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Graham Priest - Gödel's theorem and paraconsistency (UnB, 2018) - in English

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university of new york and i want to

tell you that

professor priest is considered one of

the most important philosophers recently

he has published many books especially

on topics concerning philosophical logic

and it's metaphysical and ontological

implications so professor priest

published books like towads non-being

one doubt truth to be alive in

introduction to classical logic logic a

very short introduction won many many

interesting and important books for

contemporary research on logic so he

gave ready to talks in our local seminar

logic in the airplane logic in the

airplane and it's a very important

occasion for always here to have his

presentation today on good of theorem

and para consistent so professor please

be welcome again here at the university

of Brasilia and invite you to come to

[Applause]

okay so I have to use this right so I

feel like an old rock star how we do how

do we do the translation we don't that's

makes things easy okay

and if I walk around is that okay can you

you

oh okay

can I move while I talk yeah okay well

first of all let me thank the organizers

very much for putting on this event

which is terrific

this this is a great auditorium I've

never been in anything like it so this

is fantastic

so thank you to all the people who've

made this possible and that means that

people have coughed up their money as

well as organized so I want to talk

about girdle's theorem this is the first

incompleteness theorem I'm afraid I'm

not familiar with professors or cadre

and V mean so I can't comment on how

this relates to their work I'm happy to

leave that to others to do because

girdles theorem is certainly very

important about how we reason in

particular how we reason in mathematics

and Gerda published his theorem in 1931

and I think undoubtably it's one of the

most profound results in mathematics in

the 20th century unfortunately no one

can agree on exactly what the importance

is philosophically and so I want to talk

about part of that debate power

consistent logic that is logics in which

you can have contradictions without

explosion so contradictions are kind of limited

limited

they don't go everywhere Pera consistent

logic puts a whole new light on girdle

serum which I guess most people are not

familiar with so what I want to do in

this talk is talk about the relevance of

power consistent logic for girdled serum

so talk is quite technical but let me

take it slowly and let's see what we can

do okay so that was the introduction

you've had that I want to talk about

inconsistent arithmetic and its

relevance to girdles theorem but to do

that I'll have to tell you a little bit

about girdle serum and about para

consistent logic because not everybody

knows this material so a lot of the time

I'm going to talk about this and then

we'll get on to inconsistent arithmetic

okay so first of all girdles theorem if

you open a textbook on mathematical

logic you will often find a statement of

girdles theorem is something like this

any axiomatic theory of arithmetic with

appropriate expressive capabilities is

incomplete okay that's not a good

statement this is the correct statement

the theory must be either

incomplete or inconsistent now if you're

working in classical logic if the theory

is inconsistent is trivial you can prove

everything so people can justifiably

ignore the inconsistent option but once

you move to a pair a consistent logic

this is no longer the case to assume is

that there's a formula with two free

variables the X Y which defines the

proof relation in T that is if n is the

code of a proof of a in T then B and

angle brackets a is true in the standard

model and if n is not the code of a

proof of int then it's negation is true

in the standard model third assumption

you can define probe Y as there is an X

such that X is a proof of Y then probe

is a prove predicate for T that is if

you can prove a that into you

prove that prof a and the fourth thing

you need is that there is a formula G of

the form not profit G there are some

subtleties here but I'll slide over them

okay so with those four assumptions you

can prove girdle's theorem so this is

how you do it

okay so pros you can prove G well then

you can prove not Prodigy because that

is G but if you can prove G then you can

prove property because profit proof

predicates so what we see then is that

okay now suppose that T is consistent

then you can't prove G we've just seen

that so no number is the code of a proof

of G so for any n the negation of B and

G is true in the standard model okay so

therefore the universal quantification

of that is true in the standard model

and we do a bit of logic chopping here

so that's equivalent to this so we need

a few quantify principles but that is

just the negation of prof G and that's G

so supposing that T is consistent G is

