> Chris Menzel <cmen...@remove-this.tamu.edu> writes:
>> She is a philosopher of math who knows a *lot* of set theory. To my >> knowledge, she has authored only one more or less purely technical >> paper ("Proper Classes", JSL 1983), but many of her papers assume a >> pretty deep understanding of contemporary set theory.
> If we regard /Proper Classes/ a "purely technical paper" then surely > /V=L and Maximize/ also counts?
Well, looking at "Proper Classes" again (something I might better have done before posting) I do see that there is a lot more historical and philosophical stage setting than I'd remembered, but nearly half the paper's 25 pages consist of technical mathematical logic. Comparably technical work in "V=L and Maximize" occupies scarcely three pages. Granted, though, the paper moves along beginning to end at a pretty lofty technical level.
Chris Menzel <cmen...@remove-this.tamu.edu> writes: > Well, looking at "Proper Classes" again (something I might better have > done before posting) I do see that there is a lot more historical and > philosophical stage setting than I'd remembered, but nearly half the > paper's 25 pages consist of technical mathematical logic. Comparably > technical work in "V=L and Maximize" occupies scarcely three pages.
Perhaps I too should have had a peek at /V=L and Maximize/ before posting! There's certainly less purely technical stuff than I thought.
> Granted, though, the paper moves along beginning to end at a pretty > lofty technical level.
Yep, but the same is true of /Believing the axioms I&II/ etc.
> Chris Menzel <cmen...@remove-this.tamu.edu> writes:
>> Well, looking at "Proper Classes" again (something I might better have >> done before posting) I do see that there is a lot more historical and >> philosophical stage setting than I'd remembered, but nearly half the >> paper's 25 pages consist of technical mathematical logic. Comparably >> technical work in "V=L and Maximize" occupies scarcely three pages.
> Perhaps I too should have had a peek at /V=L and Maximize/ before > posting! There's certainly less purely technical stuff than I thought.
>> Granted, though, the paper moves along beginning to end at a pretty >> lofty technical level.
> Yep, but the same is true of /Believing the axioms I&II/ etc.
Indeed, so perhaps we can say that my original assertion was correct, amdended along the lines of: "Proper Classes" is the only paper of hers a substantial portion of which consists of technical mathematical logic.
Chris Menzel <cmen...@remove-this.tamu.edu> writes: > Indeed, so perhaps we can say that my original assertion was correct, > amdended along the lines of: "Proper Classes" is the only paper of hers > a substantial portion of which consists of technical mathematical logic.
This I can live with.
Maddy is by far my favourite philosopher of set theory. Pondering venerable philosophical and metaphysical questions, the iterative conception of set, and so on, is fine and swell, but, since there really is nothing more in such things than we put there, we quickly reach a point of diminishing returns, and our philosophising becomes essentially arbitrary, unconstrained by anything we actually find in set theory. Maddy's work is refreshing precisely because she takes into account the mathematical content of set theory (as a highly fertile and advanced field of mathematics), the intellectual concerns of set theorists, the attitudes and ideas we can discern in their work, etc. reducing very effectively this arbitrariness in philosophical theoretising.
WM <mueck...@rz.fh-augsburg.de> wrote: > On 24 Jul., 17:56, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote: > > FredJeffries <FredJeffr...@gmail.com> writes: > > > From the abstract: "Our work is independent of Wette's since we have > > > failed to understand the details of his work"
> > This is not surprising. I don't think anyone's ever understood the > > details of Wette's proof.
> That's why you don't find it as boring as proofs that can be > understood by shoemakers? > Here is one of those simple proof of the inconsistency of ZF..
> A) Take {1} union it with {2}, union the result with {3}, continue > without end. The result is N = omega.
> B) Take {1} union it with {2}, remove {1}, union the result {2} with > {3}, remove {2}, union {2} with {1}, union {3} with {4}, remove {3}, > union {3} with {1, 2}, ... > union {n} with {n+1}, remove {n}, union {n} with {1, 2, 3, ..., n-1} > continue without end.
> > The result is the empty set.
> According to set theory, yes. But the real result is a set that never > is empty.
WRONG! One has an infinite sequence of all different non-empty sets whose only reasonable limit set is the empty set, provided one claims any limit exists at all.
> It has at least one element. This can also be shown by set > theory, in the following way: Renumbering (for n in N) 2n+1 by o and > 2n by e, and simultaneously executing both, adding n+1 and > subtracting > n-1, we get a set that always contains the two single elements o and > e.
Only in WM's weird world can a set change members but remain unchanged.
WM is actually dealing with infinitely many different sets, not just one set.
> Therefore its cardinality is 2.
Which of those infinite many different sets is "it"? And why aren't any of the others "it"?
> Set theory shows 0 = 2.
WM's perversion of set theory, being self-contradictory, shows everything to be both true and false.
MoeBlee <jazzm...@hotmail.com> writes: > Perhaps that evidence is not the best basis for belief that ZF is > consistent, but I don't see that it is so negligible as you deem.
But just how are we to assess the significance of evidence of this sort? This is a pet peeve, so trotting out a few hobbyhorses I'll present some thoughts.
