This is but one of many comprehensive texts about mathematics written Jean Gallier and Jocelyn Quaintance! They are all a little unpolished, but still an invaluable resource to students!

* Linear Algebra and Optimization with Applications to Machine Learning (html)

* Differential Geometry and Lie Groups (html)

* Homology, Cohomology, and Sheaf Cohomology (html)

* Proofs, Computability, Undecidability, Complexity,
and the Lambda Calculus. An Introduction (pdf)

* Aspects of Harmonic Analysis and Representation Theory (html)

* Algebra, Topology, Differential Calculus, and Optimization Theory
for Computer Science and Machine Learning (html)

* Aspects of Convex Geometry
Polyhedra, Linear Programming,
Shellings, Voronoi Diagrams,
Delaunay Triangulations (html)

* Notes on Primality Testing and Public Key Cryptography
Part 1: Randomized Algorithms
Miller-Rabin and Solovay-Strassen Tests (08/2017) (pdf)

* Spectral Graph Theory of Unsigned and Signed Graphs
Applications to Graph Clustering: A Survey (12/2014)
Posted to arXiv as paper cs.CV arXiv:1311.2492 (pdf)

* Introduction to discrete probability (12/2014) (pdf)

I'm looking at this now. It's a bit idiosyncratic but looks good. It covers standard topics in proof and recursion theory but from a more contemporary approach using natural deduction instead of Hilbert-style proof systems, and it also does complexity theory with various computation models. There are a fair number of exercises and problems, always a good thing. But it covers specialized topics at the expense of some standard ones. For example, I didn't spot any mention of the compactness theorem. So it's hard to call the book a usable general introduction to logic, despite it starting at a relatively beginner level.

It's a draft and it has some typos and some long-winded passages that could use editing, but I'm glad it is out there, and it looks to me like a good way for computer people to learn some logic and computability theory if that strikes their fancy. I've only looked at the first couple chapters but the TOC mentions that Girard's polymorphic lambda calculus makes an appearance, so there must be some type theory in there. But, my preference would be to increase the coverage of that even if it means trimming back some of the computability stuff. Another improvement might be to include some coverage and exercises in mechanized proof checking using Coq, in the spirit of Software Foundations (also by authors at Penn, https://softwarefoundations.cis.upenn.edu/ ).

Well, it depends on what you are looking for, but the traditional-style book that I'm used to is H. B. Enderton, "A Mathematical Introduction to Logic", and Enderton also has an accompanying book on set theory. I guess I'd suggest those as a companion rather than an alternative to the Gallier and Quaintance book. The G&Q book is imho more modern and more interesting from a CS-oriented perspective. It just leaves out too many important fundamentals. Alternatively, since it is still a work in progress, maybe some of the missing stuff could be added.

Boolos and Jeffrey "Logic and Computability", whatever the current edition is, is supposed to be good, but I haven't read it yet.

The G&Q book is weird in that it is about interesting topics that I'd consider slightly advanced, but written in a way that assumes almost no background, so if it's the only thing you read, you'll still be missing important background. I'd like a book that uses the G&Q approach but that includes the background. If it kept all the current contents and was expanded, it would probably end up as 2 volumes, which is fine. Or it could leave out some of the specialized stuff, but I think that stuff is there because that was what the authors wanted to write about, so I won't judge ;-).

Let ◻ be the modal operator "It is provable that", so ◻P means that P is provable.

The first part of your argument in the second link says:[

Suppose that there is a proposition UNP such that UNP ⇔ ¬◻UNP .

Then, suppose that ¬UNP .
It then follows that ¬¬◻UNP , i.e. that ◻UNP.

Therefore, ◻◻UNP .

Then, as UNP ⇔ ¬◻UNP, therefore ¬UNP ⇔ ◻UNP.

Then, replace the ◻UNP in ◻◻UNP with ¬UNP, and therefore conclude ◻¬UNP .

So, at this point, we have concluded that ◻¬UNP and also that ◻UNP.
So, you can conclude that, __under these assumptions__, that ◻⊥ .