true in the standard model so you can't

prove G but G is true in the standard

model so the theory is incomplete

alright so if the theory is inconsistent

it's incomplete but by contradiction

if it's complete it's inconsistent so

this is a proof of girdled serum which

works and we've made very little

assumption about the logic a few

assumptions about how quantifiers behave

that's all okay so as I said most people

who talk about girdles theorem assume

you must be operating in a consistent

context but I want to examine the

inconsistent possibility so what we're

going to be doing is looking at ink

consistent arithmetics and what happens

to the girdle sentence in those

arithmetics now to do that you need to

understand a little bit about power

consistent logic and some of its motive

theory so I'm going to give you the

background that you need to understand

this okay so a guide to power consists

in logic there are of course many para

consistent logics and this country is

justifiably famous for its work in power

consistent logic I'm going to describe

one system of power consistent logic a

system called LP which is simple and

robust and will do what we want it to do

today okay so the language that we're

going to use is a standard first-order

language nothing an interpretation is

going to be a structure d which is the

domain of quantification and Delta which

is the denotation function so that tells

you the denotations of the constants the

function symbols and the per etiquettes

so d is a non empty domain of

quantification the denotation already

constant is a member of D if FN is an M

plays function simple then it's D

notation is a function from intervals in

the domain to D so this is all standard

classical logic this is where it's

different because in classical logic

well come back to that the denotation of

a predicate is a pair the predicate is

going to have an extension and an anti

extension so the extension are the

things of which the predicate is true

the anti extension is the things of

interpretation of a predicate is the

pair the extension and the ant extension

I'll write them like this with a plus

and a minus

such that the extension and the anti

extension exhaust all the intervals

Devane okay so in this logic there are

no truth value gaps you can't have

something that's neither true nor false

this essentially rules this out but you

can have things which are both true and

false because the extension and the

antique stench reddit can overlap and

the identity predicate is just it's what

you'd expect it's as you get in

classical logic okay so given

interpretation the next thing we need to

do is specify the truth and the falsity

conditions of course in classical logic

once you know the truth conditions then

the falsity conditions follow because

they're just falsity is the compliment

of truth but in a para consistent logic

truth and falsity are independent so you

have to give truth conditions and

falsity conditions so first of all we

finish off denotation so if you've got a

if you've got a term function symbol and

n terms what's it's denotation well it's

exactly the same as classical logic you

apply the denotation to function simple

the denotations of the terms everything

standard there the truth that's a little

plus here of an atomic sentence well

this sentence is true just if the enter

part of the denotations of the terms are

in the extension of the predicate and

for falsity that's the little - this

formula is false if the enter poll of

the denotations of the terms is in the

anti extension of the predicate and the

extension and the anti extension can

overlap so something can be both true

and false okay

the rest are is what you've more or less

expect so a negation is true if the

thing negated is false and it's false if

the thing negated is true a conjunction

is true if both contracts are true and

false if one or other conjunct is false

a disjunction is true if one or other

disjunct is true

and false if both disjunctive false and

note that the material conditional is

defined in the way in the same way as

usual so you don't need to give separate

truth and falsity conditions because it

you define mattoon implication as this

thing okay quantifiers there are many

ways to do the semantics of quantifiers

you can use satisfaction but I've always

thought that the easiest way to do

semantics of quantifiers is just to help

yourself to enough names to ensure that

everything in the domain has a name so

what we're going to do is extend the

language so if DS in the domain let's

add this new constant KD which refers to

D so we know now that every and that

allows us to give truth and falsity

conditions for quantifies so a

universally quantified sentence is true

if every objects in their main satisfies

it says and false if some object in the

main makes it false and then for the

particular quantifier you just do the

dual thing okay and now let's have a

definition of consequence so if a is a

