Complex numbers are, usually, part of the curriculum in high school, where one learns that they can represent points on a plane, labeled by two real nimbers. So the statement about requiring three numbers is, simply, wrong. The statements about Boolean logic don’t have anything to do with the subject.

Complex numbers can be consistently defined and provide a generalization of real numbers.

It’s, also, possible to define triples of real numbers and, in fact, of any n-tuple of real numbers. That’s usually taught at university level.

Stan Sykora not quite what i asked but thank you for your answer. If you consider constructing the concept of number ground up from 1, first we might define counting, then addition, then multiplication then powers. The inverse of counting or addition (subtraction) reveals the negative numbers. The inverse of multiplication (division) reveals the rationals. The inverse of powers (roots or fractional powers) reveals the irrationals. Multiplication is repeated addition. Powers are repreated multiplication. I am not aware of any subsequent operator/inverse and number type generates by number alone. Clearly we can then go on to define tuples, vectors and matrices.

The concept of hyperoperations subsumes the operations you mention, including going beyond exponentiation --

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

Tetration in particular comes immediately after exponentiation --

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

Ross La Haye yes, thank you. These look to introduce some new syntax but amount to the standard notation of powers evaluating from right to left (left to right being pointless given (a^b)^c=a^(bc). They and their inverses do not generate any new type of number unlike i generated by (-1)^(1/2). Interesting. It appears boolean logic only has three essential dimensions.

Peter Breuer interesting. My use of the term "Boolean logic" is clearly non-standard. What I mean is non-axiomatic logic, i.e. logic which only assumes two distinct values, so is devoid of any assumption beyond the assumption of two disctinct values. It turns out that the assumption of two distinct values is sufficient to require incompleteness. This suggests to me that reality has no distinct values (nothing exist in isolation from infinity - everything is continuously entangled). Note: I did not say 2^(2^n) = 2^(2n) I said (a^b)^c = a^(bc) which is why we define a^b^c to mean a^(b^c).

Peter Breuer there are rules but they are purely definition, without any inherent meaning. Boolean logic can be reliably executed by machines.

Peter Breuer definition is simply values and operators. Definitions say how values are combined using operators to arrive at new values. What do you mean by deduce?

Peter Breuer yes, exactly. This is the distinction between predicate logic and boolean logic. What does true mean? In boolean logic true is just a definition. An operator can be defined to return a value of true or false (generally encoded as 1 or 0). Boolean logic does not have true and false until you define them. This becomes important with logical induction.

Peter Breuer I suspect we have very different mental models. I am not clear what you mean by a "truth object" but I suspect it means you have some sense that truth is pre-existing. In contrast if I wish to have a "truth object" I must define it. If I can clearly explain what I mean to a computer then I have encoded truth in the language of only two distinct values. If I cannot then my concept is unclear. Boolean logic is about consistency. It is about never allowing 1=0. We can define whatever we want but the definitions must be clear and consistent.

Peter Breuer these are mechanisms within propositional logic. Model theory strikes me as a good name for this. Models are useful in devising top-down architecture but boolean logic builds bottom-up. A propositional model can be validated using boolean logic by first clearly defining each of the words in terms of bits.

Boolean logic (at least my version of it) is not defined using propositional logic.

Computing does not build from a foundation of propositional logic, it builds directly from boolean logic (bits).

If I wish to define number, ordering and equality I define them. Nothing is assumed. I must define "

Peter Breuer I'm not sure what you are asking for. Bit-based logic is just definition. Rules are just part of the definition. Is it clear how bit-string definition works or is it something I need to explain? Would you like to know for example how a C Uint8 is defined, encoded and manipluated or how the Javascript Number class is defined encoded and manipulated, or how my own code encodes and manipulates rational and surd objects?

Peter Breuer I have not made any claim with respect to propositional logic, I have only claimed boolean (bit-based) logic does not depend on it. Give me an example of what you see as an important propositional logic rule and I will explain if and how bit-based logic deals with it. I am not certain I can do this by the way. I am interested to see your example.