With an eye on the meaning of it all in the grand scheme of things, it is a pertinent observation that consistency is a piddling correctness condition. Presumably we're not interested in consistency of ZF out of idle curiosity but because we want to use principles codified in ZF to prove mathematical stuff. But even if there are only finitely many twin primes it is entirely possible for a consistent theory to prove "there are infinitely many twin primes". We need stronger correctness guarantees, e.g. 1-consistency, arithmetical soundness, etc. And even if we accede that we have some good inductive evidence for the consistency of ZF certainly no one would claim to have any inductive evidence for the 1-consistency or arithmetical soundness of ZF -- what would that even mean?
But of course strictly speaking you didn't really say anything about the grounds on which we accept formal proofs in ZF as establishing their conclusions as true. So let's set that aside and consider the very popular idea that whatever confidence we have in the consistency of ZF stems from no one having yet found a contradiction. I submit that this is but a piece of silliness of fantastic absurdity and near incoherence. That is, I submit both that it's a piss poor reason to think ZF is consistent, and that whatever philosophical proclamations mathematicians may make their actual behaviour, their actual reasoning, doesn't at all support the view this is, in fact, what they base their confidence in the consistency of ZF in.
On the first point, the obvious observation is simply that we don't usually regard our having verified that all naturals up to, say, 10^10^10 have some decidable property P good evidence that P holds of all naturals. (Consider for example the predicate "x <= 10^10^10".) This observation can be further bolstered by noting that since consistency is undecidable the length of the shortest proof of a contradiction in an inconsistent theory can't be effectively bounded as a function of the length of (some canonical finite description of) the axioms. If we are to take the inductive evidence for consistency seriously it must be explained what in the predicate "x is not a proof of a contradiction in ZF" makes the inference from "there is no proof of contradiction in ZF of length < n" (for some presumably big n) to "there is no proof proof of contradiction in ZF" plausible. Alas, those who cite the failure of mathematicians to find a contradiction in ZF as good reason to think that ZF is consistent never do offer any such explanation. (This is of course not surprising since in general we have no idea how to gauge the plausibility of such inferences.)
Those who purport to conclude ("tentatively" etc.) that ZF is consistent on basis of inductive evidence not infrequently explain the nature of this evidence by claiming that "thousands of mathematicians have tried to find a contradiction in ZF", "ZF is thoroughly tested for contradictions, by having lots of people, including highly competent mathematicians and logicians, trying to find contradictions and failing" and so on. These claims are of course simply false. Very few competent and sane people have looked for contradictions in ZF. This is not just obnoxious pedantry. Whatever inductive evidence we have for consistency of ZF does not consist in any failed attempts to find a contradiction but rather in mathematicians proving stuff in ordinary mathematics using principles formalisable in ZF(C) (and failing to prove 0=1). But in such proofs the full power of ZF is virtually never invoked (this applies to proofs of set theoretic results as well). Or in other words, we /do/ have good inductive evidence that there is no easily discoverable proof of contradiction using, in the limited ways these principles are usually used in ordinary mathematics, principles formalisable in a weak fragment of ZF(C); but concluding from this we have good inductive evidence for consistency of ZF is arbitrary -- we might with equal justification, that is, with not much justification at all, say we have good inductive evidence that ZF + "there is a proper class of measurable cardinals" or ZF + the twin prime conjecture are consistent.
So much for the evidence being poor piss evidence. That the actual practice and reasoning of mathematicians does not square at all with the view that the inductive evidence is a good reason to think ZF is consistent we can see by reflecting on observations such as the following: If the fact that people have failed to prove 0=1 in ZF is good evidence for consistency of ZF, we have even better evidence that all famous open problems are undecidable (in ZF, in PA, ...). But very few people would be willing to accept such unpalatable conclusions. And so on. I'm confident you can come up with any number of similar observations yourself.
(Jesse's comment was about people working in /PA/ having failed to prove 0=1 being evidence that /PA/ is consistent . This is an even more bizarre notion than the standard silly "no one has yet run into a contradiction" argument for consistency of ZF. I will subject him to some exacting and acerbic pedantry on this later on.)
> Some proofs are very difficult to find, yes, but I still tend to think > that the fact no such proof has been found thus far is some evidence.
Yes, but piss poor evidence. Saying that we have "evidence that PA is consistent" is a bit odd -- we can prove that PA is consistent. That aside, the fact that no inconsistency proof has been found is a good reason to think PA is consistent only if we have good reason to think an inconsistency proof must be sufficiently simple (allowing abbreviations etc.) for us to comprehend and produce it. Do we have any such reason? Why shouldn't the inconsistency proof involve some torturous and inhumanely complicated machinations, invoking an incomprehensible instance of Pi - 10^10^10^10 + 421438543762346 induction?
"Jesse F. Hughes" wrote: > If that's sufficient to be like religion, then I agree that believing > in the consistency of PA is like religious faith.
> On the other hand, the fact is that lots of folks *work* in PA, and > over time, the fact that no inconsistency proof has been found gives > some evidence that PA is consistent. In this way, mathematics is a > bit like empirical sciences (but not too much!).
Since PA has been proved consistent, these two points don't seem very relevant to me.
-- Which of the seven heavens / Was responsible her smile / Wouldn't be sure but attested / That, whoever it was, a god / Worth kneeling-to for a while / Had tabernacled and rested.
> MoeBlee <jazzm...@hotmail.com> writes: > > Perhaps that evidence is not the best basis for belief that ZF is > > consistent, but I don't see that it is so negligible as you deem.