model of the for being a just if the

structure makes a true fames in

classical logic a is a model of a set of

formulas Sigma just if it makes every

formula in Sigma true if it's a model of

every formula and the definition of

consequence if you've got a set of

formulas a set of formulas Sigma then a

is a consequence so that just if every

model of Sigma is a model of a so this

is standard stuff from the model theory

of classical logic okay so that is a

specification of the para consistent

logic LP now some facts about it

first of all it if the extension and the

anti extension of a predicate are

disjoint then this is exactly classical

logic because the only way that these

truth conditions and falsity conditions

actually differ from classical logic is

that they've come apart and if the

extension and the edge extension don't

overlap then this is exactly okay so

classical interpretations classical

models are special cases of LP

interpretations so every classical model

is an LP model but there are more LP

models so what that means is that if

something follows in LP it follows in

um well classical logic is negation

complete in the following sense every

formula is either true or false or not both

both

so we're not talking out theories we

just talked about interpretations at the

moment okay and in classical logic every

sentence is either true or false or not

both and if you insist that the

extension and anti extension every

predicate don't intersect that's exactly

what you'll get okay yeah thank you for

the question

please feel free to ask questions of

clarification because I know that this

stuff is probably unfamiliar to many of

you so if you want me to explain some

more please ask okay so any logical

consequence in LP is a classical logic

consequence but the converse does not

follow because this inference from a and

not A to B is valid in classical logic

but it's not valid in this logic because

a can be both thrown false

where B is just plain false so this

means that the logic is a pair a

consistent logic okay so the fact that

an inference is valid in classical logic

does not imply that it's valid in LP but

there's something kind of interesting

namely if you have no premises so

there's a special case in which Sigma is

empty then the two logics coincide so LP

and classical logic have exactly the

same logical truths what they do not

have is the same consequence relation

this is a reason why you cannot identify

logic with its set of logical truths

here are two logics with the same set of

logical truths

but with a different consequence

relation okay

now so that's the background impera

consistent logic and there's one more

thing that you need to know before we

can get to arithmetic and that is a

little bit about the model theory of

para consistent logic this para

consistent logic so I need to explain to

you something about collapsed

interpretations so let a be an LP

interpretation and then it then includes

classical interpretations as well

because as I've told you every classical

interpretation is an LP interpretation

what we're going to do essentially is

collapse that interpretation let me show

you how so Tilda is an equivalence

relation on the domain of a so an

if this is a domain an equivalence

relation is something which chunks the

domain into disjoint equivalence classes

okay that's what an equivalence relation

does so tilde is an equivalence relation

and moreover it's a congruence relation

on the denotations of their function

symbols now what that means is if you

take two things in the same chunk okay

and you apply the denotation of one of

the function symbols to them you're

gonna get things which are in the same

chunk okay

so applying the function to things in

the same chunk is always going to take

you to the same chunk so mathematicians

call this a congruence relation and you

need the equivalence relation to be a

congruence relation okay now notation if

D's in the domain I'm going to write the

equivalence class of D as D in square

brackets so if d is in the domain D in

square brackets is the whole chunk okay

so now I'm going to define the collapsed

interpretation so it has a domain and a

denotation function and the domain is

just going to comprise the equivalence

classes the chunks the denotation of

every constant is going to be the chunk

which the denotation of c was already in

the denotation of the function symbol is

just going to be where you take some

members of the chunks and you apply the

function to that and the chunk that

that's in that's when you need it to be

a congruence relation and okay these are

the interesting bits let me you can read

this for yourself let me just draw your

picture of what's going on what we're

gonna do essentially is we're going to

identify all the things in one chunk so

this chunk here

is going to satisfy predicate if

something or some things in this chunk

satisfied okay so what we're really

doing is identifying the things in a

