0:06 [Music]
0:13 university of new york and i want to
0:14 tell you that
0:17 professor priest is considered one of
0:20 the most important philosophers recently
0:25 he has published many books especially
0:28 on topics concerning philosophical logic
0:31 and it's metaphysical and ontological
0:35 implications so professor priest
0:38 published books like towads non-being
0:41 one doubt truth to be alive in
0:44 introduction to classical logic logic a
0:48 very short introduction won many many
0:50 interesting and important books for
0:56 contemporary research on logic so he
0:59 gave ready to talks in our local seminar
1:01 logic in the airplane logic in the
1:04 airplane and it's a very important
1:08 occasion for always here to have his
1:11 presentation today on good of theorem
1:13 and para consistent so professor please
1:15 be welcome again here at the university
1:19 of Brasilia and invite you to come to
1:27 [Applause]
1:31 okay so I have to use this right so I
1:36 feel like an old rock star how we do how
1:39 do we do the translation we don't that's
1:42 makes things easy okay
1:46 and if I walk around is that okay can you
1:46 you
1:50 oh okay
1:55 can I move while I talk yeah okay well
1:58 first of all let me thank the organizers
2:00 very much for putting on this event
2:01 which is terrific
2:03 this this is a great auditorium I've
2:06 never been in anything like it so this
2:07 is fantastic
2:09 so thank you to all the people who've
2:10 made this possible and that means that
2:12 people have coughed up their money as
2:16 well as organized so I want to talk
2:18 about girdle's theorem this is the first
2:22 incompleteness theorem I'm afraid I'm
2:25 not familiar with professors or cadre
2:27 and V mean so I can't comment on how
2:29 this relates to their work I'm happy to
2:32 leave that to others to do because
2:34 girdles theorem is certainly very
2:37 important about how we reason in
2:40 particular how we reason in mathematics
2:46 and Gerda published his theorem in 1931
2:51 and I think undoubtably it's one of the
2:53 most profound results in mathematics in
2:57 the 20th century unfortunately no one
3:00 can agree on exactly what the importance
3:04 is philosophically and so I want to talk
3:07 about part of that debate power
3:10 consistent logic that is logics in which
3:12 you can have contradictions without
3:15 explosion so contradictions are kind of limited
3:16 limited
3:18 they don't go everywhere Pera consistent
3:21 logic puts a whole new light on girdle
3:23 serum which I guess most people are not
3:26 familiar with so what I want to do in
3:30 this talk is talk about the relevance of
3:33 power consistent logic for girdled serum
3:36 so talk is quite technical but let me
3:39 take it slowly and let's see what we can
3:41 do okay so that was the introduction
3:44 you've had that I want to talk about
3:48 inconsistent arithmetic and its
3:52 relevance to girdles theorem but to do
3:54 that I'll have to tell you a little bit
3:56 about girdle serum and about para
3:59 consistent logic because not everybody
4:01 knows this material so a lot of the time
4:04 I'm going to talk about this and then
4:08 we'll get on to inconsistent arithmetic
4:14 okay so first of all girdles theorem if
4:17 you open a textbook on mathematical
4:20 logic you will often find a statement of
4:23 girdles theorem is something like this
4:27 any axiomatic theory of arithmetic with
4:29 appropriate expressive capabilities is
4:32 incomplete okay that's not a good
4:36 statement this is the correct statement
4:39 the theory must be either
4:44 incomplete or inconsistent now if you're
4:47 working in classical logic if the theory
4:49 is inconsistent is trivial you can prove
4:52 everything so people can justifiably
4:57 ignore the inconsistent option but once
4:59 you move to a pair a consistent logic
5:01 this is no longer the case to assume is
5:04 that there's a formula with two free
5:08 variables the X Y which defines the
5:12 proof relation in T that is if n is the
5:16 code of a proof of a in T then B and
5:19 angle brackets a is true in the standard
5:21 model and if n is not the code of a
5:24 proof of int then it's negation is true
5:31 in the standard model third assumption
5:37 you can define probe Y as there is an X
5:39 such that X is a proof of Y then probe
5:42 is a prove predicate for T that is if
5:44 you can prove a that into you
5:49 prove that prof a and the fourth thing
5:51 you need is that there is a formula G of
5:55 the form not profit G there are some
5:57 subtleties here but I'll slide over them
6:00 okay so with those four assumptions you
6:03 can prove girdle's theorem so this is
6:04 how you do it
6:09 okay so pros you can prove G well then
6:11 you can prove not Prodigy because that
6:16 is G but if you can prove G then you can
6:18 prove property because profit proof
6:22 predicates so what we see then is that
6:32 okay now suppose that T is consistent
6:36 then you can't prove G we've just seen
6:39 that so no number is the code of a proof
6:45 of G so for any n the negation of B and
6:50 G is true in the standard model okay so
6:52 therefore the universal quantification
6:54 of that is true in the standard model
6:57 and we do a bit of logic chopping here
6:59 so that's equivalent to this so we need
7:02 a few quantify principles but that is
7:04 just the negation of prof G and that's G
7:09 so supposing that T is consistent G is
7:16 true in the standard model so you can't
7:18 prove G but G is true in the standard
7:21 model so the theory is incomplete
7:24 alright so if the theory is inconsistent
7:27 it's incomplete but by contradiction
7:32 if it's complete it's inconsistent so
7:34 this is a proof of girdled serum which
7:36 works and we've made very little
7:37 assumption about the logic a few
7:40 assumptions about how quantifiers behave
7:45 that's all okay so as I said most people
7:48 who talk about girdles theorem assume
7:50 you must be operating in a consistent