I don't see any logic in your discussion, nor Aristotlean nor Boolean...

Logic is much more than truth and falsehood...

Just take two bits and you have an extended Boolean Logic. Take N bits and even you can construct all the shades of false-true that you want.

Boolean Logic has the nice feature that the logical content of N bits can be transformed into a natural number... Then Mersenne numbers 2^N-1 correspond to perfect trueness: (1,1,1,...,1)...

I didn't mention a publication that will provide some hints or add more confusion... The file is added.

Ramon Carbó-Dorca your paper looks interesting.

Thanks. I feel it is a bit trivial... But it is at the level my mind can move. Perhaps some of your questions can be answered there. I have several papers using Boolean hypercubes... But this one seems adequate for the moment.

Anyway, I'll be happy if this serves to illuminate some of your doubts...

Ramon Carbó-Dorca simple is good. I dislike complexity for complexity's sake. I like your discussion with respect to determinism, causality and machine learning. As you state boolean logic (the variety executed by computers) is deterministic. The neural models they form are however often far from understandable. I would never be able to code solutions as elegant as those computers often now figure out just from a simple learning algorithm. They are less understandable than say GLMs or GAMs but can be significantly better. The learning algorithms are often simple yet the learning can be quite remarkable. Logic is however always constrained by incompleteness, both practical, and foundational. I see the power of logic in is its ability to make good predictions based on limited data. The historic skill was in the human simplification process. Increasingly computer algorithms do this for us. The more we learn to harness this machine-learning/human-learning relationship the better. Computers teach me as much or more than I teach them and always have. They constantly show me how poor my logic really is.

Peter Breuer I have read https://en.wikipedia.org/wiki/Propositional_calculus to check my understanding but your example would still be helpful. In summary most of the wikipedia article I find obvious. In boolean terms it is just a different syntax to express truth tables. Where I differ is in the concept of "closure" because my boolean definition of closure does not match the ZFC definition. Boolean (bit-based) logic is never closed because boolean logic recognises it is only ever an incomplete model. Boolean logic is only ever closed within the realm its own incompleteness. Put more simply 1/n>0 for n>0 within boolean logic but we cannot assume this implies n>0 for "all" n because "all" is not a property of boolean logic. In the limit as n tends to infinity 1/n=0. Since I can see the source of the error in ZFC I can also see how to fix it. The fix is important as our science moves into the realm of machine-learning. I will write a paper to explain and propose an alternative to ZFC which properly recognises the incompleteness issue.

Peter Breuer yes, by boolean logic I mean what you describe as boolean algebra. I do not distinguish between boolean algebra and logic. Logic is boolean algebra by my definition of the word logic. Words introduce ambiguity and/or inconsistency unless defined in terms of boolean algebra, so I find ZFC both ambiguous and inconsistent. I do however have a meta theory which sits above the incompleteness of boolean logic. That is what I will explain in terms of a replacement for ZFC and propositional logic. It is relatively easy to explain and potentially re-unites the fields of physics, mathematics and computing. It does not and cannot remove logical incompleteness, but this is also relatively easy to explain. So to rephrase my request to you. Can you provided any example of something important you feel propositional logic provides which is not provided by boolean algebra?

Peter Breuer helpful thanks.

I will write up my ZFC replacement.

It deals with your statement "As for something important provided by propositional logic, the mere fact that one can talk about propositions not just 1 or 0, is the difference. Different universe of discourse."

Is there anything else you feel propositional logic provides beyond boolean algebra?

Note: my language is non-ZFC but it is standard in terms of the usual non-rigorous English dictionary definitions (so my words mean what 99.99% of the UK population would understand by them unless I provide more specific definitons). My meta theory will explain the issue with language.

When you read what I write I suggest you assume I am using English unless I specifically state I am using some more rigorous language. I am happy to give a more precise "boolean algebra" definition of my words when appropriate and needed.