trunk so suppose for example that Alex

is in the trunk and I'm in the trunk

so the trunk is Brazilian because alex

is Brazilian and it's Australian because

I'm Australian it's not Australian

because alex is not Australian and it's

not Brazilian because I'm not Brazilian

so all the things in a trunk and the

trunk is going to have all the

properties so collapsing together just

think things to produce any a very

inconsistent object okay that's

essentially what this says okay now this

is the important fact about claps

suppose you've got any LP interpretation

and you collapse it in the way that I've

told you then okay okay skip over that

this is the important thing if a is true

before you collapse it remains true

after you collapse and if it's false

before you collapse it's false after you

collapse so you never lose truth values

when you collapse you can get new stuff

new things can become true or false but

you can't lose old stuff and this is

important because it means that if you

start with a model of some theory when

you collapse you get a model of that

theory you might get a model of more but

everything true before you collapse is

true after you collapse okay so that's

the kind of background you need to know

to follow what I'm going to say about

arithmetic let me pause for a second

because we've been through

a lot of technical material but anyone

okay so that's the background now I want

to talk about arithmetic and what I'm

going to show you is that first of all

that there are inconsistent complete

arithmetics and then we're going to talk

about what happens to the girdle

sentence in those arithmetics and then

in the last few minutes I'll just

gesture at some of the philosophical

implications of this so let L be the

usual first-order language arithmetic

you find in any textbook let n boldface

n be the set of sentences true in the

standard model so this is what logicians

call the theory of the standard model

it's just the set of sentences true in

the standard model and factor M be any

model of n so it's any model that it

makes in all the sentences and then true

and we know there are lots of those

classically no there's one standard

model but we know there are lots of

non-standard classical models okay we're

going to produce now some non-standard

power consistent models so if you take

em and you collapse it and to get M

tilde then what you get when you

collapse is a model of N and that's just

a result of the collapsing lemma okay I

didn't prove the collapsing number for

you you can take my word for it it's not

difficult to prove but for today's

purposes just take my word so if you

take a model of all the things true in

the standard model of arithmetic you get

a model of the standard model arithmetic

ok so let's collapse and models and see

what happens so suppose that M is any

non standard model of arithmetic so some

of you will know about this stuff in in

classical logic there are many

non-standard and models arithmetic

they sort of look like this okay they

have all the natural numbers and then

after those there's some junk some

non-standard numbers what we're gonna do

essentially is collapse all these guys

and make one of them okay so how do we

do that well we're going to define an

equivalence relation which is also a

Congress relation so D tilde e just if D

near standard and equally so we're going

to leave the standard numbers alone

we're not going to mess around with

those or D into your non-standard okay

so D tilde e if both D and E are

non-standard so this is a Congress

relation you can check that if you like

and we're going to collapse with that

congruence relation so what we get is a

structure that looks like this so we've

got all the natural numbers okay we've

left those alone and we've collapsed all

the non-standard numbers to a single

number which so this is the graph of the

successor relation so one is a setter of

zero and so on and everything in I is

has a successor which is itself I

because we've identified I and its

successor for every non standard natural

number okay

so our structure looks like this and

it's a model of true all the truth in

the standard model so it's complete but

it's inconsistent because something is

its own successor only this guy I and of

course the negation of that is true in

the standard model so the theory is

inconsistent but it's not trivial

because this is not true we've left 0 &

1 alone so we haven't done anything to

effect what holds of 0 & 1 and

and

in particular zero equals zero prime C R

equals one is not true in this model so

it's not trivial okay really the

inconsistencies all concerned this guy

here so that's an example of a collapsed

interpretation which shows you that you

can have a complete arithmetic I mean in

that theory in the theory of M tilde

everything is either true or false but

some things might be both okay let me

give you a second example okay so let em

this time be the standard model right

yeah the standard model arithmetic and