So, under the assumption that there is a statement UNP such that UNP ⇔ ¬◻UNP , and the additional assumption that ¬UNP, you can conclude that ◻⊥.

]

Indeed.

If a system admits a statement UNP, (and the system is strong enough to do the appropriate reasoning) (and peano arithmetic and such does admit such a statement UNP), then within the system, you can show that ¬UNP implies ◻⊥.
I.e. "if the statement that claims it cannot be proven, can be proven, then the system proves a contradiction".
This is entirely normal, and does not overturn anything.

However! While the assumption that ¬UNP allows us to derive ◻⊥, it does not allow us to derive ⊥.
We have __not__ shown that ¬UNP → ⊥, we have only shown that ¬UNP → ◻⊥.
Therefore, this is not justification in appealing to proof by contradiction, and concluding UNP (nor ◻UNP).

I've not read ProfHewitt's argument, so I'm just looking at your summary.

If ◻P means "P is a proposition that is formally provable within some implicit formal system" (perhaps there should be a subscripted letter to show which formal system because there are an infinite number of them), then I don't understand what ◻◻UNP means, because ◻UNP is not a proposition of the implicit formal system.

If we were doing this in Peano Arithmetic using Gödel numberings, then ◻P would be expression by taking the Gödel number encoding P and applying a predicate that expresses provability to that number.
This might be written as like, Provable«P» with the quotes indicating taking the Gödel number of the proposition P, in order to encode it in a way that Peano arithmetic can talk about. (Usually one uses the right angle brackets on the upper corners for the quoting, but my phone doesn’t have that on the keyboard, which is why I’m using “«“ and “»”). Statements like this are statements in the same formal system as P.

The modal logic known as “provability logic” is a nice abstracted system which captures how these things work, and allows for dealing with them in a general case.

◻◻P is perfectly sensible, and if the system is able to describe itself sufficiently well (which is not a high bar; Peano arithmetic satisfies this requirement.) then if you can derive P within the system (without additional assumptions), then, simply by including a proof that the proof of P is a proof of P, one can produce a proof in the system of ◻P,
And so, if one can derive ◻P, so too can one derive ◻◻P .

(ProfHewitt’s post also uses basically the same thing as ◻◻P , though he writes it using the turnstile symbol rather than the square symbol, and I prefer the square symbol for it, and the turnstile for something slightly different. Not saying the use of the turnstile for it is wrong though.)

You’re free to use a system that has orders of propositions, and I’m sure there are interesting things to be said about systems with such orders. (The set theory of NFU seems rather appealing, and stratified formulas seem somewhat analogous.)

(And yes, if you restrict what a system can do, it is possible to produce a system which can, in a sense, prove its own consistency. Dan Willard produced one such system. It has subtraction and division as fundamental rather than addition and multiplication.)

However, a theory is not required to have orders of propositions (Or, perhaps you might prefer saying this as “A theory can have all of its propositions be if the same order”?).

Furthermore, in a formal system modeling the mathematical properties of a syntactical system (as in, a set of rules describing allowed transformations on some strings), this also does not require having multiple orders of propositions.

And modeling what things are provable in a given formal system, is just that same kind of thing.

So, when describing in a formal system which well-formed-formulas can be derived within that system, it is not necessary (in the sense of “you don’t have to do it.”) to use multiple orders of propositions.

(Of course, in any formal proof of a wff which we interpret as having a meaning, there is perhaps a kind of gap between the string of characters which is provable within the system, and the thing which we interpret it to mean. However, if the system is sound with respect to our interpretation of it, then the things it proves will, under that interpretation of the system, correspond to a meaning which is true.
As such, it is usually safe to elide the distinction between “the string corresponding to this proposition is derivable in this formal system” and “this system proves this proposition” (where “this proposition” is taken to refer to some meaning).