> But just how are we to assess the significance of evidence of > this sort? This is a pet peeve, so trotting out a few hobbyhorses I'll > present some thoughts.
> With an eye on the meaning of it all in the grand scheme of > things, it is a pertinent observation that consistency is a piddling > correctness condition. Presumably we're not interested in consistency of > ZF out of idle curiosity but because we want to use principles codified > in ZF to prove mathematical stuff. But even if there are only finitely > many twin primes it is entirely possible for a consistent theory to > prove "there are infinitely many twin primes". We need stronger > correctness guarantees, e.g. 1-consistency, arithmetical soundness, > etc. And even if we accede that we have some good inductive evidence for > the consistency of ZF certainly no one would claim to have any inductive > evidence for the 1-consistency or arithmetical soundness of ZF -- what > would that even mean?
> But of course strictly speaking you didn't really say anything > about the grounds on which we accept formal proofs in ZF as establishing > their conclusions as true. So let's set that aside and consider the very > popular idea that whatever confidence we have in the consistency of ZF > stems from no one having yet found a contradiction. I submit that this > is but a piece of silliness of fantastic absurdity and near > incoherence. That is, I submit both that it's a piss poor reason to > think ZF is consistent, and that whatever philosophical proclamations > mathematicians may make their actual behaviour, their actual reasoning, > doesn't at all support the view this is, in fact, what they base their > confidence in the consistency of ZF in.
> On the first point, the obvious observation is simply that we > don't usually regard our having verified that all naturals up to, say, > 10^10^10 have some decidable property P good evidence that P holds of > all naturals. (Consider for example the predicate "x <= 10^10^10".) This > observation can be further bolstered by noting that since consistency is > undecidable the length of the shortest proof of a contradiction in an > inconsistent theory can't be effectively bounded as a function of the > length of (some canonical finite description of) the axioms. If we are > to take the inductive evidence for consistency seriously it must be > explained what in the predicate "x is not a proof of a contradiction in > ZF" makes the inference from "there is no proof of contradiction in ZF > of length < n" (for some presumably big n) to "there is no proof proof > of contradiction in ZF" plausible. Alas, those who cite the failure of > mathematicians to find a contradiction in ZF as good reason to think > that ZF is consistent never do offer any such explanation. (This is of > course not surprising since in general we have no idea how to gauge the > plausibility of such inferences.)
> Those who purport to conclude ("tentatively" etc.) that ZF is > consistent on basis of inductive evidence not infrequently explain the > nature of this evidence by claiming that "thousands of mathematicians > have tried to find a contradiction in ZF", "ZF is thoroughly tested for > contradictions, by having lots of people, including highly competent > mathematicians and logicians, trying to find contradictions and failing" > and so on. These claims are of course simply false. Very few competent > and sane people have looked for contradictions in ZF. This is not just > obnoxious pedantry. Whatever inductive evidence we have for consistency > of ZF does not consist in any failed attempts to find a contradiction > but rather in mathematicians proving stuff in ordinary mathematics using > principles formalisable in ZF(C) (and failing to prove 0=1). But in such > proofs the full power of ZF is virtually never invoked (this applies to > proofs of set theoretic results as well). Or in other words, we /do/ > have good inductive evidence that there is no easily discoverable proof > of contradiction using, in the limited ways these principles are usually > used in ordinary mathematics, principles formalisable in a weak fragment > of ZF(C); but concluding from this we have good inductive evidence for > consistency of ZF is arbitrary -- we might with equal justification, > that is, with not much justification at all, say we have good inductive > evidence that ZF + "there is a proper class of measurable cardinals" or > ZF + the twin prime conjecture are consistent.
> So much for the evidence being poor piss evidence. That the > actual practice and reasoning of mathematicians does not square at all > with the view that the inductive evidence is a good reason to think ZF > is consistent we can see by reflecting on observations such as the > following: If the fact that people have failed to prove 0=1 in ZF is > good evidence for consistency of ZF, we have even better evidence that > all famous open problems are undecidable (in ZF, in PA, ...). But very > few people would be willing to accept such unpalatable conclusions. And > so on. I'm confident you can come up with any number of similar > observations yourself.
> (Jesse's comment was about people working in /PA/ having failed > to prove 0=1 being evidence that /PA/ is consistent . This is an even > more bizarre notion than the standard silly "no one has yet run into a > contradiction" argument for consistency of ZF. I will subject him to > some exacting and acerbic pedantry on this later on.)
> "Wovon mann nicht sprechen kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
About the twin prime conjecture, as to whether there are finitely or infinitely many twin primes, it seems clear that there are infinitely many twin primes (i.e. primes c-1 and c+1 for some composite c), but a notion might be interpreted that for various considerations of the integers, there are too few to apply arguments that there are infinitely many, compared to the integers altogether, when their density diminishes to that of sets that are finite. Maybe to approach showing that there are infinitely many twin primes it would be easiest to start with something like the factorials, observing that either one less or one greater is always prime, showing that for infinitely many of those that there are twin primes, but that is a simplistic approach little better than brute force. (Like number theory with a prime at infinity or composite at infinity, Euclidean geometry with or without the parallel postulate, different approaches and applications might refine appropriate definition, here about varying definitions of finite.)
