There would only be a contradiction if Gödel claimed that his own theorem was unprovable. Fortunately, he was not an idiot, and therefore did not claim that. What he did claim was that some truths are unprovable (his own theorem not being among those). And the reason why some truths are unprovable was sketched in my first post in this thread, namely the theorems are all capable of being listable by an algorithm, whereas no algorithm can list all truths. Hence, there are truths that will not appear in the list of all theorems, hence, not every truth is a theorem. Again, note that, in this version, no mention is made of a specific unprovable statement, so your reasoning about the "meta-cognitive" level does not apply.

Thanks for all your explanation in this thread, really appreciate it.

(1) Gödel (well, Gödel, Church, Tarski, Rosser, etc.) showed a bit more than what you are implying. He showed that, for any consistent system containing enough arithmetic (i.e. extending Robinson's arithmetic), the set of theorems of that system is not identical to the set of truths of that system. So, if we just move one level up and claim that truth is provability in a higher system, this won't do, because the theorem will reapply at the higher level separating these sets again. — Nagase

I was just addressing the wording of the theorem: there are statements of the language of F which can neither be proved nor disproved in F. Why does it not just say that there are statements of the language of F which can neither be proved nor disproved? Is it because for any statement of the language of F there is some G that can prove or disprove it?

Could you also clarify the situation regarding my example of "there is a number greater than 0 and less than every real"? @sime's reply seems to suggest that I'm misunderstanding the status of the hyperreals. My claim was that the sentence can be provable in one system and that its inverse can be provable in another system, and that that's all there is to the sentence being either true or false.

Well, proof is relative to a system of axioms. That is, we usually define proof, relative to a theory T, as follows: a sequence of statements A1, ..., An is a proof of An in T iff for every i < n+1, either Ai is an axiom of logic, an axiom of T, or it follows from the Aj's (j < i) from the rules of logic. So it doesn't make much sense to ask about a sentence whether it is provable or unprovable tout court, whence the relativization to a given axiomatic system (unless you're asking about whether a formula is a truth of logic, in which case you don't need the relativization to a theory, though you will obviously relativize to the background logic).

As for your example, in the case of some particular theories (such as PA or real analysis), we are interested in theory primarily as a description of a particular object. For example, when in studying PA, we are generally not interested in any discretely ordered ring, rather, we are interested in the natural numbers. So when we are asking about the truths of PA, we are asking about truth in the natural numbers, not truth in any model. Similarly, when we are asking about the truth of a sentence about the reals, we are generally interested in, well, the reals, not the hyperreals. That's not to say that these other objects are not useful or interesting in their own right, just that in the case of those theories, we have an intended model in mind.

Here's a comparison that I find useful. Consider the theory of groups. It consists basically of three axioms (I'll use the additive notation because it is easier to type): x+(y+z)=(x+y)+z, x+e=x, and x+(-x)=e. Now, since there are commutative and non-commutative groups, this theory does not decide the sentence x+y=y+x, and is therefore incomplete. But this is not surprising in the least, because when studying groups, we are not studying a privileged group, so we don't expect the axioms to completely characterize this object. Indeed, the axioms were developed to be as general as possible while still characterizing an interesting class of objects (namely, groups), so it is a virtue of the theory that it is incomplete.

On the other hand, in the case of PA, the theory was crafted to completely characterize a particular object, namely the natural numbers (object here taken in a broad sense, to include structures). Moreover, in the case of second-order PA, Dedekind proved that it does completely characterize its intended object, in a sense: the theory is categorical, which means that every model of the theory is isomorphic to the standard model. Hence it completely captures the structure of the standard model. So it is surprising that there are truths about this system that it the theory is unable to prove.

Don't mention it. I'm glad this has been useful to someone...

@tim wood

There would only be a contradiction if Gödel claimed that his own theorem was unprovable. Fortunately, he was not an idiot, and therefore did not claim that. What he did claim was that some truths are unprovable (his own theorem not being among those). And the reason why some truths are unprovable was sketched in my first post in this thread, namely the theorems are all capable of being listable by an algorithm, whereas no algorithm can list all truths. Hence, there are truths that will not appear in the list of all theorems, hence, not every truth is a theorem. Again, note that, in this version, no mention is made of a specific unprovable statement, so your reasoning about the "meta-cognitive" level does not apply. — Nagase

I seem to have missed something (or maybe not). Godel's incompleteness theorem states that given an axiomatic system X capable of mathematics there will be mathematical truths, say P, that can't be proved in X. Godel doesn't seem to be saying that such statements are unprovable in every possible axiomatic system, just that it's unprovable in the axiomatic system X.

