entails the existence of a personal, subjective (first person) indeterminacy.

Then you will need to understand that the elementary arithmetical truth implement all computations (including all quantum computations, notably), in the original sense of Turing, Church, Kleene (cf the 1930s papers), which makes us, in any “here and now” indexical state, indeterminate on which computations continue us, making eventually the physical science as a statistics on all (relative) computations, among an infinity (which are indeed realised in that elementary part of arithmetic).

On of the things I seriously dislike about MW, which makes it utterly REPELLENT (Steven Weinberg's word), is that there are too many damned worlds! For example, when one considers the worlds created when putting on left or right shoe first, this just scratches the surface of how many worlds are created -- not just two worlds, but uncountably many (if space is continuous) when one considers how many distinct worlds are created just for the case of the left shoe first. I am referring to the uncountably many ways to put on one's left shoe first. For me, this is hardly an ELEGANT Cosmos, but rather a downright SILLY one! AG

When you decide to put the left shoes, in bot “quantum Mechanism” à-la Everett, or in arithmetic, only the consistent extensions must be taken into account, and if you decide to put the left shoe, you will put in all continuation that left shoe. Both the quantum and the purely Mechanist frame does not make everything happening, only the consistent (with you local belief) extension will occur, in the vast majority of extensions.

Bruno

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[Bruno Marchal]

Set theory has a long history, and the first attempt to make it precise, Frege “theory of set”, which was also the first first order logical theory, was shown inconsistent (famously by Bertrand Russell). Cantor informal theory of sets was also full of paradoxes/and contradiction. 

Frege though that those inconsistency would disappear if the language was made precise, and he invented first order logic for that reason. Then he thought that we can make a set from any closed formula, in first order logic (which in set theory is the language of first order logic + one symbol for belongness, written here by “in”).

But Russell pointed that this lead to a contradiction. Define the set E of all sets which do not belong to themselves. E = {x such that NOT(x in x)}. Then E is in E if and only if x is not in E. Contradiction. 

Nevertheless, this has led to an axiomatic theory of set, which restricts the rule of Frege, but then got many other axioms to be able to work. There are different axiomatic of set, but the most known and studied is the axiomatic of Zermelo-Fraenkel ZF, and this has become a large field of study. I can give references on many books, introductory or advanced. There has been tuns of remarkable result, most build on the work and questions by G. Cantor, notably the theories of the infinite ordinals and cardinals.

You name the empty set, zero, or 0.

I represent the intuitive number 0 by the (formal) set { }. It is the set of x such that x ≠ x (using Frege naïve idea, I can explain how we manage to do so in ZF, but I don’t want to be too much technical now.

Then we have an element or some object called x,

ZF usually used AF (the axiom of foundation, or regularity). With AF, all sets will be pure sets, that is build from { }, and the axiom of parts, or of union, etc. But you can take AF away, and works with set of objects which are not sets. 

and the set containing only x,

That is the set { x }

as well as the union of the two.

You mean the union of x and { x } ? It depends on what x is. If x is not a set, the union is not defined, a priori.

If x is a set, for example x = { 1, 2, 3 }, then x union {x} will be the set containing 1, 2, and 3, and the set {1, 2, 3}, i.e. the union will be {1, 2, 3 {1, 2, 3}}, which is a set with four element, the numbers 1, 2, 3 and the set {1, 2, 3}.

Beginners confuse often an element of a set, and a part, or subset, of a set.

We say that A is a part of a set B, or that A is a subset of B, or that A is included in B when

     For all x ((x in A) -> (x in B)).

For example, it is trivial that: For all x ((x in A) -> (x in A)), so A is a subset of A, despite being rarely, if ever (never in ZF, due to the axiom of foundation). Similarly, it is trivial that For all x ((x in { }) -> (x in A)}, and this for any A, so the empty set is included in all sets.

How many subsets does the set {a, b} have? Answer 4. Indeed there are { },{a} {b} {a, b}. In general a set with n elements will have 2^n subsets. Cantor has shown that the set of all subset of a set a, is always bigger than the set a. He manages to generalise this on the infinite sets, and discover the theory of the transfinite. Note that he dis all that just to make rigorous some statement about the topology of the set of solutions to the equations of Heat, and the use of Fourier transform.

Hope this help a little bit. Ask any question.

Bruno

Is the union also a set if x isn't? AG

 

That makes it close for the set theoretical successor relation s(x) = x union {x}. It contains (intuitively) all natural numbers.

And then you can prove in ZF that there are an infinity of higher and higher infinite cardinals, etc.

So, by abusing language, up to the faithful representation, it is correct to say that ZF is mainly PA + an axiom of infinity.

Now, interestingly, the axiom of choice is not needed by ZF to prove anything in arithmetic. So ZF and ZFC proves the same theorems of arithmetic. You can use the axiom of choice freely, as it can be proved that we can eliminate its use after. This requires a bit more mathematics to be shown. So, with respect to the arithmetical reality, ZF and ZFC are the same.

Bruno

With mechanism, physics is reduced to computer science or to arithmetic, in the same sense that for most educated people today think that biology can be reduced to physics, in principle of course. 

My first older result is simply that mechanism (the idea that we could survive with a digital brain/body) is incompatible with weak-materialism or with physicalism (the idea that there is a physical universe having a fundamental ontology, not reducible to any other science).

Bruno

like in the Washington-Moscoow thought experiment, the person is unable to predict his immediate particular personal future feeling, despite being certain (assuming Mechanism and all default hyoptheses) that it will be either like feeling to be in Moscow or like feeling to be in Washington, then you understand that mechanism entails the existence of a personal, subjective (first person) indeterminacy.

Then you will need to understand that the elementary arithmetical truth implement all computations (including all quantum computations, notably), in the original sense of Turing, Church, Kleene (cf the 1930s papers), which makes us, in any “here and now” indexical state, indeterminate on which computations continue us, making eventually the physical science as a statistics on all (relative) computations, among an infinity (which are indeed realised in that elementary part of arithmetic).

On of the things I seriously dislike about MW, which makes it utterly REPELLENT (Steven Weinberg's word), is that there are too many damned worlds! For example, when one considers the worlds created when putting on left or right shoe first, this just scratches the surface of how many worlds are created -- not just two worlds, but uncountably many (if space is continuous) when one considers how many distinct worlds are created just for the case of the left shoe first. I am referring to the uncountably many ways to put on one's left shoe first. For me, this is hardly an ELEGANT Cosmos, but rather a downright SILLY one! AG

When you decide to put the left shoes, in bot “quantum Mechanism” à-la Everett, or in arithmetic, only the consistent extensions must be taken into account, and if you decide to put the left shoe, you will put in all continuation that left shoe. Both the quantum and the purely Mechanist frame does not make everything happening, only the consistent (with you local belief) extension will occur, in the vast majority of extensions.

Bruno

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