Peter Breuer "for all x, x = 1 or x = 0 is true of the simple boolean algebra on 0,1"

This is not true for what I mean by logic so I should not use boolean algebra to describe what I mean. Perhaps you can help me with the correct word. By logical I mean anything which can at least theoretically be transformed into a computer program. Is there a well-defined word for this? If not don't worry as it is not really needed once I write up the meta-theory.

Peter Breuer I assume you are referring to the Curry-Howard correspondence.

Replace mathematics with: "You must use a computer program to formalize what you mean." and I am happy with your statement (the definition of mathematics is not clear).

My meta theory will explain the issue and how to deal with it. I mention ZFC simply because it is the most widely accepted form of Cantorian thinking. The error is in Cantorian thinking. I will write it up. Many thanks for your help.

Peter Breuer I still don't know what you mean by "rule". Since we have established that computer program and predicate logic are isomorphic (Curry-Howard) could you explain what you mean by a "rule" in terms of computer programming which I believe we share as a common language and which we both accept as non-ambiguous. Note: I do not accept ZFC terminology as non-ambiguous (if that's what you mean by mathematics). This will however cause you an incompleteness problem if there is no meta theory to replace the ambiguous term mathematics. I will write a paper to explain the replacement meta theory.

Peter Breuer rules was your word. This was your original statement "If there are no rules to your logic apart from 1 not the same as 0, that's great, but it's not logic." The discussion has moved on so are you happy your question is now answered? If we both accept Curry-Howard then presumably it removes the difficulty of what I mean by logic. Note I am not a finitist. I have no issue with irrational numbers or with calculus and neither do carefully constructed computer programs. Logic does however need a mechanism to recognise and deal with logical incompleteness.

Peter Breuer yes, perfect. I do indeed have some explaining to do. I will write my paper. Words are context specific due to logical incompleteness as I will explain.

Can you accurately define what you mean by complete and incomplete? By your definition of complete are the natural numbers complete or incomplete?

Peter Breuer so the definition of complete is context specific. In the context of Peano aritmetic the natual numbers are incomplete. Correct me if I am wrong but I believe ZFC defines the natural numbers as complete (or rather defines complete in terms of mapping from the natural numbers). Yes this is all 70 year old philosophy but it has fundamental errors. It has been debated for thousands of years not just 70 and I am not the only one who finds this 70 year old Cantorian thinking very poor. "Complete means all true statements are provable from the axioms" is a poor definition of complete. It is a defintion of "good enough" within the incomplete realm of a logical model.

Peter Breuer "If you prefer your infinite things to be classes rather than sets, go ahead! If you don't like any infinite things at all, be a Finitist!"

I don't like either of these options. I will write up the alternative including what I mean by complete and infinite.

It is indeed closely related to potential and actual infinity. Much of modern theory is circular nonsense with respect to potential and actual infinity. Godel's proof it blatently obvious given a computing perspective. Curry-Howard tells us a computing perspective of the problem is sufficient. As an aside I am now confused by what ZFC does and does not say but I reject it anyway so no big deal. Countable-infinity and infinite-set are computationally illogical definitions.

Peter Breuer interesting. Would Z be the same formal program specification language I was taught back in the 80s? I'd be interested to see your computational model if you still have it. Given that computer code is a formal specification language I could never see the point of Z which is perhaps why I have never seen it again in nearly 40 years of practical computing. From my perspective any logic has exactly the same restriction as science. No theory can ever be proved correct except within declared limits of incompleteness. Things can only ever be proven incorrect or inconsistent. Proof of correctness is always incomplete. There is a reason for this.

Peter Breuer code is specification (Curry-Howard). Yes it can be run which is a bonus but there is no requirement to run code or even for it to be runnable. for (i=1;true;i++);beep() is valid code. Not only is this valid code but modern development environments will generally highlight the unreachable statement. Yes I am aware there will be lots of academic computer science content I do not use. Nobody can know everything and knowledge is not the same as understanding. Approaches to computer programming have completely transformed in the last 40 years. I only tend to concern myself with those proven to be most productive.