If so, we would need patch system X with either the statement that's unprovable (P) or another one with which we can prove P - call this new system Y. The catch is that the problem of unprovable true statements will recur in the new system Y too and so on.

@tim wood@Nagase

Just wanted to ask you two a question:

IF proof isn't necessary for truth then, how do we distinguish between truths and falsehoods? The universal response to claims people make is to ask for proof.

If so, we would need patch system X with either the statement that's unprovable (P) or another one with which we can prove P - call this new system Y. The catch is that the problem of unprovable true statements will recur in the new system Y too and so on. — TheMadFool

Add it as a new axiom. That's my understanding too.

IF proof isn't necessary for truth then, how do we distinguish between truths and falsehoods? The universal response to claims people make is to ask for proof. — TheMadFool

I'll yield to Nagase, but it seems to me that truth is not always that important - even that truth cannot be well-defined. More it is, does the claim in question work. And words like "necessary" don't help, unless used and understood as terms of art with a determinate meaning.

The significance of claims of truth and counterclaims for proof is very much situation dependent - to the degree that (imo) it's all about the details.

It's been argued in these forums that truth is exactly and nothing more than the abstract quality that all true propositions have in common in virtue of they're being true. And what exactly is that quality of being true for each true proposition? In every case that differs. Does she or doesn't she? Did the Yankees win the 1928 world series? Is the cube root of 4 greater than the square root of 3? Three different truths to be determined. One evidential, one historical, one by proof schema. And did Homer write the Iliad and the Odyssey? Some truths are unknowable.

This in no way prevents truth and proof from being useful within systems where they are well-defined.

A couple of observations:

(1) First, note that the theorem, in the form I stated, is a bit more general, since it does not rely on any specific unprovable sentences. Thus, even if we know by construction (or other means, such as Kirby-Paris) that a given sentence is unprovable but true, and then add it to the theory, we can't be sure that we have exhausted all the truths---there could be many more true but unprovable sentences of which we are unaware, so merely adding one to the theory will not make it complete. Of course, we could simply take all the true sentences and add it to the theory---but then we wouldn't know what is the resulting theory! (In other words, there would be no algorithm to tell me whether or not a given sentence belongs to the theory)

(2) This failure to identify the truths to be added could be circumvented if we could isolate an easily identifiable set of sentences and prove that adding just this set of sentences is enough to add all the truths. For instance, a natural candidate for such a set is the set of sentences which express consistency statements. That is, we could begin with Con(PA) (a statement saying that PA is consistent), and it to PA to obtain the theory PA + Con(PA), then consider the consistency statement for this theory and add it to obtain PA + Con(PA) + Con(PA + Con(PA)), so on and so forth. By Gödel's second incompleteness theorem, we know that each resulting theory is stronger than its predecessor, so there is at least some hope that this would bear some fruit.

In fact, we can obtain a more interesting result if, instead of working with consistency statements, we work with reflection principles of the sort exemplified by Löb's theorem. That is, define Ref(T) to be the statement "For every sentence S of T, ProvT('S') -> S", where "ProvT" is the provability predicate for T. We can then consider progressions of the form PA + Ref(PA), PA + Ref(PA) + Ref(PA + Ref(PA)), etc. Now, an immediate problem appears here: for each natural number n, we can define Tn to be Tn-1 + Ref(Tn-1 + Ref(Tn-1)). And then we define T(omega) to be the union of all such Tn's. But what happens then? We can obviously continue the procedure, i.e. considering T(omega) + Ref(T(omega)). Here, however, we will need a way to codify ordinals greater than omega inside PA. This is done by what are called ordinal notations.

So how far can we go? Will the process eventually stop somewhere? It is a remarkable theorem by Feferman (cf. "Transfinite Recursive Progressions of Axiomatic Theories") that, in the case of PA, it does stop somewhere, and he gave a precise stopping point for this (if you must know, the stopping point is at $\omega^{\omega^{\omega +1}}$). That is, Feferman found that there is a way to code ordinals such that according to this code, there is a certain iteration of the addition of reflection principles that proves every true arithmetical sentence! This is often referred to as Feferman's completeness theorems and is remarkable indeed.