Some Platonists see necessary objects of the domain of discourse that aren't representable in ZF (for example in Cantor's domain principle, he expects a universal object in his set theory.) It's interesting that you defer mention of methods of proof construction in terms of model theory vis-a-vis classical methods, where the mechanics of model theory are outside the domain of discourse, it seems.
I agree that if there is a contradiction (or paradox) discoverable in ZF (or ZFC where they are co-consistent), that it should be pretty simple to express and not involve arbitrary convolutions of constants (or rather it should so framed instead of "10^10^10 is too big"). That reminds me of Goedel's "magic 85" or the 2700 steps of Chaitin and so on, Friedman's "epsilon chains of length 15". 10^^10 + 1 is a large finite natural integer.
Then, finding an internal consistency of ZF might see it necessary to take into account encompassing ZF's framework within a suitable framework of discussion and application, and for some Platonists that acknowledges the existence of mathematical objects that ZF deigns don't exist (those basically being the notional roots of paradox as to why ZF was (rather ZF's sets were) so formalized as well-founded and with a well-founded infinity that is a completion, completed infinity, when ZF is not completed (as it has no universe, except discourse broadly sweeps its objects together).
> "Jesse F. Hughes" wrote: > > If that's sufficient to be like religion, then I agree that believing > > in the consistency of PA is like religious faith.
> > On the other hand, the fact is that lots of folks *work* in PA, and > > over time, the fact that no inconsistency proof has been found gives > > some evidence that PA is consistent. In this way, mathematics is a > > bit like empirical sciences (but not too much!).
> Since PA has been proved consistent, these two points don't seem very > relevant to me.
> -- > Which of the seven heavens / Was responsible her smile / > Wouldn't be sure but attested / That, whoever it was, a god / > Worth kneeling-to for a while / Had tabernacled and rested.
Here by PA is indicated Peano Arithmetic, and while there is consistency of its axioms where zero is a natural integer and its successors natural integers and addition and multiplication are the normal operations, that there is closure of those operations for those inputs, it's incomplete, which Goedel tells us means that there are true features about the objects of the theory that are not theorems of the theory. Now, by no means would I want to represent these notions of other authors but for example modernly with Paris and Kirby's implication via incompleteness of a nonstandard countable natural integers (not uncountable hyperintegers), or along the lines of Boucher's F as discussed on sci.logic, there's a big point at infinity in the natural integers from the small axiomatics, not just because of classical intuition or application by number theorists.
Aatu Koskensilta <aatu.koskensi...@uta.fi> writes: > "Jesse F. Hughes" <je...@phiwumbda.org> writes:
>> Some proofs are very difficult to find, yes, but I still tend to think >> that the fact no such proof has been found thus far is some evidence.
> Yes, but piss poor evidence. Saying that we have "evidence that PA is > consistent" is a bit odd -- we can prove that PA is consistent. That > aside, the fact that no inconsistency proof has been found is a good > reason to think PA is consistent only if we have good reason to think an > inconsistency proof must be sufficiently simple (allowing abbreviations > etc.) for us to comprehend and produce it. Do we have any such reason? > Why shouldn't the inconsistency proof involve some torturous and > inhumanely complicated machinations, invoking an incomprehensible > instance of Pi - 10^10^10^10 + 421438543762346 induction?
Because it just doesn't.
Look, I know stuff.
-- Jesse F. Hughes "It's easy folks. Just talk about my approach to your favorite mathematician. If they can't be interested in it, they've demonstrated a lack of mathematical skill." -- James Harris
Aatu Koskensilta wrote: > Saying that we have "evidence that PA is > consistent" is a bit odd -- we can prove that PA is consistent.
Did you mean absolute consistency: no PA's syntactical theorem of the form F /\ ~F? How could one by syntactical means prove the consistency of a formal system T in general?
>> How could one by syntactical means prove the consistency of a formal >> system T in general?
> There is no general method. What a consistency proof, if such can be > had, for a formal theory T looks like necessarily depends on the details > of T.
> On Jul 26, 3:00 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
> > MoeBlee <jazzm...@hotmail.com> writes: > > > Perhaps that evidence is not the best basis for belief that ZF is > > > consistent, but I don't see that it is so negligible as you deem.
> > But just how are we to assess the significance of evidence of > > this sort? This is a pet peeve, so trotting out a few hobbyhorses I'll > > present some thoughts.
> > With an eye on the meaning of it all in the grand scheme of > > things, it is a pertinent observation that consistency is a piddling > > correctness condition. Presumably we're not interested in consistency of > > ZF out of idle curiosity but because we want to use principles codified > > in ZF to prove mathematical stuff. But even if there are only finitely > > many twin primes it is entirely possible for a consistent theory to > > prove "there are infinitely many twin primes". We need stronger > > correctness guarantees, e.g. 1-consistency, arithmetical soundness, > > etc. And even if we accede that we have some good inductive evidence for > > the consistency of ZF certainly no one would claim to have any inductive > > evidence for the 1-consistency or arithmetical soundness of ZF -- what > > would that even mean?