let I'm going to define a new congruence

relation so N and P are natural numbers

greater than zero and D tilde a just if

D and E are less than N and D equals E

so for things less than n we're not

going to freak around with them so we're

leaving the things less than n alone but

if D and E are great things when then D

tilde e just if de identical mod P so in

other words we're doing sort clock

arithmetic ok so in clock arithmetic if

you add 12 to a number B at the same

number okay one o'clock in the morning

you had 12 yet one o'clock in the

afternoon to house we've got so really

anything that differs by twelve is you

get the same okay that's clock

arithmetic or what mathematicians called

modular arithmetic so we're going to

take all the numbers greater than n and

identify them mod P so P does not mean

prime P means period because what does

this structure look like well okay the

structure of the collapse model looks

like this so we haven't done anything

with the numbers less than n we've left

those alone right but we've made sure

that there's a loop after that

such that once you take any number and

you add P to it you get back to where

you started so there's a cycle there's

an initial tail where things behave

classically and then there's a cycle

can I compare the numbers that come

after n to the numbers before like n

anything true in the standard model is

true in this model so for example in the

standard model 0 is less than everything

so 0 is going to be less than all these

guys right so everything behaves

perfectly normally here but when you get

to something it's going to happen

because n is going to be H less than

itself ok so that of course is

inconsistent but that's it's an

inconsistent model so strange things

happen once you get to the cycle but

everything down here is normal okay so

what do we know about this structure

well first of all something is it's P

time successor so if you there are some

things such that if you add them you get

back to where you started Shh ok

however the model is inconsistent the

model is not trivial because again we

hadn't fiddled around with 0 and 1 so in

this model it is not true that 0 is 1

there are other contradictions but not

concerning 0 and 1 ok now I give you

this model because it has a particularly

the first model I gave you was not

finite this one is okay so have a look

at it okay the domain of the collateral

interpretation is finite that means that

the truths of this model and the false

to this model are decidable because if

you've got any finite structure there's

a way of deciding whether something is

true or false in it okay undecidability

comes from the fact that you're dealing

with infinite domains but if you've got

a finite domain then the universal

quantifier is just a big conjunction the

particular quantifier is just a big

disjunction so you can apply truth

tables so the set of things which are

true in this collapse model are

decidable and that means they're

axiomatic because decidable sets are

axiomatic that's a sort of small

definition in in recursion theory now

this particular interpretation is finite

and axiomatic there are infinite

interpretations which are also axiomatic

but that's harder to prove sight gave

you this simple example but what you

have now seen is that there are

axiomatic theories which are complete in

other words everything is either true or

false in the interpretation but

inconsistent some things can be both

true and false so these models are

models of inconsistent complete

arithmetics but they're not trivial okay

you can't it's not the case of

everything holds in these models all

right so

I started by saying that once you move

to a para consistent logic you can have

inconsistent but complete arithmetics

and you've now just seen what these are

like or some of them are like okay so

the next question you might want to ask

is what happens to the girdle sentence

in these things so we know that the

you've got a model of everything true in

the standard model what happens to the

girdle sentence remember in the in the

consistent case it's going to be such

that neither it nor its indication is

proof or what do you think is going to

happen in the inconsistent case let's

see okay so let t be any axiomatic

arithmetic which contains whatever is

true in the standard model and we've now

okay let prophets be the arithmetic or

predicate which defines T in the

standard model so this theory is

axiomatic so in particular the proof

predicate is arithmetic that means you

suppose what that means is if you can

prove a and T then improv a is true in

the standard model so you can prove it

in T and if you can't prove a entry then

not provi is true in the standard model

because prov defines the proof predicate

in the standard model and so it's

negation is true not prevail is true and

team okay so this is a fact this is a

consequence of the fact you're dealing

with an axiomatic theory whose proof

predicate is representable in standard