7:52 context but I want to examine the
7:56 inconsistent possibility so what we're
7:57 going to be doing is looking at ink
8:00 consistent arithmetics and what happens
8:02 to the girdle sentence in those
8:06 arithmetics now to do that you need to
8:08 understand a little bit about power
8:11 consistent logic and some of its motive
8:13 theory so I'm going to give you the
8:15 background that you need to understand
8:21 this okay so a guide to power consists
8:23 in logic there are of course many para
8:26 consistent logics and this country is
8:29 justifiably famous for its work in power
8:32 consistent logic I'm going to describe
8:35 one system of power consistent logic a
8:38 system called LP which is simple and
8:40 robust and will do what we want it to do
8:45 today okay so the language that we're
8:47 going to use is a standard first-order
8:52 language nothing an interpretation is
8:55 going to be a structure d which is the
8:58 domain of quantification and Delta which
9:01 is the denotation function so that tells
9:05 you the denotations of the constants the
9:07 function symbols and the per etiquettes
9:11 so d is a non empty domain of
9:14 quantification the denotation already
9:18 constant is a member of D if FN is an M
9:20 plays function simple then it's D
9:22 notation is a function from intervals in
9:25 the domain to D so this is all standard
9:28 classical logic this is where it's
9:31 different because in classical logic
9:34 well come back to that the denotation of
9:38 a predicate is a pair the predicate is
9:40 going to have an extension and an anti
9:43 extension so the extension are the
9:45 things of which the predicate is true
9:48 the anti extension is the things of
10:00 interpretation of a predicate is the
10:03 pair the extension and the ant extension
10:04 I'll write them like this with a plus
10:05 and a minus
10:09 such that the extension and the anti
10:11 extension exhaust all the intervals
10:14 Devane okay so in this logic there are
10:16 no truth value gaps you can't have
10:18 something that's neither true nor false
10:20 this essentially rules this out but you
10:22 can have things which are both true and
10:24 false because the extension and the
10:27 antique stench reddit can overlap and
10:31 the identity predicate is just it's what
10:32 you'd expect it's as you get in
10:40 classical logic okay so given
10:43 interpretation the next thing we need to
10:45 do is specify the truth and the falsity
10:48 conditions of course in classical logic
10:51 once you know the truth conditions then
10:54 the falsity conditions follow because
10:56 they're just falsity is the compliment
11:00 of truth but in a para consistent logic
11:03 truth and falsity are independent so you
11:04 have to give truth conditions and
11:10 falsity conditions so first of all we
11:12 finish off denotation so if you've got a
11:17 if you've got a term function symbol and
11:20 n terms what's it's denotation well it's
11:23 exactly the same as classical logic you
11:25 apply the denotation to function simple
11:27 the denotations of the terms everything
11:34 standard there the truth that's a little
11:36 plus here of an atomic sentence well
11:38 this sentence is true just if the enter
11:41 part of the denotations of the terms are
11:44 in the extension of the predicate and
11:48 for falsity that's the little - this
11:52 formula is false if the enter poll of
11:53 the denotations of the terms is in the
11:56 anti extension of the predicate and the
11:58 extension and the anti extension can
12:02 overlap so something can be both true
12:04 and false okay
12:07 the rest are is what you've more or less
12:10 expect so a negation is true if the
12:12 thing negated is false and it's false if
12:15 the thing negated is true a conjunction
12:17 is true if both contracts are true and
12:20 false if one or other conjunct is false
12:23 a disjunction is true if one or other
12:25 disjunct is true
12:28 and false if both disjunctive false and
12:31 note that the material conditional is
12:33 defined in the way in the same way as
12:35 usual so you don't need to give separate
12:37 truth and falsity conditions because it
12:40 you define mattoon implication as this
12:45 thing okay quantifiers there are many
12:47 ways to do the semantics of quantifiers
12:49 you can use satisfaction but I've always
12:52 thought that the easiest way to do
12:54 semantics of quantifiers is just to help
12:56 yourself to enough names to ensure that
12:59 everything in the domain has a name so
13:02 what we're going to do is extend the
13:08 language so if DS in the domain let's
13:13 add this new constant KD which refers to
13:18 D so we know now that every and that
13:21 allows us to give truth and falsity
13:24 conditions for quantifies so a
13:26 universally quantified sentence is true
13:28 if every objects in their main satisfies
13:32 it says and false if some object in the
13:36 main makes it false and then for the
13:38 particular quantifier you just do the
13:45 dual thing okay and now let's have a
13:50 definition of consequence so if a is a
13:54 model of the for being a just if the
13:57 structure makes a true fames in
14:01 classical logic a is a model of a set of
14:05 formulas Sigma just if it makes every
14:07 formula in Sigma true if it's a model of
14:11 every formula and the definition of
14:13 consequence if you've got a set of
14:17 formulas a set of formulas Sigma then a
14:19 is a consequence so that just if every
14:21 model of Sigma is a model of a so this
14:23 is standard stuff from the model theory
14:28 of classical logic okay so that is a
14:30 specification of the para consistent
14:35 logic LP now some facts about it
14:40 first of all it if the extension and the
14:42 anti extension of a predicate are
14:47 disjoint then this is exactly classical
14:51 logic because the only way that these
14:54 truth conditions and falsity conditions
14:57 actually differ from classical logic is