> > But of course strictly speaking you didn't really say anything > > about the grounds on which we accept formal proofs in ZF as establishing > > their conclusions as true. So let's set that aside and consider the very > > popular idea that whatever confidence we have in the consistency of ZF > > stems from no one having yet found a contradiction. I submit that this > > is but a piece of silliness of fantastic absurdity and near > > incoherence. That is, I submit both that it's a piss poor reason to > > think ZF is consistent, and that whatever philosophical proclamations > > mathematicians may make their actual behaviour, their actual reasoning, > > doesn't at all support the view this is, in fact, what they base their > > confidence in the consistency of ZF in.
> > On the first point, the obvious observation is simply that we > > don't usually regard our having verified that all naturals up to, say, > > 10^10^10 have some decidable property P good evidence that P holds of > > all naturals. (Consider for example the predicate "x <= 10^10^10".) This > > observation can be further bolstered by noting that since consistency is > > undecidable the length of the shortest proof of a contradiction in an > > inconsistent theory can't be effectively bounded as a function of the > > length of (some canonical finite description of) the axioms. If we are > > to take the inductive evidence for consistency seriously it must be > > explained what in the predicate "x is not a proof of a contradiction in > > ZF" makes the inference from "there is no proof of contradiction in ZF > > of length < n" (for some presumably big n) to "there is no proof proof > > of contradiction in ZF" plausible. Alas, those who cite the failure of > > mathematicians to find a contradiction in ZF as good reason to think > > that ZF is consistent never do offer any such explanation. (This is of > > course not surprising since in general we have no idea how to gauge the > > plausibility of such inferences.)
> > Those who purport to conclude ("tentatively" etc.) that ZF is > > consistent on basis of inductive evidence not infrequently explain the > > nature of this evidence by claiming that "thousands of mathematicians > > have tried to find a contradiction in ZF", "ZF is thoroughly tested for > > contradictions, by having lots of people, including highly competent > > mathematicians and logicians, trying to find contradictions and failing" > > and so on. These claims are of course simply false. Very few competent > > and sane people have looked for contradictions in ZF. This is not just > > obnoxious pedantry. Whatever inductive evidence we have for consistency > > of ZF does not consist in any failed attempts to find a contradiction > > but rather in mathematicians proving stuff in ordinary mathematics using > > principles formalisable in ZF(C) (and failing to prove 0=1). But in such > > proofs the full power of ZF is virtually never invoked (this applies to > > proofs of set theoretic results as well). Or in other words, we /do/ > > have good inductive evidence that there is no easily discoverable proof > > of contradiction using, in the limited ways these principles are usually > > used in ordinary mathematics, principles formalisable in a weak fragment > > of ZF(C); but concluding from this we have good inductive evidence for > > consistency of ZF is arbitrary -- we might with equal justification, > > that is, with not much justification at all, say we have good inductive > > evidence that ZF + "there is a proper class of measurable cardinals" or > > ZF + the twin prime conjecture are consistent.
> > So much for the evidence being poor piss evidence. That the > > actual practice and reasoning of mathematicians does not square at all > > with the view that the inductive evidence is a good reason to think ZF > > is consistent we can see by reflecting on observations such as the > > following: If the fact that people have failed to prove 0=1 in ZF is > > good evidence for consistency of ZF, we have even better evidence that > > all famous open problems are undecidable (in ZF, in PA, ...). But very > > few people would be willing to accept such unpalatable conclusions. And > > so on. I'm confident you can come up with any number of similar > > observations yourself.
> > (Jesse's comment was about people working in /PA/ having failed > > to prove 0=1 being evidence that /PA/ is consistent . This is an even > > more bizarre notion than the standard silly "no one has yet run into a > > contradiction" argument for consistency of ZF. I will subject him to > > some exacting and acerbic pedantry on this later on.)
> > "Wovon mann nicht sprechen kann, darüber muss man schweigen" > > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
> About the twin prime conjecture, as to whether there are finitely or > infinitely many twin primes, it seems clear that there are infinitely > many twin primes (i.e. primes c-1 and c+1 for some composite c), but a > notion might be interpreted that for various considerations of the > integers, there are too few to apply arguments that there are > infinitely many, compared to the integers altogether, when their > density diminishes to that of sets that are finite. Maybe to approach > showing that there are infinitely many twin primes it would be easiest > to start with something like the factorials, observing that either one > less or one greater is always prime, showing that for infinitely many > of those that there are twin primes, but that is a simplistic approach > little better than brute force. (Like number theory with a prime at > infinity or composite at infinity, Euclidean geometry with or without > the parallel postulate, different approaches and applications might > refine appropriate definition, here about varying definitions of > finite.)
> Some Platonists see necessary objects of the domain of discourse that > aren't representable in ZF (for example in Cantor's domain principle, > he expects a universal object in his set theory.) It's interesting > that you defer mention of methods of proof construction in terms of > model theory vis-a-vis classical methods, where the mechanics of model > theory are outside the domain of discourse, it seems.
> I agree that if there is a contradiction (or paradox) discoverable in > ZF (or ZFC where they are co-consistent), that it should be pretty > simple to express and not involve arbitrary convolutions of constants > (or rather it should so framed instead of "10^10^10 is too big"). > That reminds me of Goedel's "magic 85" or the 2700 steps of Chaitin > and so on, Friedman's "epsilon chains of length 15". 10^^10 + 1 is a > large finite natural integer.