arithmetic okay so let that G be the

girdle sentence let's see what happens

to it well either you can prove G or you

come in either case you can prove

prov gee because if you can prove g well

G is not property and if you can't prove

G well then you can prove not prove G ok

thats by this condition so i in either

case then you can prove not probe G but

of course there is G so by the first

condition of a proof predicate you can

prove G so you can prove G and you can

prove its negation so the girdle

undecidable sentence is both provable

and its negation is approval so in the

consistent case you can prove neither

the girdle sentence nor its negation in

this inconsistent arithmetics you can

prove the girdle sentence and its

negation alright so let me spend the

last 10 minutes talking about

philosophical issues this is much hard

work okay but we've done with all the

technical details you'll be happy to

know okay so let's think some more about

the girdle sentence alright we know that

you can prove neither it nor its

negation in the incomplete case and you

can prove both it and it's negation in

the inconsistent case now which of these

is actually more plausible well look

behind girdle's proof there is a paradox

it's rather like the liar paradox except

that it involves proof rather than truth

so let G be the sentence G is improvable

and prov here I don't mean the proved

ability predicate of some formal system

I mean just you know something like it's

provably true that in that language so

take a sentence which

hey I'm not provable alright so suppose

you can prove oh well then it's true

because anything you can prove is true

and G is not property so if G isn't

proved but it is proof 'l so by you know

standard logical move you've just proved

it that is you've proved G that is

you've proved that G is proof but

because you just proved G okay so you

have a paradox G is both provable and

not provable this is the analog of the

liar paradox but it's for proved ability

rather than truth so of course there's

an enormous question about how you solve

the liar paradox the literature is

enormous but dilute this such as myself

hold that the correct way to solve the

liar paradox is to just accept that the

liar sentence is both true and false so

if the sentence L is L is false then

that sentence is both true and false now

if you accept a dilute solution to the

liar paradox then the natural thing is

to accept a dialectic solution to this

paradox okay

now we're dealing with proved ability

not truth but the two would seem to be

very closely related so if you accept a

dialectic solution to this paradox which

is sometimes called girdles paradox if

you accept a dialectic solution to

girdles paradox what you really want is

an inconsistent arithmetic one such that

both G and not G hold so this is one

reason why you might prefer an

inconsistent arithmetic because it gives

you the same kind of solution to girdles

paradox as you get for the liar paradox

if you are a daily theist that's one

reason second reason this is sometimes

called herbs principle okay

so consider the sentence if you can

prove a then I now that damn well looks

as though it's true should be okay

because hey whatever you can prove is

true so you expect that to be true but

you cannot prove this inconsistent in

consistent arithmetic so this is love's

okay the herb serum says if you can

prove this you can already proved I so

you cannot have this for every a in

classical arithmetic even though it's

down well intuitively true okay you can

only prove some instances of it know me

the ones for which you've independently

got a proof of a and that seems odd I

mean no this thing is true in the

standard model so okay so you can add it

to the arithmetic but if you do that

you're messing around with the axioms so

you got a you can add this provided

you're dealing with proof in the old

sense but if you add this as an axiom

and you've got proof in the new sense

well then you're back with the situation

so I mean you cannot have this principle

in generality in a consistent arithmetic

but you can have it so here's why okay

so for any a you can either prove it or

you can't in the first case you can

prove the lower principle that's just

because this is material conditional if

you can prove a you can prove this

conditional and in the second case by

the properties of the prove multi

predicate if you can't prove a you can

prove not prove a and so again you can

prove this because this is a material

conditional and you've just established

the ghost

the anti Sweden so what we've just seen

is that herbs principle holds in these

inconsistent arithmetics so you cannot

have this inconsistent arithmetics

although it seems you ought to have it

because hey it seems to be true but you

can have it any consists north mistakes

so these are two reasons why you might

prefer an inconsistent arithmetic to a

consistent arithmetic so last couple of

minutes I've been telling you that