15:00 that they've come apart and if the
15:02 extension and the edge extension don't
15:07 overlap then this is exactly okay so
15:09 classical interpretations classical
15:12 models are special cases of LP
15:17 interpretations so every classical model
15:19 is an LP model but there are more LP
15:24 models so what that means is that if
15:26 something follows in LP it follows in
16:02 um well classical logic is negation
16:04 complete in the following sense every
16:06 formula is either true or false or not both
16:07 both
16:09 so we're not talking out theories we
16:11 just talked about interpretations at the
16:14 moment okay and in classical logic every
16:16 sentence is either true or false or not
16:21 both and if you insist that the
16:23 extension and anti extension every
16:26 predicate don't intersect that's exactly
16:29 what you'll get okay yeah thank you for
16:30 the question
16:32 please feel free to ask questions of
16:35 clarification because I know that this
16:37 stuff is probably unfamiliar to many of
16:39 you so if you want me to explain some
16:44 more please ask okay so any logical
16:46 consequence in LP is a classical logic
16:50 consequence but the converse does not
16:52 follow because this inference from a and
16:56 not A to B is valid in classical logic
16:59 but it's not valid in this logic because
17:00 a can be both thrown false
17:05 where B is just plain false so this
17:07 means that the logic is a pair a
17:13 consistent logic okay so the fact that
17:15 an inference is valid in classical logic
17:19 does not imply that it's valid in LP but
17:20 there's something kind of interesting
17:25 namely if you have no premises so
17:27 there's a special case in which Sigma is
17:33 empty then the two logics coincide so LP
17:36 and classical logic have exactly the
17:39 same logical truths what they do not
17:42 have is the same consequence relation
17:44 this is a reason why you cannot identify
17:47 logic with its set of logical truths
17:50 here are two logics with the same set of
17:51 logical truths
17:52 but with a different consequence
17:56 relation okay
18:00 now so that's the background impera
18:03 consistent logic and there's one more
18:04 thing that you need to know before we
18:07 can get to arithmetic and that is a
18:09 little bit about the model theory of
18:12 para consistent logic this para
18:15 consistent logic so I need to explain to
18:16 you something about collapsed
18:23 interpretations so let a be an LP
18:25 interpretation and then it then includes
18:27 classical interpretations as well
18:29 because as I've told you every classical
18:32 interpretation is an LP interpretation
18:34 what we're going to do essentially is
18:36 collapse that interpretation let me show
18:41 you how so Tilda is an equivalence
18:43 relation on the domain of a so an
18:59 if this is a domain an equivalence
19:01 relation is something which chunks the
19:04 domain into disjoint equivalence classes
19:06 okay that's what an equivalence relation
19:11 does so tilde is an equivalence relation
19:16 and moreover it's a congruence relation
19:18 on the denotations of their function
19:23 symbols now what that means is if you
19:26 take two things in the same chunk okay
19:29 and you apply the denotation of one of
19:30 the function symbols to them you're
19:33 gonna get things which are in the same
19:35 chunk okay
19:39 so applying the function to things in
19:40 the same chunk is always going to take
19:42 you to the same chunk so mathematicians
19:45 call this a congruence relation and you
19:48 need the equivalence relation to be a
19:56 congruence relation okay now notation if
19:59 D's in the domain I'm going to write the
20:01 equivalence class of D as D in square
20:04 brackets so if d is in the domain D in
20:09 square brackets is the whole chunk okay
20:11 so now I'm going to define the collapsed
20:16 interpretation so it has a domain and a
20:18 denotation function and the domain is
20:21 just going to comprise the equivalence
20:26 classes the chunks the denotation of
20:29 every constant is going to be the chunk
20:34 which the denotation of c was already in
20:38 the denotation of the function symbol is
20:39 just going to be where you take some
20:41 members of the chunks and you apply the
20:45 function to that and the chunk that
20:46 that's in that's when you need it to be
20:50 a congruence relation and okay these are
20:55 the interesting bits let me you can read
20:57 this for yourself let me just draw your
21:00 picture of what's going on what we're
21:02 gonna do essentially is we're going to
21:06 identify all the things in one chunk so
21:08 this chunk here
21:11 is going to satisfy predicate if
21:14 something or some things in this chunk
21:18 satisfied okay so what we're really
21:20 doing is identifying the things in a
21:25 trunk so suppose for example that Alex
21:27 is in the trunk and I'm in the trunk
21:36 so the trunk is Brazilian because alex
21:39 is Brazilian and it's Australian because
21:42 I'm Australian it's not Australian
21:44 because alex is not Australian and it's
21:47 not Brazilian because I'm not Brazilian
21:52 so all the things in a trunk and the
21:54 trunk is going to have all the
21:59 properties so collapsing together just
22:01 think things to produce any a very
22:06 inconsistent object okay that's
22:13 essentially what this says okay now this
22:15 is the important fact about claps
22:19 suppose you've got any LP interpretation
22:22 and you collapse it in the way that I've
22:27 told you then okay okay skip over that
22:30 this is the important thing if a is true
22:34 before you collapse it remains true
22:38 after you collapse and if it's false
22:40 before you collapse it's false after you
22:44 collapse so you never lose truth values
22:47 when you collapse you can get new stuff
22:49 new things can become true or false but
22:53 you can't lose old stuff and this is
22:55 important because it means that if you