Unfortunately, there is a catch (there's a Brazilian song that says that a ripe fruit hanging near a well-trodden path must be either rotten or its tree full of wasps...). There is more than one way to code ordinals, and Feferman's proof depends heavily on the choice of code. Indeed, he also showed (together with Clifford Spector) that there are infinitely many ways of coding ordinals for which this result is not true. Moreover, to discover the correct coding is as difficult as discovering what are all the true sentences of PA (this is basically because he employs a rather bizarre code that ensures that in every iteration of the construction we "sneak in" a true formula). Hence, there is no real algorithm for extending PA in such a way to obtain a complete extension using reflection principles. The obstacle is the same as before, at some point we don't know anymore what is the theory we are obtaining by this procedure. (For those curious about this, the book by Torkel Franzén, Inexhaustibility: A Non-Exhaustive Treatment is still the best, though be warned that it is technically demanding).

(3) So what is going on? Again, the point is that provability (in the sense of obtaining a proof that we can recognize as such) is different from truth. Note that, in spite of my talk of models in a previous post, this need not be cashed out in strictly platonist terms. One can think about this in terms of the principles we are committed to when reasoning about arithmetic (where the vocabulary of principles and commitment can be taken in a broadly Sellarsian fashion). These principles can be taken to be encoded in the axioms for second-order arithmetic, and here the problem becomes even more evident: second-order logic does not have a complete proof-procedure. So, again, the semantic content of our principles invariably outstrip our capacity to prove things from them.

(4) Lastly, about your question, it is ambiguous. "To distinguish" can be taken to mean "how do we establish that something is true?", in which case you are right that the answer is "by proving it". But it can also be taken to mean "how do we characterize a sentence as true", in which case the answer is not "by proving it", but rather according to Tarski's satisfaction definition (or, if you will, you can say that a sentence of arithmetic is true if it follows semantically from the second-order axioms of PA).

I don't think Löb's theorem supports the constructivist position. That's because truth is generally taken, prima facie to obey the capture and release principles: if T('S'), then S (release), and, if S, then T('S') (capture). But what Löb's theorem shows is that proof does not obey the release principle. So there is at least something suspicious going on here. — Nagase

Constructively, Prov doesn't fail either the capture or release properties of a truth predicate with respect to decided sentences, rather Prov doesn't supply negative truth value when classifying undecided sentences that no axiomatic system can decide without either begging the question, or by performing a potentially infinite proof search equivalent to doing the same in Peano arithmetic.

Perhaps part of the confusion/suspicion comes from overlooking the following symmetry in what Peano arithmetic cannot derive

PA |-/- (For All S: S --> Prov('S')) ( since compilation is potentially infinite)
PA |-/- (For All S: Prov('S') --> S ) ( since decompilation is potentially infinite)

Löb's theorem only deduces the existence of an as-of-yet undecided object on the assumption that the decompilation process of it's respective code terminates. And yet the undecided object will only compile into a code in the first place, if it is decidable. Therefore Löb's theorem does not have constructively relevant implications.

Why would you want, let alone expect, a truth predicate to capture and release the properties of potentially infinite objects whose existence is potentially non-demonstrable?

Moreover, one can show that the addition of a minimally adequate truth-predicate to PA (one that respects the compositional nature of truth) is not conservative over PA. Call this theory CT (for compositional truth). Then CT⊢∀x(Sent(x)→(Prov(x)→T(x)))CT⊢∀x(Sent(x)→(Prov(x)→T(x))), where "T" is the truth predicate. As a corollary, CT proves the consistency of PA. So truth, unlike provability, is not conservative over PA. — Nagase

In other words, introducing new axioms to represent undecided formulas generally permits the derivation of new sentences in a vacuous manner.

Finally, you have yet to reply to my argument regarding the computability properties of the two predicates, namely that one does have an algorithm for listing all the theorems of PA, whereas one does not have an algorithm for listing all the truths of PA. So the two cannot be identical. — Nagase

The set of theorems of PA isn't recursive due to the halting problem, meaning that any proposed test of theoremhood by a "truth predicate" is bound to be either incomplete or to contain an infinite number of mistakes.

Consequently, the "truth" of PA consists of the explicit construction of each and every theorem, doing everything the hard way.

Edit: I rushed this post, so came back and rectified some mistakes.