arithmetic is inconsistent right that's

sort of shocking claim so presumably you

want to object so you may have many

objections okay we can those let me just

give you a few possible objections an

inconsistent arithmetic doesn't prove

enough okay this one's a non-starter

because everything you can prove

classically everything true in the

standard model is true in this

arithmetic so it's not the case that you

haven't got enough that's the worry with

the incomplete case if there's a worry

it's not that you haven't got enough

it's that you got too much okay so

second objection

it proves too much why might you think

it proves too much well you might have a

very general reason for this because you

might think that the principle of

non-contradiction is true okay no

sentence could be both true and false so

if you believe that then of course

you're going to reject dialy thism

because dire theism exactly denies this

now over the last thirty years there's

been a debate on whether or not you how

good the principle of non-contradiction

is and one thing that's transpired in

the debate is that it's actually kind of

hard to defend the principle of

non-contradiction without begging the question

so I mean if you believe in the

principle of non-contradiction then

you've certainly proved too much because

you've got inconsistencies and where the

debate goes now is well whether you

should accept the principle of

non-contradiction that's a big issue as

I've indicated and I'm not going to go

into that in this lecture if you want to

take it up in the discussion or in the

tea room that's fine by me okay but you

might have very general objections to

inconsistency that's one thing but you

might have more limited objections maybe

you know inconsistencies can be true

sometimes but hey you don't want them in

arithmetic because and you might think

well you know inconsistent arithmetic

would ruin the applications of recipe so

you know arithmetic is not just a pure

mathematical theory we use the rhythm

click all the time when we count and you

know add up and do science and so on and

you might think that the fact that you

can prove certain inconsistencies is

going to ruin arithmetic for its applications

applications

ok now let's think about this in all

these inconsistent models everything

behaves fine for a while and then things

going consistent you know in due course

for some number let's call it n so up to

n everything is consistent then after n

strange inconsistent things happen

big number of atoms in the cosmos it's

bigger than the number of games of chess

you can play if every atom in the cosmos

were a chess piece okay it's so bloody

enormous it has no practical

significance whatsoever because bigger

than anything with a physical meaning so

you know that standard arithmetic is

okay until you

to this bloody enormous number which has

no physical significance okay

so what it means is that if the least

inconsistent number is so bloody big

then it's not going to affect

applications because all the

applications concern things which have

physical significance okay so the

inconsistent arithmetic on that

assumption is not going to have I mean

it's got to be an interesting fact about

pure mathematics but it's not going to

affect any of the applications of pure

mathematics okay so that that's the end

I've been talking about girdle's theorem

and in particular the girdle sentence

this sentence is not provable and [Music]

[Music]

inconsistent in consistent logics

consistent arithmetics neither the

girdle sentence norris negation is

provable so the theory is incomplete but

what we've now seen is that there are

inconsistent arithmetics which are

complete but inconsistent and in

particular then you can prove the girdle

sentence and its negation okay and then

just at the end I raised a few thoughts

about what this might mean

philosophically so the technical stuff

is not contestable the the technical

stuff is just you know beyond argument

these are just results of pure model

theory what is of course contentious is

what this means and whether one should

take an inconsistent arithmetic as the

correct arithmetic and not the standard

model okay which is just a chunk or

gives you a chunk and what's true about

numbers but a proper chunk so there's

all the stuff true in the standard model

and then some more okay now that's

contentious and

no more than start a discussion with

respect to that we can take that up in

questions or in the Tea Room or by

correspondence that's fine but at least

now I hope in this talk I've opened your

eyes to some new possibilities and you

know philosophical discussions about the

philosophy of mathematics take off from

technical results well I've given you

some technical results today which you

may not have known about before and this

I think opens a whole new page on the

discussion of the philosophical

significance of girdle serum and of

course where that goes well time will tell

tell

thank you [Applause]

[Applause]