22:59 start with a model of some theory when
23:00 you collapse you get a model of that
23:04 theory you might get a model of more but
23:06 everything true before you collapse is
23:11 true after you collapse okay so that's
23:14 the kind of background you need to know
23:16 to follow what I'm going to say about
23:21 arithmetic let me pause for a second
23:22 because we've been through
23:25 a lot of technical material but anyone
23:37 okay so that's the background now I want
23:41 to talk about arithmetic and what I'm
23:43 going to show you is that first of all
23:45 that there are inconsistent complete
23:49 arithmetics and then we're going to talk
23:50 about what happens to the girdle
23:53 sentence in those arithmetics and then
23:55 in the last few minutes I'll just
23:57 gesture at some of the philosophical
24:04 implications of this so let L be the
24:06 usual first-order language arithmetic
24:12 you find in any textbook let n boldface
24:15 n be the set of sentences true in the
24:18 standard model so this is what logicians
24:20 call the theory of the standard model
24:22 it's just the set of sentences true in
24:27 the standard model and factor M be any
24:30 model of n so it's any model that it
24:33 makes in all the sentences and then true
24:34 and we know there are lots of those
24:36 classically no there's one standard
24:38 model but we know there are lots of
24:42 non-standard classical models okay we're
24:43 going to produce now some non-standard
24:48 power consistent models so if you take
24:51 em and you collapse it and to get M
24:54 tilde then what you get when you
24:56 collapse is a model of N and that's just
24:59 a result of the collapsing lemma okay I
25:00 didn't prove the collapsing number for
25:04 you you can take my word for it it's not
25:06 difficult to prove but for today's
25:08 purposes just take my word so if you
25:11 take a model of all the things true in
25:13 the standard model of arithmetic you get
25:15 a model of the standard model arithmetic
25:18 ok so let's collapse and models and see
25:24 what happens so suppose that M is any
25:28 non standard model of arithmetic so some
25:30 of you will know about this stuff in in
25:31 classical logic there are many
25:34 non-standard and models arithmetic
25:39 they sort of look like this okay they
25:44 have all the natural numbers and then
25:46 after those there's some junk some
25:54 non-standard numbers what we're gonna do
25:57 essentially is collapse all these guys
26:00 and make one of them okay so how do we
26:06 do that well we're going to define an
26:08 equivalence relation which is also a
26:14 Congress relation so D tilde e just if D
26:16 near standard and equally so we're going
26:17 to leave the standard numbers alone
26:18 we're not going to mess around with
26:22 those or D into your non-standard okay
26:25 so D tilde e if both D and E are
26:29 non-standard so this is a Congress
26:31 relation you can check that if you like
26:34 and we're going to collapse with that
26:40 congruence relation so what we get is a
26:42 structure that looks like this so we've
26:44 got all the natural numbers okay we've
26:48 left those alone and we've collapsed all
26:50 the non-standard numbers to a single
26:52 number which so this is the graph of the
26:56 successor relation so one is a setter of
26:59 zero and so on and everything in I is
27:02 has a successor which is itself I
27:06 because we've identified I and its
27:09 successor for every non standard natural
27:11 number okay
27:15 so our structure looks like this and
27:19 it's a model of true all the truth in
27:23 the standard model so it's complete but
27:26 it's inconsistent because something is
27:29 its own successor only this guy I and of
27:32 course the negation of that is true in
27:34 the standard model so the theory is
27:37 inconsistent but it's not trivial
27:41 because this is not true we've left 0 &
27:45 1 alone so we haven't done anything to
27:47 effect what holds of 0 & 1 and
27:47 and
27:51 in particular zero equals zero prime C R
27:53 equals one is not true in this model so
27:56 it's not trivial okay really the
27:58 inconsistencies all concerned this guy
28:02 here so that's an example of a collapsed
28:05 interpretation which shows you that you
28:08 can have a complete arithmetic I mean in
28:12 that theory in the theory of M tilde
28:15 everything is either true or false but
28:18 some things might be both okay let me
28:22 give you a second example okay so let em
28:26 this time be the standard model right
28:29 yeah the standard model arithmetic and
28:34 let I'm going to define a new congruence
28:37 relation so N and P are natural numbers
28:44 greater than zero and D tilde a just if
28:47 D and E are less than N and D equals E
28:49 so for things less than n we're not
28:50 going to freak around with them so we're
28:54 leaving the things less than n alone but
28:58 if D and E are great things when then D
29:04 tilde e just if de identical mod P so in
29:07 other words we're doing sort clock
29:11 arithmetic ok so in clock arithmetic if
29:15 you add 12 to a number B at the same
29:17 number okay one o'clock in the morning
29:18 you had 12 yet one o'clock in the
29:21 afternoon to house we've got so really
29:24 anything that differs by twelve is you
29:25 get the same okay that's clock
29:27 arithmetic or what mathematicians called
29:31 modular arithmetic so we're going to
29:34 take all the numbers greater than n and
29:39 identify them mod P so P does not mean
29:43 prime P means period because what does
29:46 this structure look like well okay the
29:48 structure of the collapse model looks
29:50 like this so we haven't done anything
29:52 with the numbers less than n we've left
29:55 those alone right but we've made sure
30:00 that there's a loop after that
30:04 such that once you take any number and
30:05 you add P to it you get back to where
30:11 you started so there's a cycle there's
30:14 an initial tail where things behave