When interpreting statements made in a system “about” that system, there are kind of two levels of interpretation, kinda-sorta. First, when we interpret a proof within the system that some statement(s) is(are) (not) provable in the system, we first have to interpret the string we have derived as corresponding to the meaning of a claim about the system, namely, a claim about what strings can be derived in the system. At that point, we also interpret what those strings that we interpret the first string as referring to, would mean.
This can perhaps sometimes be a little confusing.
Keeping this distinction in mind should make it clear why there is no need for orders of propositions when dealing with a system referring to what it can and can’t prove.)

Are you claiming that (e.g.) ZFC (which does not have orders for propositions) is not "foundations", or that it isn't consistent?

Or by "foundations" are you referring to a particular system you are proposing as a foundational system, and which you have named "foundations"?

You appear to justify the argument on the basis of the idea of a liar sentence.

As exemplified in NFU , it is not necessary to give strict orders to things, as long as you put restrictions on how things are constructed. TST has linearly ordered types, but NFU has no need to introduce these types and orders, as just requiring that formulas be stratified is sufficient.

There is no liar sentence in Peano Arithmetic. It isn't a well-formed-formula. Partitioning propositions into orders is not needed in order to prevent it being a well-formed-formula. Just, don't include anything in your rules for what counts as a wff which would let you define it.

(you may object that, what if one just does the Godel numbering thing to do some quine-ing, and uses that to produce a liar sentence, but you can't express "The proposition [some number] encodes, is false" in PA (see Tarski's undefinability theorem) .)

It isn't like UNK is defined in PA as "a statement UNK such that UNK iff not(provable('UNK'))". That wouldn't be a wff in PA. Rather, it is some long expression involving a bunch of quantifiers over natural numbers, and also a bunch of large numbers, and a bunch of arithmetical relations, and happens to be such that one can prove (in PA) that [UNK iff not(provable('UNK')] .

> If ◻P means "P is a proposition that is formally provable within some implicit formal system" (perhaps there should be a subscripted letter to show which formal system because there are an infinite number of them)

It doesn't mean that. The formal system is explicit: it's the system we're using to write statements like `◻P` (it's a form of modal logic https://en.wikipedia.org/wiki/Modal_logic ).

Prove I’mUnprovable using proof by contradictions as follows:

In order to obtain a contradiction, hypothesize
¬I’mUnprovable. Therefore ⊢I’mUnprovable
(using I’mUnprovable⇔⊬I’mUnprovable). Consequently,
⊢⊢I’mUnprovable using ByProvabilityOfProofs
{⊢∀[Ψ:Proposition<i>] (⊢Ψ)⇒⊢⊢Ψ}. However,
⊢¬I’mUnprovable (using I’mUnprovable ⇔⊬I’mUnprovable),
which is the desired contradiction in foundations.

Consequently, I’mUnprovable has been proved to be

a theorem using proof by contradiction in which

¬I’mUnprovable is hypothesized and a contradiction derived.

In your notation, the proof shows that following holds:

¬I’mUnprovable ⇒ ⊥

Why do you think that the proof is for the following?

Edit: if you had as an axiom, or could otherwise prove within the system, that ¬◻⊥, I.e. that the system is consistent, then you could conclude from ◻P and ◻¬P that ⊥, by first concluding ◻⊥, and then combining this with ¬◻⊥ .

And in this case, you could indeed say that this is a contradiction, and therefore reject the assumption of ¬UNK, and then conclude UNK without assumptions.

So, if one could show ¬◻⊥ (or if the system had it as an axiom), the reasoning would go through. (This is the thing that is missing.)

Therefore, you could then prove ◻UNK, and therefore ¬UNK, and therefore (having already shown UNK) would have a contradiction, ⊥.

So, this is a way of showing that, if you can show ¬◻⊥ (and if there is a statement UNK), then the system is inconsistent. Which is just one of Gödel’s theorems: a strong system can’t prove its own consistency without being inconsistent.