I think you are focusing too much on the fact that theoremhood is not strongly representable in PA, with the consequence that you are ignoring the fact that it is weakly representable in PA. Indeed, while theoremhood is not computable, it is computably enumerable. In other words, there is an algorithm which lists all and only theorems of PA. It exploits the fact that, given your favorite proof system, whether or not a sequence of formulas is a proof of a sentence of PA is decidable. Call the algorithm which decides that "Check Proof". Here's an algorithm which lists all the theorems of PA, relative, of course, to some Gödel coding:

Step 1: Check whether n is the Gödel number of a sequence of formulas of PA (starting with 0). If YES, go to the next step. Otherwise, go to the next number (i.e. n+1).

Step 2: Decode the sequence of formulas and use Check Proof to see if it is a proof. If YES, go to the next step. Otherwise, go back to Step 1 using as input n+1.

Step 3: Erase all the formulas in the sequence except the last. Go back to Step 1, using as input n+1.

This (horrible) algorithm lists all the theorems, i.e. if S is a theorem of PA, it will eventually appear in this list. Obviously, this cannot be used to decide whether or not a given formula is a theorem, since, if it is not a theorem, then we will never know it isn't, since the list is endless. But, again, it can be used to list all the theorems. My point is that there is nothing comparable for the truths, i.e. there is no algorithm that lists all the truths. In fact, by Tarski's theorem, there can be no such algorithm. So, again, the two lists (the list of all the theorems, the list of all the truths) are not the same, whence the concepts are different.

The upshot of all this is that, in my opinion, constructivists should resist the temptation of reducing truth to provability. Instead, they should follow Dummett and Heyting (on some of their most sober moments, anyway) and declare truth to be a meaningless notion. If truth were reducible to provability, then it would be a constructively respectable notion. But it isn't (because of the above considerations). So the constructivist should reject it. (Unsurprisingly, most constructivists who tried to explicate truth in terms of provability invariably ended up in a conceptual mess---cf. Raatikainen's article "Conceptions of truth in intuitionism" for an analysis that corroborates this point.)

thanks for the reply, will get back later.

If we cannot prove a theorem we need to expand our alphabet!

The upshot of all this is that, in my opinion, constructivists should resist the temptation of reducing truth to provability. — Nagase

Not only intuitionists (who would require an actual construction), but other practitioners as well. :cool:

think you are focusing too much on the fact that theoremhood is not strongly representable in PA, with the consequence that you are ignoring the fact that it is weakly representable in PA. Indeed, while theoremhood is not computable, it is computably enumerable. In other words, there is an algorithm which lists all and only theorems of PA. It exploits the fact that, given your favorite proof system, whether or not a sequence of formulas is a proof of a sentence of PA is decidable. Call the algorithm which decides that "Check Proof". Here's an algorithm which lists all the theorems of PA, relative, of course, to some Gödel coding:

Step 1: Check whether n is the Gödel number of a sequence of formulas of PA (starting with 0). If YES, go to the next step. Otherwise, go to the next number (i.e. n+1).

Step 2: Decode the sequence of formulas and use Check Proof to see if it is a proof. If YES, go to the next step. Otherwise, go back to Step 1 using as input n+1.

Step 3: Erase all the formulas in the sequence except the last. Go back to Step 1, using as input n+1.

This (horrible) algorithm lists all the theorems, i.e. if S is a theorem of PA, it will eventually appear in this list. Obviously, this cannot be used to decide whether or not a given formula is a theorem, since, if it is not a theorem, then we will never know it isn't, since the list is endless. But, again, it can be used to list all the theorems. My point is that there is nothing comparable for the truths, i.e. there is no algorithm that lists all the truths. In fact, by Tarski's theorem, there can be no such algorithm. So, again, the two lists (the list of all the theorems, the list of all the truths) are not the same, whence the concepts are different. — Nagase

I'm still confused as to where and how we disagree. I suspect the issue might be mostly terminological, due to your use of modal notions versus my deflationary/constructive terminology. However it is possibly worth recalling that PA must have the following theorem for any provability predicate

PA |-- Prov('G') --> ~G for any Godel sentence G.

In other words, there cannot be an exhaustive and infallible enumeration of theoremhood within PA. - and here we aren't referring to the failure of ~Prov to enumerate the non-theorems - rather we are referring to the inability of Prov to correctly enumerate all and every derivable theorem.