30:16 classically and then there's a cycle
30:32 can I compare the numbers that come
30:36 after n to the numbers before like n
30:46 anything true in the standard model is
30:49 true in this model so for example in the
30:53 standard model 0 is less than everything
30:55 so 0 is going to be less than all these
30:58 guys right so everything behaves
31:01 perfectly normally here but when you get
31:03 to something it's going to happen
31:07 because n is going to be H less than
31:11 itself ok so that of course is
31:14 inconsistent but that's it's an
31:16 inconsistent model so strange things
31:18 happen once you get to the cycle but
31:26 everything down here is normal okay so
31:28 what do we know about this structure
31:34 well first of all something is it's P
31:36 time successor so if you there are some
31:40 things such that if you add them you get
31:44 back to where you started Shh ok
31:50 however the model is inconsistent the
31:53 model is not trivial because again we
31:58 hadn't fiddled around with 0 and 1 so in
32:02 this model it is not true that 0 is 1
32:04 there are other contradictions but not
32:09 concerning 0 and 1 ok now I give you
32:11 this model because it has a particularly
32:19 the first model I gave you was not
32:23 finite this one is okay so have a look
32:26 at it okay the domain of the collateral
32:31 interpretation is finite that means that
32:34 the truths of this model and the false
32:38 to this model are decidable because if
32:42 you've got any finite structure there's
32:44 a way of deciding whether something is
32:48 true or false in it okay undecidability
32:49 comes from the fact that you're dealing
32:52 with infinite domains but if you've got
32:56 a finite domain then the universal
32:59 quantifier is just a big conjunction the
33:02 particular quantifier is just a big
33:04 disjunction so you can apply truth
33:10 tables so the set of things which are
33:12 true in this collapse model are
33:14 decidable and that means they're
33:20 axiomatic because decidable sets are
33:22 axiomatic that's a sort of small
33:26 definition in in recursion theory now
33:30 this particular interpretation is finite
33:34 and axiomatic there are infinite
33:37 interpretations which are also axiomatic
33:41 but that's harder to prove sight gave
33:43 you this simple example but what you
33:46 have now seen is that there are
33:50 axiomatic theories which are complete in
33:52 other words everything is either true or
33:56 false in the interpretation but
33:58 inconsistent some things can be both
34:01 true and false so these models are
34:05 models of inconsistent complete
34:08 arithmetics but they're not trivial okay
34:09 you can't it's not the case of
34:12 everything holds in these models all
34:14 right so
34:19 I started by saying that once you move
34:22 to a para consistent logic you can have
34:25 inconsistent but complete arithmetics
34:27 and you've now just seen what these are
34:31 like or some of them are like okay so
34:33 the next question you might want to ask
34:37 is what happens to the girdle sentence
34:40 in these things so we know that the
34:44 you've got a model of everything true in
34:46 the standard model what happens to the
34:51 girdle sentence remember in the in the
34:55 consistent case it's going to be such
34:57 that neither it nor its indication is
35:00 proof or what do you think is going to
35:04 happen in the inconsistent case let's
35:12 see okay so let t be any axiomatic
35:14 arithmetic which contains whatever is
35:16 true in the standard model and we've now
35:24 okay let prophets be the arithmetic or
35:26 predicate which defines T in the
35:28 standard model so this theory is
35:32 axiomatic so in particular the proof
35:34 predicate is arithmetic that means you
35:40 suppose what that means is if you can
35:43 prove a and T then improv a is true in
35:45 the standard model so you can prove it
35:51 in T and if you can't prove a entry then
35:54 not provi is true in the standard model
35:57 because prov defines the proof predicate
35:59 in the standard model and so it's
36:02 negation is true not prevail is true and
36:05 team okay so this is a fact this is a
36:06 consequence of the fact you're dealing
36:08 with an axiomatic theory whose proof
36:11 predicate is representable in standard
36:15 arithmetic okay so let that G be the
36:18 girdle sentence let's see what happens
36:23 to it well either you can prove G or you
36:28 come in either case you can prove
36:31 prov gee because if you can prove g well
36:34 G is not property and if you can't prove
36:37 G well then you can prove not prove G ok
36:41 thats by this condition so i in either
36:44 case then you can prove not probe G but
36:48 of course there is G so by the first
36:50 condition of a proof predicate you can
36:56 prove G so you can prove G and you can
36:59 prove its negation so the girdle
37:04 undecidable sentence is both provable
37:07 and its negation is approval so in the
37:09 consistent case you can prove neither
37:13 the girdle sentence nor its negation in
37:16 this inconsistent arithmetics you can
37:19 prove the girdle sentence and its
37:26 negation alright so let me spend the
37:29 last 10 minutes talking about
37:40 philosophical issues this is much hard
37:42 work okay but we've done with all the
37:44 technical details you'll be happy to
37:50 know okay so let's think some more about
37:53 the girdle sentence alright we know that
37:55 you can prove neither it nor its
38:00 negation in the incomplete case and you
38:02 can prove both it and it's negation in
38:07 the inconsistent case now which of these
38:12 is actually more plausible well look
38:17 behind girdle's proof there is a paradox
38:20 it's rather like the liar paradox except
38:23 that it involves proof rather than truth
38:28 so let G be the sentence G is improvable
38:31 and prov here I don't mean the proved
38:33 ability predicate of some formal system
38:36 I mean just you know something like it's