> Then, finding an internal consistency of ZF might see it necessary to > take into account encompassing ZF's framework within a suitable > framework of discussion and application, and for some Platonists that > acknowledges the existence of mathematical objects that ZF deigns > don't exist (those basically being the notional roots of paradox as to > why ZF was (rather ZF's sets were) so formalized as well-founded and > with a well-founded infinity that is a completion, completed infinity, > when ZF is not completed (as it has no universe, except discourse > broadly sweeps its objects together).
Eh, about twin primes it is false that one more or less than a factorial, or a product of successive primes for that manner, is prime, although there are "more" primes among those than of integers at uniform random. The factorizations of those composites then are of course of elements larger in value than those comprising the unique prime factorization. Making statements about the distributions of subsets of the integers not dense in them, besides generally about asymptotic density, was at one point roundly castigated for example on sci.math, yet now it's accepted that is a natural usage of those terms. That's an example of, if not a game-changer like "ZF is inconsistent", at least "half of the integers are even." Basically at some point objections to that stopped.
Besides "philosophical objections to Cantor's theory" and that proof- theoretic statements around ZF involve its universe of objects, there are as well applicable axiomatizations of number systems representing for a Platonist truisms that aren't theorems of ZF.
On Jul 24, 9:58 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:
> Marshall says...
> >Mostly, though, I don't think "unsensible" is justly applied to > >Virgil.
> I'm guessing that Aatu meant the way that Virgil responds to trolls > and crackpots, inadvertently encouraging them. I doubt that he is > talking about Virgil's mathematics abilities.
Bleah. My behavior was terrible. Apologies to everyone who had to see my peevishness, and especially to Aatu.
> >> Then how did you go about it when you said "we can prove that PA is > >> consistent"?
> > I tappity-tapped the keys on my laptop.
> Great!
See the Notes at the end of Chapter 10 of Troelstra and Schwichtenberg's Basic Proof Theory.
-- Which of the seven heavens / Was responsible her smile / Wouldn't be sure but attested / That, whoever it was, a god / Worth kneeling-to for a while / Had tabernacled and rested.
On Jul 25, 12:00 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> Transfer Principle <lwal...@lausd.net> writes: > > WM already has enough posters to tell > > him how wrong he is and how he's a "crank," so why would I > > want to do that too? > I haven't demanded that you tell him how wrong he is. I've asked why > it is that you think others' willingness to tell him he's wrong is an > indication that they are intellectually dishonest [....] That is, they > would deny ZFC is inconsistent, even if a valid proof were found.
OK, think about it this way. We know that WM claims that ZFC is inconsistent, and some posters call him "crank." Now there are at least two reasons why they might do so (and once again, I present these without mentioning "-isms"):
1) WM's proof of ~Con(ZFC) is invalid. 2) They don't _want_ ZFC to be proved inconsistent.
Now someone who calls WM a "crank" is more likely than one who doesn't call him that to follow 1), and someone who calls WM a "crank" is more likely than one who doesn't call him that to follow 2). So the use of the word "crank" to describe WM is insufficient to distinguish between 1) and 2). There is insufficent data period to distinguish between 1) and 2). If someone gave a valid proof of ~Con(ZFC), we could distinguish between 1) and 2), as those who follow 2) would call the prover of ~Con(ZFC) a "crank," but those who follow 1) wouldn't.
But maybe there's a way to distinguish between 1) and 2), without actually having a proof of ~Con(ZFC), which might not appear for decades or centuries if ever. I know that I'll never have the brainpower to produce a proof of ~Con(ZFC), but what if I (or someone else) were to write what appears on the surface to be two proofs in ZFC, one of P, and one of ~P. But one of the proofs contains a flaw, but unlike previously flawed proofs posted on sci.math, the error isn't easy to spot in seconds, but one has to inspect both proofs for hours until the subtle error is found.
This might be able to distinguish between 1) and 2). Someone who follows 1) might check the proofs for errors before calling the poster a "crank," while a follower of 2) would just declare the poster a "crank" without checking the proof.
Of course, there are those who would immediately assume the proof is wrong if my name, or WM's name, or the name of someone already declared "crank" were attached to the proof, but would search the proof if the name of someone not called "crank" were attached. This is why, if I actually had what I thought was proofs of P and ~P -- even if I knew that the proofs were likely to have a subtle flaw -- I would post the proofs under a different name and full newsreader, in order to remove this possible source of bias.
So the answer to Hughes's question is that a "crank"-caller is more likely than a non-"crank"-caller to be "intellectually dishonest" and ignore all proofs that one's favorite set theory is inconsistent, but there's insufficient data to know who will actually be "intellectually dishonest." Until someone either produces a valid proof of ~Con(ZFC), or at least makes a more valiant effort than the ones called "cranks" to do so, we'll never know how sci.math posters will react.
On Jul 25, 7:09 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
> Transfer Principle <lwal...@lausd.net> writes: > > So if, for example, the Axiom of Infinity is the axiom that would > > cause the contradiction, as many believe (including both WM and > > Nelson), then the working mathematician would try to avoid the axiom > > (say, by sticking to finitistic mathematics) until the foundational > > mathematicians fixed the theory. > Finitism is a highly restrictive conception of mathematics. Recall that > on the finitist conception, statements of the form "there exists a > natural n such that ..." are not meaningful, that induction can only be > applied to primitive recursive predicates, and so on. It is extremely > unlikely that even if infinity was found to be problematic people would > fall back to finitism.