well again I enjoyed very much the talk

and what I have to say is small and

limited it's just I want to ask you if

you know of some works I have met new

years ago some people because I was

working about non-standard mathematics

as I told you at some lunch and some

people in that school we're trying to

they realize that you have a

non-standard discourse what we needed

was in an infinitely large natural

number it was enough because if we have

an infinitely large natural number then

it's that basis and so did the issue was

how should we introduce an infinitely

large number casually Robinson was doing

that with using cultural products or sub

suggests that and so people realized

that it was something to do that that

only happened a role of arithmetic

without the whole machinery or sensitive

and some people that's that one way of

doing that was using builders becoming

this accomplice theorem there is a

figure from you which is from 1980 I

think which does that P defines on the

basis of through the sentence with a

little tricky term good acceptance to

say the sentence of the farmer exists X

such that a of X and this sentence but

we know for sure that for any actual K a

of K is not true so we have introduced

in that way and it's only girl in

conflicted what the it's a question

separately because otherwise I'll forget

okay so I'm not familiar with the paper

that you mentioned and so I'd be very

happy if you'd give me the reference okay

okay

secondly I am familiar with some work in

this area so there's a beautiful

construction by the mathematician Conway

called surreal numbers which allows you

to have infinite numbers infinitesimally

and a lot of other things so that that's

a beautiful construction so that I am

familiar with now I think in in all the

systems that we're talking about they

are consistent okay

and as long as you remain consistent

you're gonna get the sort of issues that

I've been talking about so what what

I've been talking about is what happens

when you sort of transgress beyond a

consistent and the constructions that

remain retain consistency I can have

exactly the same kind of features as

peon arithmetic of course they have

other interesting properties but they're

the same as P an arithmetic in this in

the sense that they're consistent what

I've been doing is saying hey what

happens when you go beyond the consistent

is holding is going to hold in the

standard model where we process the

arithmetic signal one sentence which

defines the theory in the model

because everything in aim is into well

you can prove in today provide so it's

true because we are assuming that the

theory Zac somatic so it's proof it's

definable in the standard model by a

signal one sentence and anything you can

tell the standard model you can okay now

when we apply with tick we apply a

consistent rhythm check and if what I

said about the listing cuz this numbers

right then you know there's no funny

business until you get so a very large

number suppose smaller so it's only you

know 10 to the 10 to the 10 or something

like that then maybe this has physical

significance so maybe there are some

physical phenomena we need is actually

consistent arithmetic yep

okay like that however mathematics and

physics are very strange things and pure

mathematics is developed

which finds an application physics only

time this was exactly which tell you that told you about the stretch of

that told you about the stretch of reality so my question is that any

reality so my question is that any possible application for inconsistent

possible application for inconsistent now but hey who knows what's going to

now but hey who knows what's going to happen in two hundred years time I'll be

happen in two hundred years time I'll be around to see it I wish I could come

around to see it I wish I could come back and find out

is definition so we'll work in standard first-order logic when you haven't got

first-order logic when you haven't got modal prices and you know that friction

modal prices and you know that friction goes all the way back to girdle of

goes all the way back to girdle of course so at least as far as that goes

course so at least as far as that goes being very traditional okay now of

being very traditional okay now of course that doesn't mean you can't have

course that doesn't mean you can't have a modal logic where is effectively a

a modal logic where is effectively a modal operator we know you can do that

modal operator we know you can do that okay so there are systems of modal logic

okay so there are systems of modal logic where there's an operator prod and it

where there's an operator prod and it behaves very much like the predicate

behaves very much like the predicate prod in say piano arithmetic and you get

prod in say piano arithmetic and you get a kind of unusual modal logic I think

a kind of unusual modal logic I think it's usually called G if so investigate

it's usually called G if so investigate like people like Boulos and what makes

like people like Boulos and what makes it unusual

it unusual that it the accessibility relation for

that it the accessibility relation for the modal operator is kind of unusual

the modal operator is kind of unusual and you have logical truths which are

and you have logical truths which are kind of unusual because you had this

kind of unusual because you had this unusual constraint so I mean no that's

unusual constraint so I mean no that's correct because what you have you're not

correct because what you have you're not going to get the analog of you've got

going to get the analog of you've got the analog of lube serum okay so you're

the analog of lube serum okay so you're not going to have to take people to be a

not going to have to take people to be a modal operator you're not going to have

modal operator you're not going to have it

it if probe a then I because that will be

if probe a then I because that will be Lib serum and you haven't got that okay

Lib serum and you haven't got that okay now you can certainly do them now

now you can certainly do them now interesting question what happens if you

interesting question what happens if you do the same thing for the incan

do the same thing for the incan systematics ah don't know but you can

systematics ah don't know but you can certainly do it what you're gonna get is

certainly do it what you're gonna get is a modal logic it's going to be a pair a

a modal logic it's going to be a pair a consistent modal logic but you know we

consistent modal logic but you know we know these things what pair consists of

know these things what pair consists of modal logic and you're gonna get well

modal logic and you're gonna get well that's a really interesting question I

that's a really interesting question I don't know the answer what you are going

don't know the answer what you are going to get is reflexivity okay because

to get is reflexivity okay because you've got NURBS principle back but

you've got NURBS principle back but you're going to get some other stuff too

you're going to get some other stuff too what you're going to get you know what

what you're going to get you know what the theory is going to be in general I

the theory is going to be in general I don't know that's a really interesting

don't know that's a really interesting question

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YouTube Transcript:Graham Priest - Gödel's theorem and paraconsistency (UnB, 2018) - in English

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Graham Priest - Gödel's theorem and paraconsistency (UnB, 2018) - in English