38:41 provably true that in that language so
38:42 take a sentence which
38:46 hey I'm not provable alright so suppose
38:50 you can prove oh well then it's true
38:53 because anything you can prove is true
38:57 and G is not property so if G isn't
39:01 proved but it is proof 'l so by you know
39:04 standard logical move you've just proved
39:11 it that is you've proved G that is
39:13 you've proved that G is proof but
39:16 because you just proved G okay so you
39:20 have a paradox G is both provable and
39:22 not provable this is the analog of the
39:25 liar paradox but it's for proved ability
39:34 rather than truth so of course there's
39:36 an enormous question about how you solve
39:39 the liar paradox the literature is
39:42 enormous but dilute this such as myself
39:45 hold that the correct way to solve the
39:49 liar paradox is to just accept that the
39:52 liar sentence is both true and false so
39:56 if the sentence L is L is false then
40:00 that sentence is both true and false now
40:03 if you accept a dilute solution to the
40:06 liar paradox then the natural thing is
40:07 to accept a dialectic solution to this
40:12 paradox okay
40:13 now we're dealing with proved ability
40:16 not truth but the two would seem to be
40:23 very closely related so if you accept a
40:25 dialectic solution to this paradox which
40:28 is sometimes called girdles paradox if
40:31 you accept a dialectic solution to
40:34 girdles paradox what you really want is
40:36 an inconsistent arithmetic one such that
40:41 both G and not G hold so this is one
40:43 reason why you might prefer an
40:45 inconsistent arithmetic because it gives
40:51 you the same kind of solution to girdles
40:56 paradox as you get for the liar paradox
40:58 if you are a daily theist that's one
41:05 reason second reason this is sometimes
41:07 called herbs principle okay
41:10 so consider the sentence if you can
41:17 prove a then I now that damn well looks
41:20 as though it's true should be okay
41:22 because hey whatever you can prove is
41:26 true so you expect that to be true but
41:29 you cannot prove this inconsistent in
41:35 consistent arithmetic so this is love's
41:39 okay the herb serum says if you can
41:44 prove this you can already proved I so
41:49 you cannot have this for every a in
41:54 classical arithmetic even though it's
41:58 down well intuitively true okay you can
42:01 only prove some instances of it know me
42:03 the ones for which you've independently
42:05 got a proof of a and that seems odd I
42:07 mean no this thing is true in the
42:10 standard model so okay so you can add it
42:13 to the arithmetic but if you do that
42:16 you're messing around with the axioms so
42:20 you got a you can add this provided
42:22 you're dealing with proof in the old
42:24 sense but if you add this as an axiom
42:25 and you've got proof in the new sense
42:27 well then you're back with the situation
42:31 so I mean you cannot have this principle
42:34 in generality in a consistent arithmetic
42:39 but you can have it so here's why okay
42:42 so for any a you can either prove it or
42:47 you can't in the first case you can
42:49 prove the lower principle that's just
42:51 because this is material conditional if
42:54 you can prove a you can prove this
42:59 conditional and in the second case by
43:00 the properties of the prove multi
43:02 predicate if you can't prove a you can
43:05 prove not prove a and so again you can
43:06 prove this because this is a material
43:08 conditional and you've just established
43:09 the ghost
43:12 the anti Sweden so what we've just seen
43:15 is that herbs principle holds in these
43:18 inconsistent arithmetics so you cannot
43:20 have this inconsistent arithmetics
43:22 although it seems you ought to have it
43:25 because hey it seems to be true but you
43:28 can have it any consists north mistakes
43:31 so these are two reasons why you might
43:34 prefer an inconsistent arithmetic to a
43:37 consistent arithmetic so last couple of
43:42 minutes I've been telling you that
43:45 arithmetic is inconsistent right that's
43:50 sort of shocking claim so presumably you
43:52 want to object so you may have many
43:56 objections okay we can those let me just
44:01 give you a few possible objections an
44:03 inconsistent arithmetic doesn't prove
44:05 enough okay this one's a non-starter
44:08 because everything you can prove
44:10 classically everything true in the
44:11 standard model is true in this
44:15 arithmetic so it's not the case that you
44:17 haven't got enough that's the worry with
44:22 the incomplete case if there's a worry
44:24 it's not that you haven't got enough
44:28 it's that you got too much okay so
44:29 second objection
44:35 it proves too much why might you think
44:40 it proves too much well you might have a
44:43 very general reason for this because you
44:44 might think that the principle of
44:46 non-contradiction is true okay no
44:49 sentence could be both true and false so
44:52 if you believe that then of course
44:54 you're going to reject dialy thism
44:59 because dire theism exactly denies this
45:02 now over the last thirty years there's
45:05 been a debate on whether or not you how
45:07 good the principle of non-contradiction
45:11 is and one thing that's transpired in
45:13 the debate is that it's actually kind of
45:15 hard to defend the principle of
45:18 non-contradiction without begging the question
45:23 so I mean if you believe in the
45:24 principle of non-contradiction then
45:26 you've certainly proved too much because
45:30 you've got inconsistencies and where the
45:33 debate goes now is well whether you
45:34 should accept the principle of
45:36 non-contradiction that's a big issue as
45:39 I've indicated and I'm not going to go
45:42 into that in this lecture if you want to
45:44 take it up in the discussion or in the
45:48 tea room that's fine by me okay but you
45:50 might have very general objections to
45:53 inconsistency that's one thing but you