We can of course construct an infallible Prov by defining it to enumerate only the godel-numbers that have been independently determined to be proofs via brute-force checking. In which case Prov along with the godel-numbering system are merely redundant accountancy of what we've already derived, and in which case Prov sacrifices exhaustivity for infallibility.

Alternatively, prov could be an 'a priori' algorithmic 'guess' as to theoremhood , in which case it can be exhaustive , e.g. by guessing "True" to every godel number, but at the cost of infallibility, in guessing both correctly and incorrectly as to theoremhood.

But we're at least both in agreement it seems, that there isn't an algorithm for listing all and only the actual theorems of PA - which is precisely the reason why truth should be constructively identified with actual derivability, as opposed to being identified with whatever is unreliably or incompletely indicated by a 'provability'/'truth' predicate.

So where and how do we diverge?

The upshot of all this is that, in my opinion, constructivists should resist the temptation of reducing truth to provability. Instead, they should follow Dummett and Heyting (on some of their most sober moments, anyway) and declare truth to be a meaningless notion. If truth were reducible to provability, then it would be a constructively respectable notion. But it isn't (because of the above considerations). So the constructivist should reject it. (Unsurprisingly, most constructivists who tried to explicate truth in terms of provability invariably ended up in a conceptual mess---cf. Raatikainen's article "Conceptions of truth in intuitionism" for an analysis that corroborates this point.) — Nagase

Dummett's overall arguments sound roughly similar to my deflationary position of truth in mathematical logic'; software engineers don't say that the operations of a software library has no truth value, rather they define truth practically in terms of software-testing, without appeals to the choice axioms, or the Law of excluded middle.

What i think classical philosophers overlook is that the absolute consistency of PA isn't knowable, or even intelligible. As an absolute notion, undecidability is meaningless.

is that the absolute consistency of PA isn't knowable, or even intelligible. As an absolute notion, undecidability is meaningless. — sime

It seems to me that at the level of the absolute, you have being and not-being - and that's it! Kindly, patiently, and briefly educate me on the utility and value - and meaning - of your observation. I get (I'm pretty sure I get) Godel, at the level of his paper on undecidable propositions.

He built a "machine" that produces true propositions that cannot by the machine be proved true. They're known to be true by extra-machine argument and proof. A consequence is that the consistency of the machine's production cannot by the machine be proved consistent. None of this suggests that PA is itself inconsistent. For PA to be inconsistent there must be a false proposition that is true, yes? Fortunately there's no need to examine transfinitely many propositions one-by-one, one merely needs try to build one, and it's hard to see how that could be done.

I'm not asking you to produce one, but if you think about it a bit, it does not seem to be possible. Maybe, then, there's a meta-mathematical proof for the consistency of PA. Is there one?

I'll be very explicit, then: there is, in fact, an algorithm that lists all and only the theorems of PA. This algorithm therefore provides an exhaustive and infallible enumeration of theoremhood of PA. It is exhaustive, i.e. every theorem appears in the list. And it is infallible, i.e. every formula in the list is in fact a theorem. Since for some reason you apparently missed it from my last post, I will reproduce it here again. Choose your favorite Gödel numbering for formulas and sequences. Given this Gödel numbering, there will be an algorithm, call it DecodeS, which, given a number m, first decided whether or not m is the Gödel numbering of a sequence of formulas and, if it is, returns the sequence of formulas for which m is a Gödel number. We also have an algorithm, Check Proof, which, given a sequence of formulas, decided whether or not the sequence of formulas is a proof in PA. Given these, the algorithm is as follows:

Step 1: Input n (starting with 0). Use DecodeS to check if n is the Gödel number of a sequence of a formulas. If YES, go to the next step, otherwise, start again with n+1.

Step 2: Use DecodeS to print the sequence of formulas coded by n. Apply Check Proof to this sequence. If the result is YES, go to the next step. Otherwise, go back to step 1 with input n+1.

Step 3: Erase all the formulas in the sequence except the last. Go back to step 1 with n+1.

Call the above three step algorithm Theorem List. I claim that Theorem List is an exhaustive and infallible enumeration of theoremhood in PA. It is obviously infallible, since Check Proof is infallible. It is also exhaustive, since the algorithm will basically go through every sequence of formulas of PA, so, if P is a theorem of PA, it s bound to find a proof for it eventually.

