# Re: Question about PA and 1p

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On 17 Jan 2012, at 20:06, Stephen P. King wrote:```
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```On 1/17/2012 5:08 AM, Bruno Marchal wrote:
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SNIP

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- I disagree that set theory might be more primitive than arithmetic. Why? First because arithmetic has been proved more primitive than set theory, and less primitive than logic. With logic we cannot define numbers. with set, we can define numbers, even all of them (N, Z, Q, ... octonions, etc.). The natural numbers are often defined by the von Neuman finite ordinal:
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0 = { }
1 = {{{}}}  = {0}
2 = {{}, {{}} } = {0, 1}
3 = {{}, {{}},{{}}},,{{} ,{{}} } } = {0, 1, 2}
...
n = {0, 1, 2, ..., n-1}
etc.

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And you can define addition by the disjoint union cardinal, and multiplication by the cardinal of cartesian product, and then, you can *prove* the laws of addition and the laws of multiplication. With arithmetic you cannot recover any axioms of set theory, except for the hereditarily finite sets.
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I am confused. It seems to me that you are admitting that sets are more primitive than Arithmetic since what you wrote here is a demonstration of how numbers supervene on set theoretic operations. The fact that we can define the natural numbers via the von Neuman finite ordinals is the equivalent of claiming that the natural numbers emerge from the von Neuman finite ordinals (up to isomorphism!), so I am confused by what you are claiming here! But whichever is the most primitive, it is not more primitive than the neutral foundation of existence in itself.
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I just meant that sets are more complicated that natural numbers, so by assuming sets you assume more than by assuming just the natural numbers. With comp we have that assuming arithmetic is enough. Sets, real numbers and the physical world are recovered in the epistemology of relative natural numbers (that is a number + a universal numbers). I have no clue what you mean by "neutral foundation of existence in itself".
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I have another problem with set theories. There is no clear "standard model". For arithmetic there is. The set of Gödel number of true arithmetical sentences is a highly complex set, but it is still well defined. That is not the case for the set of Gödel numbers of set theoretical sentences. I cannot be realist about sets. Yet another problem: in the quantified self-reference logic on set theory, B(P(x)), or [ ] P(x) has no easy meaning, and suffer from all the Quine-Marcus critics of quantified model logic, where on the contrary in the quantified self-reference logic on arithmetic BP(x) is crystal clear, and defeat all those critics, by showing transparent counter-example. I will confess you, Stephen, that I have never really believe in Set Theories. Set theories looks just like quite imaginative Löbian numbers to me. But I know well ZF, and appreciate it as an interesting logical object.
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- As I said, I don't take the word "Existence" as a theory. I have no clue what you mean by that. I was asking for a theory. You say that by taking (N, +, *) as a primitive structure, I am no more neutral monist, due to the use of + and *. This is not correct. It would make neutral monism empty. We alway need ontological terms (here 0, s(0) etc.) and laws relating those terms (here addition and multiplication).
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No, I am not making Existence as a theory, it is merely a postulate of my overall theory
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Theories are made of postulates. I don't see a postulate. Existence of what?
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(if you can call what I have been discussing a "theory"). I am using the notion of Existence as it is defined in Objectivist Epistemology. For example, as explained in this video lecture: http://www.youtube.com/watch?v=GfOS7xfxezA&feature=player_embedded Neutral monist takes is empty in the sense that it shows the coherent implication that the most basic ontological level cannot be considered to have some definite set of properties to the exclusion of others.
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This does not make sense for me. Sorry. (I am not a philosopher). You might have to elaborate. Something primitive without any property cannot explain anything. That is why physicist postulate particles and forces, and mathematicians postulate numbers and laws or operations, or set and belonging relations.
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- All you arguments with the term "physical" are going through in arithmetic, given that you agree that "physical" is not primitive. For example, the physical world is not required to make sense of what is a universal machine. It is required for human chatting on the net, but such a physical world is provided by arithmetic. Including concurrency.
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WE simply might have to agree to disagree.
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You cannot disagree with a theorem in a theory. You have to find a flaw in the proof, or you have to disagree with the premise. If you disagree with what I say above, I take it that you say "no" to the digitalist doctor, and defend a non computationalist theory of mind.
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- I don't do philosophy. I offer you a technical result only. I still don't know if you grasped it, or if you have any problem with it.
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You result has deep philosophical implications and as a student of philosophy I am very interested in it.
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I do think that genuine philosophers should appreciate it :)

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If you agree to assume that the brain works like a material machine, then arithmetic is enough and more than arithmetic is necessarily useless: it can only make the mind body problem unecessarily more complex. Primitive matter (time, space) becomes like invisible horse. Not epiphenomena, but epinomena.
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Again, we have to agree to disagree on this. The necessity of physical implementation cannot be dismissed otherwise the scientific method itself is empty and useless.
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"physical" is what I attempt to explain in a non circular way. That is why, even before studying comp, I did not postulate an ontological physical reality. I don't see at all why the scientific methods need an ontological physical reality. Its existence is not a fact, or show me a paper proving its existence. It is only a tradition since Aristotle. The founders of science, the greeks, where actually taking some distance with what is only an animal extrapolation from experience. Comp shows that such an extrapolation is not sensical.
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```Without the definiteness that the physical world
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With QM, such definiteness can be doubted. But comp explains completely (up to the comparison with facts) that there is no ontological physical reality. The physical implementations does not disappear, but are themselves made definable in arithmetic (or better through arithmetic, because it refers to the non arithmetical notion of arithmetical truth). Again, it would be more efficacious to read UDA and point on where you think a non valid step might have been made, given that what I say is just the UDA conclusion. I think that if you are set realist, the step-8 might not be necessary, or could be simplified. Perhaps.
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offers us is accepted there can be only idle speculation, we saw this kind of thinking in the Scholastics and know well how that was such a terrible waste of time. So why are you advocating a return to that? Ideal monism was pushed hard by Bishop Berkeley and failed back in the 18th century, its flaw - that the material world becomes causally ineffective epiphenomena - is not solved by your result, it is only more explicitly shown. You seem to think that it is a virtue. No, sorry, it is not a virtue for the simple reason that it makes the falsification of the theory impossible thus rendering it useless as an explanation.
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Physics is entirely given by the logic Z1* and X1*, so it is hard to imagine something more refutable than that. And Z1* and X1* are derived from comp (by UDA) + the classical theory of knowledge. Even if that theory is not correct, it provides a counter-example on what you say above. Comp is empirically refutable. This does not mean that we take the empirical reality as a primary given (it is not even for biological reason) Also, all that talk contradicts your neutral monism, given that you too don't take the physical as primitive. It looks like you contradict yourself a little bit.
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```    My alternative hypothesis
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I don't see it. I still don't see what you take as primitively existing, and what you derive from that.
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You have to find a way to express your idea so that simple minded scientist can understand them especially when you refer to scientists theories or math.
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has the chance of being falsified as it predicts that the physical worlds actually observe must be representable as Boolean algebras (up to isomorphisms).
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This is even weirder coming from someone who insists nature is not boolean (?).
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Bruno

http://iridia.ulb.ac.be/~marchal/

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