45:55 might have more limited objections maybe
45:57 you know inconsistencies can be true
46:00 sometimes but hey you don't want them in
46:05 arithmetic because and you might think
46:07 well you know inconsistent arithmetic
46:10 would ruin the applications of recipe so
46:12 you know arithmetic is not just a pure
46:14 mathematical theory we use the rhythm
46:15 click all the time when we count and you
46:19 know add up and do science and so on and
46:22 you might think that the fact that you
46:25 can prove certain inconsistencies is
46:28 going to ruin arithmetic for its applications
46:29 applications
46:37 ok now let's think about this in all
46:40 these inconsistent models everything
46:45 behaves fine for a while and then things
46:47 going consistent you know in due course
46:52 for some number let's call it n so up to
46:55 n everything is consistent then after n
46:59 strange inconsistent things happen
47:14 big number of atoms in the cosmos it's
47:16 bigger than the number of games of chess
47:19 you can play if every atom in the cosmos
47:22 were a chess piece okay it's so bloody
47:24 enormous it has no practical
47:27 significance whatsoever because bigger
47:31 than anything with a physical meaning so
47:33 you know that standard arithmetic is
47:34 okay until you
47:36 to this bloody enormous number which has
47:40 no physical significance okay
47:43 so what it means is that if the least
47:46 inconsistent number is so bloody big
47:49 then it's not going to affect
47:50 applications because all the
47:53 applications concern things which have
47:58 physical significance okay so the
48:00 inconsistent arithmetic on that
48:04 assumption is not going to have I mean
48:05 it's got to be an interesting fact about
48:08 pure mathematics but it's not going to
48:11 affect any of the applications of pure
48:16 mathematics okay so that that's the end
48:23 I've been talking about girdle's theorem
48:26 and in particular the girdle sentence
48:30 this sentence is not provable and [Music]
48:32 [Music]
48:36 inconsistent in consistent logics
48:39 consistent arithmetics neither the
48:41 girdle sentence norris negation is
48:46 provable so the theory is incomplete but
48:49 what we've now seen is that there are
48:53 inconsistent arithmetics which are
48:57 complete but inconsistent and in
49:00 particular then you can prove the girdle
49:04 sentence and its negation okay and then
49:07 just at the end I raised a few thoughts
49:09 about what this might mean
49:12 philosophically so the technical stuff
49:15 is not contestable the the technical
49:18 stuff is just you know beyond argument
49:22 these are just results of pure model
49:25 theory what is of course contentious is
49:28 what this means and whether one should
49:32 take an inconsistent arithmetic as the
49:34 correct arithmetic and not the standard
49:37 model okay which is just a chunk or
49:39 gives you a chunk and what's true about
49:41 numbers but a proper chunk so there's
49:43 all the stuff true in the standard model
49:46 and then some more okay now that's
49:47 contentious and
49:49 no more than start a discussion with
49:52 respect to that we can take that up in
49:55 questions or in the Tea Room or by
49:57 correspondence that's fine but at least
50:00 now I hope in this talk I've opened your
50:03 eyes to some new possibilities and you
50:06 know philosophical discussions about the
50:08 philosophy of mathematics take off from
50:11 technical results well I've given you
50:14 some technical results today which you
50:16 may not have known about before and this
50:19 I think opens a whole new page on the
50:21 discussion of the philosophical
50:25 significance of girdle serum and of
50:27 course where that goes well time will tell
50:28 tell
50:31 thank you [Applause]
50:38 [Applause]
50:53 well again I enjoyed very much the talk
50:58 and what I have to say is small and
51:02 limited it's just I want to ask you if
51:09 you know of some works I have met new
51:14 years ago some people because I was
51:18 working about non-standard mathematics
51:23 as I told you at some lunch and some
51:26 people in that school we're trying to
51:28 they realize that you have a
51:30 non-standard discourse what we needed
51:33 was in an infinitely large natural
51:35 number it was enough because if we have
51:38 an infinitely large natural number then
51:52 it's that basis and so did the issue was
51:55 how should we introduce an infinitely
51:58 large number casually Robinson was doing
52:01 that with using cultural products or sub
52:04 suggests that and so people realized
52:06 that it was something to do that that
52:07 only happened a role of arithmetic
52:11 without the whole machinery or sensitive
52:15 and some people that's that one way of
52:17 doing that was using builders becoming
52:20 this accomplice theorem there is a
52:23 figure from you which is from 1980 I
52:29 think which does that P defines on the
52:34 basis of through the sentence with a
52:37 little tricky term good acceptance to
52:40 say the sentence of the farmer exists X
52:47 such that a of X and this sentence but
52:52 we know for sure that for any actual K a
52:55 of K is not true so we have introduced
52:59 in that way and it's only girl in
53:06 conflicted what the it's a question
53:47 separately because otherwise I'll forget
53:50 okay so I'm not familiar with the paper
53:52 that you mentioned and so I'd be very
53:54 happy if you'd give me the reference okay
53:55 okay
53:57 secondly I am familiar with some work in
53:59 this area so there's a beautiful
54:02 construction by the mathematician Conway
54:04 called surreal numbers which allows you
54:06 to have infinite numbers infinitesimally
54:09 and a lot of other things so that that's
54:12 a beautiful construction so that I am
54:17 familiar with now I think in in all the
54:19 systems that we're talking about they
54:22 are consistent okay
54:24 and as long as you remain consistent
54:26 you're gonna get the sort of issues that