Peter Breuer your statements are logically incorrect but it would appear you only have an academic (Godel like) understanding of what logic and incompleteness is. The problem with academic understanding is that it is based on human thought which is prone to under and over interpretation. Output all integers is an illogical instruction since the integers are incomplete so there is no such thing as "all" integers within any consistent logic. Someone with practical rather than academic understanding of logic knows this instantly. A compiler also confirms this. It is not a matter of opinion. A non-thinking machine can confirm it. You can invent illogical logic if you wish. I am more interested in explaining what logic really says. I will find a reference to the philosopher who first explained what I have just said in terms of experiential and theoretical understanding. It actually relates to logic itself although the philosopher would probably not have recognised this. I can explain why.

Peter Breuer note your comment with respect to maxint is irrelevant. It relates to the difference between practical, inherent and foundational incompleteness (my words). See Preprint Logic & Infinity

Peter Breuer An early draft Preprint Singularity Theory

Peter Breuer I don't think you understand it. It is a new idea so may take some getting used to. This may help Preprint Singularity Theory -Reality versus Logic

Okay. It is clear to me but gibberish to you. That is fine. ZFC is gibberish to me but I think clear to you. The point is 1/0=∞ is an error in logic as you state. Infinity does not lie within logic. By my definition reality is infinity, something which lies beyond logic. Logic works well until we move close to the logical definition of infinity (or of continuity). Logic can't cope with this but reality can because reality is infinite and thus complete. Infinite is self-consistent so cannot be composed out of 1 and 0. Any logic which has distinct concepts like 1 and 0 is required to remian incomplete.

Peter Breuer don't worry. The problem is the distinction between abstract and real but it needs a very clear understanding that abstract is not real and the two are not quite the same to understand what I am saying. Real does not allow 0 and 1 to be distinct. It does not allow anything to be distinct. Everything is part of reality. It is genuine infinity not the can't count to it form that abstract logic has. Abstract logic must correct for the faulty assumption of discretness, which it can partially do. Relativity partially does this. It corrects for the fact time and space are not distinct.

Peter Breuer I'm confident I am correct now but I just need to prove it experimentally. Real infinity is 1 - always. Real infinity does not have any other value. You cannot add or remove stuff from reality. That is why we see a constant speed of light.

B(100,0.5) makes no sense because of the faulty 0 1 assumption.

The low end probabilies are too low. It cannot be correct. Work out how often you would expect 100 heads and how long that would take. Sure its not likely but it cant take 3000 times the age of the universe with 1 billion experiments a second. If that were true we would never see any result. There is no difference to the actual permutation we get and the 100 one - there are just lots of ways to do it.

Binomail is wrong because we have not corrected for real infinity. We have assumed independence and real infinity can't allow it.

The P(X70) zone is far too low. An experiment should prove we get these results too often. I need to explain it all more clearly.

Einstein was right but he missed why. Quantum theory is wrong.

What I am saying is relativity does not just apply to gravity. It applies to everything and it applies to everything because reality is real infinity (not a poor abstract model of infinity). Reality does not have any discreteness. Only logical models (including the one our mind forms) have discreteness. Most of the time it is fine. things are so close to discrete it makes no difference. We are in the bounds where the model and reality are near enough for the variation to be expected. But we can design experiments which go beyond these bounds and show the model must be adjusted for 0 1 non-indpendence. It will apply to any formulae we have for the real world. They will work fine in the sweet spot (the one we generally care about) but they will fail as the model goes to foolishly small or large values, which it will. Logic can create really daft values. 10^10^10^10 - is insane. It dwarfs real infinity but strictly real infinity does not have any numbers - you get the whole thing. Always.

Singularity Theory suggest..... We need to be clearer about the distinction between abstract and real. If we allow 1 to exist in the abstract world then 1 cannot exist in the real world (nothing exists in isolation). Conversely infinity cannot then exist in the abstract world but must exist in the real world. There is such a thing as real infinity but it is not the abstract concept of infinity. We can calculate abstract 1 based limits on how large infinity can be and how much uncertainty must be involved. The binomial distrubution can be used to prove this is correct. I just now need to do the maths and write some code to validate it. We can then fix quantum theory.