By the way, because the proposition I'mUnprovable does not

exist in foundations, it is OK for foundations to prove

their own consistency as follows:

Consistency of a theory can be formally defined as follows:

Consistent⇔¬∃[Ψ] ⊢Ψ∧¬Ψ

Contra [Gödel 1931], a foundational theory can prove its own

consistency as shown in the following theorem:

Classical theorem. ⊢Consistent

Classical proof. In order to obtain a contraction,
hypothesize ¬Consistent. Consequently,
¬¬∃[Ψ] ⊢Ψ∧¬Ψ, which classically implies ∃[Ψ]⊢Ψ∧¬Ψ by
double negation elimination. Consequently, there is a
proposition Ψ0 such that ⊢Ψ0∧¬Ψ0 (by eliminating the
existential quantifier in ∃[Ψ]⊢Ψ∧¬Ψ). By the principle
of ByTheoremUse {⊢∀[Φ] (⊢Φ)⇒Φ} with Ψ0 for Φ, Ψ0∧¬Ψ0,
which is the desired contradiction.

However, the proof does not carry conviction that a

contradiction cannot be derived because the proof is valid

even if the theory is inconsistent. Consistency of the

mathematical theories Actors and Ordinals is established by

proving each theory has a unique-up-to-isomorphism model

ByTheoremUse is not a valid principle, at least for any system that includes Peano Arithmetic.
(It is essentially the assumption that the system is not only consistent, but sound. It is therefore no surprise that ByTheoremUse would imply Consistent.)

By Löb’s theorem, ByTheoremUse would imply that for all propositions P, that P holds. I.e. it implies ⊥.

Perhaps I misunderstood what you mean by ByTheoremUse .

If you mean that, if in a context/environment with certain givens, one can derive a conclusion, then one can apply that in other cases,
Or if you just mean modus ponens, or cut elimination,
Then ok, that’s fine. That’s valid. (Though it doesn’t justify the step you cited it in.)

But you can’t, within the system, go from ◻P to P. That isn’t a valid rule of inference.

There’s a distinction between “therefore P” and “therefore ◻P”, and you cannot use the latter as the former.

You seem to equivocate better “P is provable” and “therefore P”

Like, suppose I was writing a computer program in a strongly typed language, and something required an argument of type A->B , and I tried to pass in a string which has the text of a function with that type. Obviously that wouldn’t type check.

(1) does not assume good faith. (2) is not the strongest possible interpretation of what Hewitt said. Listing them here is flamebait and reflects poorly on the community. Shame.

Having skimmed through the paper accessible at the end of the first link, I am leaning towards (1). It contains a barrage of eye-assaulting notation, lots of digressions and repetition, and basically no formal substance (which, for a logic/computation paper, is bizarre).

> Sorry that mathematical notation is causing you problems

For somebody who elsewhere in this thread literally asked HN discussion participants to "not degrade into personal attacks" this was an unbelievably arrogant remark.

What about giving us some summary on how the overall community of researchers in the theory of computation area are thinking of your breakthrough insights? I'm sure there are some CS Ph.D.s in here that are interested in some context.

I suggest you bring the discussion to the appropriate academic forums like conferences and workshops. Doing this via wikipedia and hacker news instead makes it look like your academic peers have decided it's not worth their time so that you are coming here instead.

Once the academic community with researchers in the same field accepts your breakthrough results, it will be easier to convince us that it's legit. And it will come here all by itself, without you having to do active promotion.

I am disappointed at several comments on this thread. Professor Hewitt, I personally think you've enriched the community by putting these claims here. They've led me to read more about several things, which I will list here for anyone else who might be looking for related topics: The difference between indeterminism and nondeterminism, bounded and unbounded nondeterminism, the actor model of computation, local arbitration versus global consensus, and state machines versus configuration-based computation.

You've responded to the ad hominem attacks with class. I hope you do not take them as a reflection of the HN community at large; please keep coming back and posting (provocatively) as you have done, because it induces learning.

That book doesn't touch any nondeterministic behavior at all. So, the nondeterministic extension of the Church-Turing Thesis is absolutely irrelevant here. Your other cited "developments" seem senseful only in your personal universe.