54:29 I've been talking about so what what
54:31 I've been talking about is what happens
54:34 when you sort of transgress beyond a
54:37 consistent and the constructions that
54:40 remain retain consistency I can have
54:41 exactly the same kind of features as
54:44 peon arithmetic of course they have
54:46 other interesting properties but they're
54:49 the same as P an arithmetic in this in
54:51 the sense that they're consistent what
54:53 I've been doing is saying hey what
54:54 happens when you go beyond the consistent
56:53 is holding is going to hold in the
56:55 standard model where we process the
56:58 arithmetic signal one sentence which
57:02 defines the theory in the model
57:06 because everything in aim is into well
57:10 you can prove in today provide so it's
57:13 true because we are assuming that the
57:17 theory Zac somatic so it's proof it's
57:18 definable in the standard model by a
57:21 signal one sentence and anything you can
58:12 tell the standard model you can okay now
58:15 when we apply with tick we apply a
58:18 consistent rhythm check and if what I
58:20 said about the listing cuz this numbers
58:21 right then you know there's no funny
58:23 business until you get so a very large
58:28 number suppose smaller so it's only you
58:29 know 10 to the 10 to the 10 or something
58:32 like that then maybe this has physical
58:37 significance so maybe there are some
58:44 physical phenomena we need is actually
58:47 consistent arithmetic yep
58:54 okay like that however mathematics and
59:03 physics are very strange things and pure
59:05 mathematics is developed
59:07 which finds an application physics only
60:16 time this was exactly which tell you that told you about the stretch of
60:17 that told you about the stretch of reality so my question is that any
60:24 reality so my question is that any possible application for inconsistent
60:26 possible application for inconsistent now but hey who knows what's going to
60:29 now but hey who knows what's going to happen in two hundred years time I'll be
60:32 happen in two hundred years time I'll be around to see it I wish I could come
60:34 around to see it I wish I could come back and find out
62:07 is definition so we'll work in standard first-order logic when you haven't got
62:09 first-order logic when you haven't got modal prices and you know that friction
62:14 modal prices and you know that friction goes all the way back to girdle of
62:15 goes all the way back to girdle of course so at least as far as that goes
62:20 course so at least as far as that goes being very traditional okay now of
62:24 being very traditional okay now of course that doesn't mean you can't have
62:26 course that doesn't mean you can't have a modal logic where is effectively a
62:30 a modal logic where is effectively a modal operator we know you can do that
62:32 modal operator we know you can do that okay so there are systems of modal logic
62:36 okay so there are systems of modal logic where there's an operator prod and it
62:40 where there's an operator prod and it behaves very much like the predicate
62:42 behaves very much like the predicate prod in say piano arithmetic and you get
62:46 prod in say piano arithmetic and you get a kind of unusual modal logic I think
62:50 a kind of unusual modal logic I think it's usually called G if so investigate
62:52 it's usually called G if so investigate like people like Boulos and what makes
62:55 like people like Boulos and what makes it unusual
62:56 it unusual that it the accessibility relation for
62:58 that it the accessibility relation for the modal operator is kind of unusual
63:00 the modal operator is kind of unusual and you have logical truths which are
63:02 and you have logical truths which are kind of unusual because you had this
63:04 kind of unusual because you had this unusual constraint so I mean no that's
63:12 unusual constraint so I mean no that's correct because what you have you're not
63:15 correct because what you have you're not going to get the analog of you've got
63:16 going to get the analog of you've got the analog of lube serum okay so you're
63:19 the analog of lube serum okay so you're not going to have to take people to be a
63:21 not going to have to take people to be a modal operator you're not going to have
63:23 modal operator you're not going to have it
63:23 it if probe a then I because that will be
63:26 if probe a then I because that will be Lib serum and you haven't got that okay
63:28 Lib serum and you haven't got that okay now you can certainly do them now
63:32 now you can certainly do them now interesting question what happens if you
63:35 interesting question what happens if you do the same thing for the incan
63:36 do the same thing for the incan systematics ah don't know but you can
63:40 systematics ah don't know but you can certainly do it what you're gonna get is
63:43 certainly do it what you're gonna get is a modal logic it's going to be a pair a
63:45 a modal logic it's going to be a pair a consistent modal logic but you know we
63:47 consistent modal logic but you know we know these things what pair consists of
63:51 know these things what pair consists of modal logic and you're gonna get well
63:52 modal logic and you're gonna get well that's a really interesting question I
63:54 that's a really interesting question I don't know the answer what you are going
63:56 don't know the answer what you are going to get is reflexivity okay because
63:58 to get is reflexivity okay because you've got NURBS principle back but
64:01 you've got NURBS principle back but you're going to get some other stuff too
64:04 you're going to get some other stuff too what you're going to get you know what
64:08 what you're going to get you know what the theory is going to be in general I
64:10 the theory is going to be in general I don't know that's a really interesting
64:11 don't know that's a really interesting question
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