What if I were to restate the question ike this: What if someone proved that ZF is inconsistent, but ZF-Infinity avoided the contradiction found in ZF? Since ZF-Infinity is "highly restrictive," would someone keep working in ZF even though it would be inconsistent, because at least it's less restrictive than ZF-Infinity?
Furthermore, what if every single attempt to find a replacement axiom for Infinity also resulted in an inconsistent theory, so that the only way to avoid all the contradictions was ZF-Infinity? Would mathematicians use the inconsistent theory ZF anyway, because at least it's less restrictive than ZF-Infinity?
If so, then this is exactly the type of mathematician that I've been trying to describe all along. They obviously don't _want_ ZF to be inconsistent (because the alternative is too restrictive) and if ZF turned out to be inconsistent, they would continue using infinite sets anyway (because, once again, the alternative is too restrictive). They already call those like WM who don't like infinite sets "cranks," and would continue to do so even if ZF were found inconsistent.
Notice that not all "cranks" are like WM, who would like to avoid infinite sets altogether. Other "cranks" still want there to be infinite sets, but such that they don't work the same way that infinite sets in ZF work. It is possible that ZF is inconsistent yet a theory in which infinite sets work the way that these "cranks" desire might avoid the contradiction. (Remember the Katz theories from earlier this year?) If so, then we would have a theory which would avoid both the contradictions to be found in ZF _and_ the high restrictions of finitism.
But WM wants to avoid infinite sets altogether. He obviously doesn't care that finitism is highly restrictive, since to him, the counterintuitions of infinite sets outweighs the high restrictions of the lack thereof.
On Jul 25, 7:31 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
> Chris Menzel <cmen...@remove-this.tamu.edu> writes: > > She is a philosopher of math who knows a *lot* of set theory. To my > > knowledge, she has authored only one more or less purely technical > > paper ("Proper Classes", JSL 1983), but many of her papers assume a > > pretty deep understanding of contemporary set theory. > If we regard /Proper Classes/ a "purely technical paper" then surely > /V=L and Maximize/ also counts?
Ah yes, "V=L and Maximize." I've mentioned this in previous posts as well.
ZF+"V=L" is a simple theory. We know that it proves CH, so that c=aleph_1. ZF+"Maximize," on the other hand, is a more complex theory. Not only does it prove ~CH, but in this theory, c is very large. (I'm not quite sure what it means to "maximize" c, since if there's a model of ZF in which c=kappa for some cardinal kappa, then there's another model in which c=kappa+. Still, I know that c isn't supposed to be, say aleph_2, in ZF+"Maximize" -- it's intended to be very large.)
We know that most set theorists prefer the theory ZF+"Maximize" to ZF+"V=L." Why? IIRC, Maddy tells us that most set theorists find ZF+"V=L" too restrictive -- the same reason Aatu gives for ZF itself being preferable to ZF-Infinity, the latter being too restrictive.
Of course, there's a difference between the restrictive theories ZF+"V=L" and ZF-Infinity. We know that if ZF is consistent, then so are both ZF+"V=L" (Goedel) and ZF+"Maximize" (Cohen forcing). But it's possible for ZF to be inconsistent without ZF-Infinity also being inconsistent, so one might choose ZF-Infinity if ZF turns out to be inconsistent, even though it's restrictive. But one can never choose the restrictive ZF+"V=L" over the more permissive ZF+"Maximize" for reasons due to consistency, since if one's inconsistent so is the other.
I often mention ZF+"V=L" and ZF+"Maximize" when discussing Platonism, especially when mentioning one's own conception of sets. If a Platonist can choose ZF+"Maximize" over ZF+"V=L" because the former is a better match for one's own conception, then why can't another Platonist choose ZF-Infinity over ZF, if he or she believes that the former is a better match?
But due to all this confusion about what a Platonist even is, I'll no longer discuss the relationship between Platonism and the potential inconsistency of ZF here.
On Jul 27, 1:51 pm, Transfer Principle <lwal...@lausd.net> wrote:
> OK, think about it this way. We know that WM claims that > ZFC is inconsistent, and some posters call him "crank." Now > there are at least two reasons why they might do so (and > once again, I present these without mentioning "-isms"):
> 1) WM's proof of ~Con(ZFC) is invalid.
Not ONLY that.
> 2) They don't _want_ ZFC to be proved inconsistent.
> Now someone who calls WM a "crank" is more likely than one > who doesn't call him that to follow 1), and someone who calls > WM a "crank" is more likely than one who doesn't call him that > to follow 2). So the use of the word "crank" to describe WM is > insufficient to distinguish between 1) and 2).
Or you could just see what people actually SAY as to why they call him a crank.
> There is insufficent > data period to distinguish between 1) and 2). If someone gave a > valid proof of ~Con(ZFC), we could distinguish between 1) and > 2), as those who follow 2) would call the prover of ~Con(ZFC) a > "crank," but those who follow 1) wouldn't.
> But maybe there's a way to distinguish between 1) and 2), > without actually having a proof of ~Con(ZFC), which might not > appear for decades or centuries if ever. I know that I'll never > have the brainpower to produce a proof of ~Con(ZFC), but > what if I (or someone else) were to write what appears on the > surface to be two proofs in ZFC, one of P, and one of ~P. But > one of the proofs contains a flaw, but unlike previously flawed > proofs posted on sci.math, the error isn't easy to spot in > seconds, but one has to inspect both proofs for hours until > the subtle error is found.