In other words, the set of theorems of PA is what we call computably enumerable. This is a well-known fact (and the proof is clearly constructive---it is not like the concept of r. e. sets is somehow constructively suspect), so I'm surprised that you are still insisting that it is somehow impossible to generate an exhaustive and infallible list when the above demonstrates that it is not only possible but actual (and it also exhibits the algorithm in question!). Now, you claimed that this alleged impossibility somehow followed from the fact that PA |- Prov('G') --> ~G, but I don't see the relevance of this for listing all and only the theorems.

Yes, there is a proof of the consistency of PA, though whether or not it is finitistically acceptable is debatable. Gentzen proved that the consistency of PA can be proved in PRA + Epsilon_0 induction, i.e. primitive recursive arithmetic augmented by the principle that the ordinal epsilon_0 is well-ordered (it should be noted that PRA + Epsilon_0 induction and PA are incomparable in strength, so the result is not a triviality). See this very nice article by Timothy Chow for more on the topic.

It is obviously infallible, since Check Proof is infallible. — Nagase

Infallible maybe, but certainly incomplete, in the elementary sense that it only includes - if even it does that - just those propositions that stand as the conclusion of proofs. What will it do for propositions that aren't the conclusions of proofs? Call them non-theorems? And if by theorem you mean exactly that - proved propositions - then it would appear you miss G and G' and G'', and so forth.

As an old retired mathematician not conversant with contemporary set theory, could you clarify your recent posts by explaining how they relate to common, everyday theorems arising in classical areas of mathematics, like classical analysis. Are you saying the collection of proofs of all such theorems in this context are computatively enumerable? :chin:

If by G you mean the Gödel sentence, then, yes, the algorithm will miss it. But that's because the algorithm lists all the theorems of PA, and the Gödel sentence is not a theorem of PA!

If a theory is such that: (i) it has a reasonable proof system (i.e. one can check by an algorithm whether or not a sequence of formulas is a proof) and (ii) is recursively axiomatized (i.e. there is an algorithm which tells whether or not something is an axiom of the system), then that theory will have a computably enumerable set of theorems.

And yes, this applies also to classical mathematics such as classical analysis, to the extent that it can be formalized in a reasonable proof system (be it by formalizing in second-order arithmetic or by formalizing it in first-order ZFC or something similar).

I've never come across, nor contemplated tracing the proof of something like the fundamental theorem of calculus back to its set theoretic roots. As I've mentioned, in a set theory class sixty years ago we did define the exponential function, however, from the PAs.

I've never come across, nor contemplated tracing the proof of something like the fundamental theorem of calculus back to its set theoretic roots. — jgill

What do you mean to communicate here? Do you doubt it could be done? Do you wonder how it's done? Could you produce at least a high-level drilldown of the basic ideas if called upon to do so? Would you like to see what that would look like? Do you wish math would go back to the mid nineteenth century before the age of formalization? I'm wondering what is the purpose of your post. I know the formalisms, I'd be glad to outline them if that's what you're curious about.

Or are you asking why anyone would bother with such pedantry? It's those pesky nineteenth century guys again, noticing that their lack of rigor was causing trouble. For example before the great age of epsilons and deltas, even Cauchy famously wrote an erroneous proof by failing to distinguish between pointwise and uniform convergence.

tl;dr: Did you want to see an outline of a proof of FTC directly from ZF? Or are you wondering why anyone cares?

What do you mean to communicate here? — fishfry

What is upsetting you? I have simply stated something I haven't done. There are no unstated jabs at modern mathematics. And yes, I am curious to see what that might look like. Nagase gave a nice reply to a question I raised. Since I once went through the process of attaining the exponential function through set theory I am curious how one would prove something like the fundamental theorem from "scratch". But I am not curious enough to work through all the details.

Do you wish math would go back to the mid nineteenth century before the age of formalization? — fishfry

Of course not. I know you think I'm a dinosaur of a mathematician, and I admit I am. I have even avoided holomorphic functions at times in order to investigate more general non-linear conditions in complex dynamics. That really puts me out to pasture. But you know what? It's been a delightful journey and it still is.