Yeah - its a long time since I studied this kind of stuff but I'm pretty sure the Church–Turing thesis doesn't say anything about non-determinism or not - being more relevant to complexity than computability?

The original thesis concerns only computable functions, which are deterministic by definition. It doesn't even consider their complexity. There are extended variations of the thesis, though. The complexity variation means that any computable function in one model of computation can have only a polynomial slowdown factor in comparison with another model of computation. It's believed to be false since the raise of quantum computations.

Ignore the downvotes: there is value in reading and attempting to understand some of this material, despite the inflammatory way the claims are made. Carl Hewitt is presumably in his 70s, so I'll forgive him for being an old man who probably doesn't give a shit what people think.

In particular, I find the discussion of "paradox" attacks to be quite illuminating. Bertrand Russell attempted to remove self-referential and self-applicable paradoxes through the embellishments of types and orders. Hewitt argues that Gödel basically "cheated" in his proofs because the Diagonalization Lemma violated restrictions on orders.

I'm not qualified to evaluate that argument on the basis of mathematical reasoning, but I do believe it points to a mismatch between the way computation is modeled in the abstract and the way it happens in reality. In the purely abstract world of natural numbers, there's really only one kind of thing: the natural number. Statements about natural numbers can be represented by natural numbers, statements about statements about natural numbers can also be represented by natural numbers, etc. While a given natural number can be parsed to compute if it is a valid encoding of a proposition of a certain order, it is always and forever a natural number and nothing else.

However, I'm not entirely convinced that this captures the nature of computation in reality. When computation is embodied in a physical system, is it possible that the output of some computation can itself manifest computational properties that are not mere compositions of the lower level computational model? Where the interactions between the "higher level" objects simply follow a different set of rules and thus form the foundation of a new formalism?

The notion of types seems to be a way to capture this, by introducing the possibility of distinguishing a "thing" from its abstract representation or description. If everything exists only within the world of the abstract representation, then a thing is indeed no different from its description, as any manipulation of the thing is equivalent to some manipulation of its description. But when the thing is, in fact, a physical object, it is clear that construction of the thing can fundamentally alter the model of computation. Why is this?

I suspect that it is because there are no "pure" or "inert" abstractions in the real world; everything is in fact performing computation all the time. So physical constructions are not just compositions of objects, but also compositions of the computational capabilities of those objects. I realize this probably sounds like navel-gazing at this point, but sometimes that can be a useful activity.

SSRN is not a peer reviewed journal. It's essentially just a preprint server where anybody can dump whatever they please. So it's more like the personal blog that the parent comment referred to than a peer reviewed journal they were asking you to cite.

Please cite peer reviewed works. As an accomplished academic of 70+ years I'm assuming you are very familiar with that process.

Here are some more book titles copied from his website: (https://www.cis.upenn.edu/~jean/home.html)

* Linear Algebra and Optimization with Applications to Machine Learning (html)

* Differential Geometry and Lie Groups (html)

* Homology, Cohomology, and Sheaf Cohomology (html)

* Proofs, Computability, Undecidability, Complexity, and the Lambda Calculus. An Introduction (pdf)

* Aspects of Harmonic Analysis and Representation Theory (html)

* Algebra, Topology, Differential Calculus, and Optimization Theory for Computer Science and Machine Learning (html)

* Aspects of Convex Geometry Polyhedra, Linear Programming, Shellings, Voronoi Diagrams, Delaunay Triangulations (html)

* Notes on Primality Testing and Public Key Cryptography Part 1: Randomized Algorithms Miller-Rabin and Solovay-Strassen Tests (08/2017) (pdf)

* Spectral Graph Theory of Unsigned and Signed Graphs Applications to Graph Clustering: A Survey (12/2014) Posted to arXiv as paper cs.CV arXiv:1311.2492 (pdf)

* Introduction to discrete probability (12/2014) (pdf)

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