> This might be able to distinguish between 1) and 2). Someone > who follows 1) might check the proofs for errors before calling > the poster a "crank," while a follower of 2) would just declare > the poster a "crank" without checking the proof.
> Of course, there are those who would immediately assume > the proof is wrong if my name, or WM's name, or the name of > someone already declared "crank" were attached to the proof, > but would search the proof if the name of someone not called > "crank" were attached. This is why, if I actually had what I > thought was proofs of P and ~P -- even if I knew that the proofs > were likely to have a subtle flaw -- I would post the proofs under > a different name and full newsreader, in order to remove this > possible source of bias.
> So the answer to Hughes's question is that a "crank"-caller > is more likely than a non-"crank"-caller to be "intellectually > dishonest" and ignore all proofs that one's favorite set theory > is inconsistent,
WHAT?! Your "reasoning" toward that conclusion is RIDICULOUS.
> but there's insufficient data to know who will > actually be "intellectually dishonest." Until someone either > produces a valid proof of ~Con(ZFC), or at least makes a more > valiant effort than the ones called "cranks" to do so, we'll never > know how sci.math posters will react.
On Jul 27, 2:56 pm, MoeBlee <jazzm...@hotmail.com> wrote:
> On Jul 27, 1:51 pm, Transfer Principle <lwal...@lausd.net> wrote: > > Now someone who calls WM a "crank" is more likely than one > > who doesn't call him that to follow 1), and someone who calls > > WM a "crank" is more likely than one who doesn't call him that > > to follow 2). So the use of the word "crank" to describe WM is > > insufficient to distinguish between 1) and 2). > Or you could just see what people actually SAY as to why they call him > a crank.
OK, then. Let's see what MoeBlee wrote last week:
> So why don't people (some who go on and on for years and DECADES, over > even THOUSANDS and THOUSANDS of posts) making uninformed, misinformed, > mal-informed, illogical, often incoherent, dogmatic, unresponsive > arguments for various propositions, some of which are plainly untrue > and some just unproven (and some, for their incoherence, not even > propositoins) not deserve to be called 'cranks'?
A proof that "uninformed" or "illogical" would fall under my 1) -- the proof is invalid. I ignore the word "dogmatic," since just as MoeBlee considers "cranks" to be "dogmatic," the "cranks" consider MoeBlee and those who agree with him to be "dogmatic," so that cancels out. "Unresponsive" -- a bit odd that MoeBlee would call "cranks" "unresponsive" since they seem to respond all the time -- maybe just not in the way he would like them too. "Not even" propositions sounds like the "not even true" phrase I've mentioned before.
This is why MoeBlee calls WM a "crank." Now if MoeBlee were the only poster who ever called WM a "crank," we'd be all set, but unfortunately other posters besides MoeBlee call him a "crank," and I still don't know why they call him a "crank," or what percentage of posters who call him a "crank" would agree with MoeBlee's reason for doing so.
Now here MoeBlee asks, do such posters deserve to be called "cranks?" Maybe -- but I won't, unless I add scare quotes and/or the "so-called" qualifier. My argument was that those who would call them "cranks" then would follow path 2) above deserve the label _more_, and if this scenario ever occurs, I'll drop all scare quotes with my labels.
> > So the answer to Hughes's question is that a "crank"-caller > > is more likely than a non-"crank"-caller to be "intellectually > > dishonest" and ignore all proofs that one's favorite set theory > > is inconsistent, > WHAT?! Your "reasoning" toward that conclusion is RIDICULOUS.
The claim here is that suppose there are two posters, one who calls WM a "crank," and one who doesn't. If we know nothing else about the two posters, which of them is more likely to defend ZFC even if proved inconsistent? This is a question about _probability_, not definiteness. We clearly can't conclude from the label "crank" alone who _will_ ignore a proof of ~Con(ZFC). I'm only asking who is more _likely_ to do so.
WM calls that ZFC is inconsistent. There is a nonzero probability that someone who calls WM a "crank" will call everyone who claims ZFC inconsistent a "crank" (and would therefore ignore the proof that ZFC is inconsistent.) It doesn't matter how small this probability is -- as long as it is nonzero, it's enough to raise the probability that a "crank"-caller will ignore the proof of ~Con(ZFC) above that of the non-"crank"-caller -- which is all I claimed.
In particular, I never claimed that _MoeBlee_ would ignore a valid proof of ~Con(ZFC). As this and other posts suggest, MoeBlee would accept a _valid_ proof of ~Con(ZFC) and attempt to find a theory which would avoid the contradiction, and he calls WM a "crank" not merely because he claims ~Con(ZFC), but because WM's proof is _wrong_, among other reasons (since he wrote "Not ONLY that").
But there's no reason to assume that MoeBlee's reaction to a proof of ~Con(ZFC) is representative of most sci.math posters' reaction to such a proof. I believe that some posters will have a reaction similar to the one that MoeBlee describes and others will have a reaction similar to what I fear -- and that those who will have the reaction I fear are among (i.e., a proper subset of the set of) posters who currently call WM a "crank" right now.