I apologize if I have offended you. Let's agree that modern set theory as well as other modern areas of mathematics are very important subjects that some old timers may not fully appreciate. :cool:

I know you think I'm a dinosaur of a mathematician — jgill

I have never held such an opinion. When have I ever said such a thing? If you can link or quote anything I've ever written to that effect I'll stand corrected, but as I have never said any such thing you will not be able to. I have noticed that you very often refer to your status as a retired math professor; and that you often announce your ignorance of virtually everything outside of your specialty, and most of the standard undergrad math curriculum. You've said this probably a dozen times in the past couple of months, most recently in the post I just replied to. Perhaps you're projecting or perhaps you have feelings about this. I am certain I've never commented on these matters at all. You are the one who keeps bringing them up.

I don't understand why you post that you have no idea how to prove FTC from first principles, but then get angry when I ask if you are curious to learn how. I admit I don't get where you're coming from. I was, and still am, curious as to why you would post to the effect that you don't know something, without having the intention of either learning it or objecting to it. And that's what I asked.

From ZF the axiom of infinity gives us $\mathbb N$. We use an equivalence relation on $\mathbb N \times \mathbb N$ to form the integers; and another one on the integers to form the rationals. We build the reals as Dedekind cuts of rationals. One the reals are built we can make rigorous definitions of limits, continuity, derivatives, and Riemann integrals; and then any textbook proof of FTC will suffice.

Integers from naturals: http://web.math.ucsb.edu/~padraic/ucsb_2014_15/ccs_proofs_f2014/ccs_proofs_f2014_lecture5.pdf

Rationals from integers: https://en.wikipedia.org/wiki/Field_of_fractions

Reals from rationals: https://en.wikipedia.org/wiki/Dedekind_cut

you often announce your ignorance of virtually everything outside of your specialty, and most of the standard undergrad math curriculum. — fishfry

Really? I've taught analytic geometry, the calculus sequence, advanced calculus, intro to real analysis (grad), point set topology, statistics and probability, history of mathematics, mathematics of finance, metric spaces, intro to computer science, complex variables, differential equations, special topics in analysis, math for engineering technology, linear algebra, etc.

I have taught neither abstract algebra nor set theory. The one faculty colleague who was a set theorist became more of a math educator and journal editor. We never developed a set theory course due to a lack of student interest as well as the fact that most of us were not competent in that subject. As you have seen.

I have not kept up with more than a small handful of the myriad of directions math has taken during the last century. I got my degree fifty years ago and teaching, administrating, and on and off research kept me busy, along with a family and a serious outdoor avocation. One makes choices.

Please stop the ad hominem attacks.

Please stop the ad hominem attacks. — jgill

I apologize.

I only wish to tell you that with your background, every single thing that's been discussed on this forum that you think is beyond you, is actually trivially within your capabilities and knowledge.

I only wish to tell you that with your background, every single thing that's been discussed on this forum that you think is beyond you, is actually trivially within your capabilities and knowledge. — fishfry

Fair enough. When I speak of a topic being "beyond me" it's a cop-out for not having the mental energy at my age (83) to study it, or just a complete lack of interest. I appreciate your comment.

Every so often, however, something a bit out of my purview will intrigue me and I will make an effort to understand it. For example, a couple of years ago the notion of a functional integral sparked my interest, having read of Feynman's Sum of All Paths concept. My brief exposure to the concept fifty years ago was shallow and uncompelling.

That was a delightful exploration, starting with the basic Wikipedia definition, and I wrote a short math note about functional integrals in spaces of complex contours. I enjoy writing math programs, especially graphics, and I came up with some nice imagery. That was fun.

I should not be making dismissive comments about set theory. You, fdrake, Nagase, and a few others have clearly explained ideas in this subject, and it is a powerful link between math and philosophy, and a vital part of the mathematical galaxy. I apologize, and if I slip up in the future you should nail me!

Most of my research efforts have been in classical analysis, very basic dynamical systems in the complex plane, trying to determine convergence/divergence of certain sequences. At one time this was a popular topic, but modern analysis has moved the focus more toward algebraic systems and generalizations.

But I remain attached to the old-fashioned, nuts and bolts, stuff. For example, my latest efforts concern the iteration of linear fractional transformations (f(z)=(az+b)/(cz+d)) when the attracting fixed points are functions of time and are no longer "fixed". Like predator and prey, do the iterates "catch up" with the roving attractors? Modern theory dealing with LFTs is more geometrical and algebraic.

OK. Enough rambling. Thank you for your comments.

Hopefully this thread can go back to its original focus. Sorry for the diversion. :sad: