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Re: The seven step-Mathematical preliminaries
I think that resorting to calling the biggest natural number BIGGEST,
rather than specifying exactly what that number is, is a tell-tale sign
that the ultrafinitist knows that any specification for BIGGEST will
immediately reveal that it is not the biggest because one could always
add one more.
Quentin Anciaux wrote:
You have to explain why the exception is needed in the first place...
The rule is true until the rule is not true anymore, ok but you have
to explain for what sufficiently large N the successor function would
yield next 0 and why or to add that N and that exception to the
successor function as axiom, if not you can't avoid N+1. But torgny
doesn't evacuate N+1, merely it allows his set to grows undefinitelly
as when he has defined BIGGEST, he still argues BIGGEST+1 makes sense
, is a natural number but not part of the set of natural number, this
is non-sense, assuming your special successor rule BIGGEST+1 simply
does not exists at all.
I can understand this overflow successor function for a finite data
type or a real machine registe but not for N. The successor function
is simple, if you want it to have an exception at biggest you should
justify it.
Regards,
Quentin
2009/6/9 Brent Meeker meekerdb@...:
Quentin Anciaux wrote:
2009/6/9 Torgny Tholerus torgny@...:
Jesse Mazer skrev:
Date: Sat, 6 Jun 2009 21:17:03 +0200
From: torgny@...
To: everything-list@...
Subject: Re: The seven step-Mathematical preliminaries
My philosophical argument is about the mening of the word "all". To be
able to use that word, you must associate it with a value set.
What's a "value set"? And why do you say we "must" associate it in
this way? Do you have a philosophical argument for this "must", or is
it just an edict that reflects your personal aesthetic preferences?
Mostly that set is "all objects in the universe", and if you stay
inside the
universe, there is no problems.
*I* certainly don't define numbers in terms of any specific mapping
between numbers and objects in the universe, it seems like a rather
strange notion--shall we have arguments over whether the number 113485
should be associated with this specific shoelace or this specific
kangaroo?
When I talk about "universe" here, I do not mean our physical universe.
What I mean is something that can be called "everything". It includes
all objects in our physical universe, as well as all symbols and all
words and all numbers and all sets and all other universes. It includes
everything you can use the word "all" about.
It includes all set, but no all set as it N includes all natural
number but not all natural number... excuse-me but this is non-sense.
Either N exists and has an infinite number of member and is
incompatible with an ultrafinitist view or N does not exists because
obviously N cannot be defined in an ultra-finitist way,
That's not obvious to me. You're assuming that N exists apart from
whatever definition of it is given and that it is the infinite set
described by the Peano axioms or equivalent. But that's begging the
question of whether a finite set of numbers that we would call "natural
numbers" can be defined. To avoid begging the question we need some
definition of "natural" that doesn't a priori assume the set is finite
or infinite; something like, "A set of numbers adequate to do all
arithmetic we'll ever need" (unfortunately not very definite). The
problem is the successor axiom, if we modify it to S{n}=n+1 for n e N
except for the case n=N where S{N}=0 and choose sufficiently large N it
might satisfy the "natural" criteria.
Brent
any set that
contains a finite number of natural number (and still you haven't
defined what it is in an ultrafinitist way) are not the set N.
Also any operation involving two number (addition/multiplication) can
yield as result a number which has the same property as the departing
number (being a natural number) but is not natural number... Also
induction and inference cannot work in such a context.
For you to be able to use the word "all", you must define the "domain"
of that word. If you do not define the domain, then it will be
impossible for me and all other humans to understand what you are
talking about.
Well you are the first and only human I know who don't understand
"all" as everybody else does.
Quentin Anciaux
--
Torgny Tholerus
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Re: The seven step-Mathematical preliminaries
Let me correct...
Assuming your special successor rule BIGGEST+1 simply is 0 and is well
defined and *is part* of the previously defined set of natural number
(defined as 0,...,BIGGEST) unlike what Torgny argues.
Regards,
Quentin
2009/6/9 Quentin Anciaux < allcolor@...>:
> You have to explain why the exception is needed in the first place...
>
> The rule is true until the rule is not true anymore, ok but you have
> to explain for what sufficiently large N the successor function would
> yield next 0 and why or to add that N and that exception to the
> successor function as axiom, if not you can't avoid N+1. But torgny
> doesn't evacuate N+1, merely it allows his set to grows undefinitelly
> as when he has defined BIGGEST, he still argues BIGGEST+1 makes sense
> , is a natural number but not part of the set of natural number, this
> is non-sense, assuming your special successor rule BIGGEST+1 simply
> does not exists at all.
>
> I can understand this overflow successor function for a finite data
> type or a real machine registe but not for N. The successor function
> is simple, if you want it to have an exception at biggest you should
> justify it.
>
> Regards,
> Quentin
>
> 2009/6/9 Brent Meeker < meekerdb@...>:
>>
>> Quentin Anciaux wrote:
>>> 2009/6/9 Torgny Tholerus < torgny@...>:
>>>
>>>> Jesse Mazer skrev:
>>>>
>>>>>
>>>>>> Date: Sat, 6 Jun 2009 21:17:03 +0200
>>>>>> From: torgny@...
>>>>>> To: everything-list@...
>>>>>> Subject: Re: The seven step-Mathematical preliminaries
>>>>>>
>>>>>> My philosophical argument is about the mening of the word "all". To be
>>>>>> able to use that word, you must associate it with a value set.
>>>>>>
>>>>> What's a "value set"? And why do you say we "must" associate it in
>>>>> this way? Do you have a philosophical argument for this "must", or is
>>>>> it just an edict that reflects your personal aesthetic preferences?
>>>>>
>>>>>
>>>>>> Mostly that set is "all objects in the universe", and if you stay
>>>>>>
>>>>> inside the
>>>>>
>>>>>> universe, there is no problems.
>>>>>>
>>>>> *I* certainly don't define numbers in terms of any specific mapping
>>>>> between numbers and objects in the universe, it seems like a rather
>>>>> strange notion--shall we have arguments over whether the number 113485
>>>>> should be associated with this specific shoelace or this specific
>>>>> kangaroo?
>>>>>
>>>> When I talk about "universe" here, I do not mean our physical universe.
>>>> What I mean is something that can be called "everything". It includes
>>>> all objects in our physical universe, as well as all symbols and all
>>>> words and all numbers and all sets and all other universes. It includes
>>>> everything you can use the word "all" about.
>>>>
>>>
>>> It includes all set, but no all set as it N includes all natural
>>> number but not all natural number... excuse-me but this is non-sense.
>>> Either N exists and has an infinite number of member and is
>>> incompatible with an ultrafinitist view or N does not exists because
>>> obviously N cannot be defined in an ultra-finitist way,
>>
>> That's not obvious to me. You're assuming that N exists apart from
>> whatever definition of it is given and that it is the infinite set
>> described by the Peano axioms or equivalent. But that's begging the
>> question of whether a finite set of numbers that we would call "natural
>> numbers" can be defined. To avoid begging the question we need some
>> definition of "natural" that doesn't a priori assume the set is finite
>> or infinite; something like, "A set of numbers adequate to do all
>> arithmetic we'll ever need" (unfortunately not very definite). The
>> problem is the successor axiom, if we modify it to S{n}=n+1 for n e N
>> except for the case n=N where S{N}=0 and choose sufficiently large N it
>> might satisfy the "natural" criteria.
>>
>> Brent
>>
>>
>>> any set that
>>> contains a finite number of natural number (and still you haven't
>>> defined what it is in an ultrafinitist way) are not the set N.
>>>
>>> Also any operation involving two number (addition/multiplication) can
>>> yield as result a number which has the same property as the departing
>>> number (being a natural number) but is not natural number... Also
>>> induction and inference cannot work in such a context.
>>>
>>>
>>>> For you to be able to use the word "all", you must define the "domain"
>>>> of that word. If you do not define the domain, then it will be
>>>> impossible for me and all other humans to understand what you are
>>>> talking about.
>>>>
>>>
>>> Well you are the first and only human I know who don't understand
>>> "all" as everybody else does.
>>>
>>> Quentin Anciaux
>>>
>>>
>>>> --
>>>> Torgny Tholerus
>>>>
>>>>
>>>
>>>
>>>
>>>
>>
>>
>> >>
>>
>
>
>
> --
> All those moments will be lost in time, like tears in rain.
>
--
All those moments will be lost in time, like tears in rain.
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RE: The seven step-Mathematical preliminaries
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> Date: Tue, 9 Jun 2009 18:38:23 +0200 > From: torgny@...> To: everything-list@...> Subject: Re: The seven step-Mathematical preliminaries > > > Jesse Mazer skrev: >> >> >>> Date: Sat, 6 Jun 2009 21:17:03 +0200 >>> From: torgny@...>>> To: everything-list@...>>> Subject: Re: The seven step-Mathematical preliminaries >>> >>> My philosophical argument is about the mening of the word "all". To be >>> able to use that word, you must associate it with a value set. >> >> What's a "value set"? And why do you say we "must" associate it in >> this way? Do you have a philosophical argument for this "must", or is >> it just an edict that reflects your personal aesthetic preferences? >> >>> Mostly that set is "all objects in the universe", and if you stay >> inside the >>> universe, there is no problems. >> >> *I* certainly don't define numbers in terms of any specific mapping >> between numbers and objects in the universe, it seems like a rather >> strange notion--shall we have arguments over whether the number 113485 >> should be associated with this specific shoelace or this specific >> kangaroo? > > When I talk about "universe" here, I do not mean our physical universe. > What I mean is something that can be called "everything". It includes > all objects in our physical universe, as well as all symbols and all > words and all numbers and all sets and all other universes. It includes > everything you can use the word "all" about. > > For you to be able to use the word "all", you must define the "domain" > of that word. If you do not define the domain, then it will be > impossible for me and all other humans to understand what you are > talking about. OK, so how do you say I should define this type of "universe"? Unless you are demanding that I actually give you a list which spells out every symbol-string that qualifies as a member, can't I simply provide an abstract *rule* that would allow someone to determine in principle if a particular symbol-string they are given qualifies? Or do you have a third alternative besides spelling out every member or giving an abstract rule?
Jesse
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Re: The seven step-Mathematical preliminaries
Quentin Anciaux wrote:
> You have to explain why the exception is needed in the first place...
>
> The rule is true until the rule is not true anymore, ok but you have
> to explain for what sufficiently large N the successor function would
> yield next 0 and why or to add that N and that exception to the
> successor function as axiom, if not you can't avoid N+1. But torgny
> doesn't evacuate N+1, merely it allows his set to grows undefinitelly
> as when he has defined BIGGEST, he still argues BIGGEST+1 makes sense
> , is a natural number but not part of the set of natural number, this
> is non-sense, assuming your special successor rule BIGGEST+1 simply
> does not exists at all.
>
> I can understand this overflow successor function for a finite data
> type or a real machine registe but not for N. The successor function
> is simple, if you want it to have an exception at biggest you should
> justify it.
You don't justify definitions. How would you justify Peano's axioms as being
the "right" ones? You are just confirming my point that you are begging the
question by assuming there is a set called "the natural numbers" that exists
independently of it's definition and it satisfies Peano's axioms. Torgny is
denying that and pointing out that we cannot know of infinite sets that exist
independent of their definition because we cannot extensively define an infinite
set, we can only know about it what we can prove from its definition.
So the numbers modulo BIGGEST+1 and Peano's numbers are both mathematical
objects. The first however is more definite than the second, since Godel's
theorems don't apply. Which one is called the *natural* numbers is a convention
which might not have any practical consequences for sufficiently large BIGGEST.
Brent
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Re: The seven step-Mathematical preliminaries
2009/6/9 Brent Meeker < meekerdb@...>:
>
> Quentin Anciaux wrote:
>> You have to explain why the exception is needed in the first place...
>>
>> The rule is true until the rule is not true anymore, ok but you have
>> to explain for what sufficiently large N the successor function would
>> yield next 0 and why or to add that N and that exception to the
>> successor function as axiom, if not you can't avoid N+1. But torgny
>> doesn't evacuate N+1, merely it allows his set to grows undefinitelly
>> as when he has defined BIGGEST, he still argues BIGGEST+1 makes sense
>> , is a natural number but not part of the set of natural number, this
>> is non-sense, assuming your special successor rule BIGGEST+1 simply
>> does not exists at all.
>>
>> I can understand this overflow successor function for a finite data
>> type or a real machine registe but not for N. The successor function
>> is simple, if you want it to have an exception at biggest you should
>> justify it.
>
> You don't justify definitions.
then you say it is an axiom, no problem with that.
> How would you justify Peano's axioms as being the "right" ones?
You don't, and either I misexpressed myself or you did not understood.
> You are just confirming my point that you are begging the
> question by assuming there is a set called "the natural numbers" that exists
> independently of it's definition and it satisfies Peano's axioms.
No, I have a definition for a set called the set of natural number,
this set is defined by the peano's axioms and the set defined by these
axioms is unbounded and it is called the set of natural number. Any
upper limit bounded set containing natural number is not N but a
subset of N.
http://en.wikipedia.org/wiki/Natural_number#Formal_definitionsThe set Torgny is talking about is not N, like a dog is not a cat, he
can call it whatever he likes but not N.
But merely what I want to point out is that the definition he use is
inconsistent unlike yours which is simply modulo arithmetics.
http://en.wikipedia.org/wiki/Modular_arithmetic
> Torgny is
> denying that and pointing out that we cannot know of infinite sets that exist
> independent of their definition because we cannot extensively define an infinite
> set, we can only know about it what we can prove from its definition.
>
> So the numbers modulo BIGGEST+1 and Peano's numbers are both mathematical
> objects. The first however is more definite than the second, since Godel's
> theorems don't apply. Which one is called the *natural* numbers is a convention
> which might not have any practical consequences for sufficiently large BIGGEST.
>
> Brent
>
>
> >
>
--
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Re: The seven step-Mathematical preliminaries
2009/6/9 Quentin Anciaux < allcolor@...>:
> 2009/6/9 Brent Meeker < meekerdb@...>:
>>
>> Quentin Anciaux wrote:
>>> You have to explain why the exception is needed in the first place...
>>>
>>> The rule is true until the rule is not true anymore, ok but you have
>>> to explain for what sufficiently large N the successor function would
>>> yield next 0 and why or to add that N and that exception to the
>>> successor function as axiom, if not you can't avoid N+1. But torgny
>>> doesn't evacuate N+1, merely it allows his set to grows undefinitelly
>>> as when he has defined BIGGEST, he still argues BIGGEST+1 makes sense
>>> , is a natural number but not part of the set of natural number, this
>>> is non-sense, assuming your special successor rule BIGGEST+1 simply
>>> does not exists at all.
>>>
>>> I can understand this overflow successor function for a finite data
>>> type or a real machine registe but not for N. The successor function
>>> is simple, if you want it to have an exception at biggest you should
>>> justify it.
>>
>> You don't justify definitions.
>
> then you say it is an axiom, no problem with that.
And your axiom can't just say there is a BIGGEST number without having
a rule to either find it or discriminate it or setting the value
arbitrarily.
BIGGEST must be a well defined number not a boundary that you can't
reach... because if it was the case you're no more an ultrafinitist
and N is not a problem.
>> How would you justify Peano's axioms as being the "right" ones?
>
> You don't, and either I misexpressed myself or you did not understood.
>
>> You are just confirming my point that you are begging the
>> question by assuming there is a set called "the natural numbers" that exists
>> independently of it's definition and it satisfies Peano's axioms.
>
> No, I have a definition for a set called the set of natural number,
> this set is defined by the peano's axioms and the set defined by these
> axioms is unbounded and it is called the set of natural number. Any
> upper limit bounded set containing natural number is not N but a
> subset of N.
>
> http://en.wikipedia.org/wiki/Natural_number#Formal_definitions>
> The set Torgny is talking about is not N, like a dog is not a cat, he
> can call it whatever he likes but not N.
> But merely what I want to point out is that the definition he use is
> inconsistent unlike yours which is simply modulo arithmetics.
>
> http://en.wikipedia.org/wiki/Modular_arithmetic>
>
>
>> Torgny is
>> denying that and pointing out that we cannot know of infinite sets that exist
>> independent of their definition because we cannot extensively define an infinite
>> set, we can only know about it what we can prove from its definition.
>>
>> So the numbers modulo BIGGEST+1 and Peano's numbers are both mathematical
>> objects. The first however is more definite than the second, since Godel's
>> theorems don't apply. Which one is called the *natural* numbers is a convention
>> which might not have any practical consequences for sufficiently large BIGGEST.
>>
>> Brent
>>
>>
>> >>
>>
>
>
>
> --
> All those moments will be lost in time, like tears in rain.
>
--
All those moments will be lost in time, like tears in rain.
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Re: The seven step-Mathematical preliminaries
A good model of the naturalist math that Torgny is talking about is the
overflow mechanism in computers.
For example in a 64 bit machine you may define overflow for positive
integers as 2^^64 -1. If negative integers are included then the
biggest positive could be 2^^32-1.
Torgny would also have to define the operations +, - x / with specific
exceptions for overflow.
The concept of BIGGEST needs to be tied with the kind of operations
you want to apply to the numbers.
George
Brent Meeker wrote:
Quentin Anciaux wrote:
You have to explain why the exception is needed in the first place...
The rule is true until the rule is not true anymore, ok but you have
to explain for what sufficiently large N the successor function would
yield next 0 and why or to add that N and that exception to the
successor function as axiom, if not you can't avoid N+1. But torgny
doesn't evacuate N+1, merely it allows his set to grows undefinitelly
as when he has defined BIGGEST, he still argues BIGGEST+1 makes sense
, is a natural number but not part of the set of natural number, this
is non-sense, assuming your special successor rule BIGGEST+1 simply
does not exists at all.
I can understand this overflow successor function for a finite data
type or a real machine registe but not for N. The successor function
is simple, if you want it to have an exception at biggest you should
justify it.
You don't justify definitions. How would you justify Peano's axioms as being
the "right" ones? You are just confirming my point that you are begging the
question by assuming there is a set called "the natural numbers" that exists
independently of it's definition and it satisfies Peano's axioms. Torgny is
denying that and pointing out that we cannot know of infinite sets that exist
independent of their definition because we cannot extensively define an infinite
set, we can only know about it what we can prove from its definition.
So the numbers modulo BIGGEST+1 and Peano's numbers are both mathematical
objects. The first however is more definite than the second, since Godel's
theorems don't apply. Which one is called the *natural* numbers is a convention
which might not have any practical consequences for sufficiently large BIGGEST.
Brent
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RE: The seven step-Mathematical preliminaries
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> Date: Tue, 9 Jun 2009 12:54:16 -0700 > From: meekerdb@...> To: everything-list@...> Subject: Re: The seven step-Mathematical preliminaries > > You don't justify definitions. How would you justify Peano's axioms as being > the "right" ones? You are just confirming my point that you are begging the > question by assuming there is a set called "the natural numbers" that exists > independently of it's definition and it satisfies Peano's axioms.
What do you mean by "exists" in this context? What would it mean to have a well-defined, non-contradictory definition of some mathematical objects, and yet for those mathematical objects not to "exist"?
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Re: The seven step-Mathematical preliminaries
Jesse Mazer wrote:
>
>
> > Date: Tue, 9 Jun 2009 12:54:16 -0700
> > From: meekerdb@...
> > To: everything-list@...
> > Subject: Re: The seven step-Mathematical preliminaries
> >
>
> > You don't justify definitions. How would you justify Peano's axioms
> as being
> > the "right" ones? You are just confirming my point that you are
> begging the
> > question by assuming there is a set called "the natural numbers"
> that exists
> > independently of it's definition and it satisfies Peano's axioms.
>
> What do you mean by "exists" in this context? What would it mean to
> have a well-defined, non-contradictory definition of some mathematical
> objects, and yet for those mathematical objects not to "exist"?
A good question. But if one talks about some mathematical object, like
the natural numbers, having properties that are unprovable from their
defining set of axioms then it seems that one has assumed some kind of
existence apart from the particular definition. Everybody believes
arithmetic, per Peano's axioms, is consistent, but we know that can't be
proved from Peano's axioms. So it seems we are assigning (or betting
on, as Bruno might say) more existence than is implied by the definition.
When Quentin insists that Peano's axioms are the right ones for the
natural numbers, he is either just making a statement about language
conventions, or he has an idea of the natural numbers that is
independent of the axioms and is saying the axioms pick out the right
set of natural numbers.
Brent
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Re: The seven step-Mathematical preliminaries
2009/6/10 Brent Meeker < meekerdb@...>:
>
> Jesse Mazer wrote:
>>
>>
>> > Date: Tue, 9 Jun 2009 12:54:16 -0700
>> > From: meekerdb@...
>> > To: everything-list@...
>> > Subject: Re: The seven step-Mathematical preliminaries
>> >
>>
>> > You don't justify definitions. How would you justify Peano's axioms
>> as being
>> > the "right" ones? You are just confirming my point that you are
>> begging the
>> > question by assuming there is a set called "the natural numbers"
>> that exists
>> > independently of it's definition and it satisfies Peano's axioms.
>>
>> What do you mean by "exists" in this context? What would it mean to
>> have a well-defined, non-contradictory definition of some mathematical
>> objects, and yet for those mathematical objects not to "exist"?
>
> A good question. But if one talks about some mathematical object, like
> the natural numbers, having properties that are unprovable from their
> defining set of axioms then it seems that one has assumed some kind of
> existence apart from the particular definition. Everybody believes
> arithmetic, per Peano's axioms, is consistent, but we know that can't be
> proved from Peano's axioms. So it seems we are assigning (or betting
> on, as Bruno might say) more existence than is implied by the definition.
>
> When Quentin insists that Peano's axioms are the right ones for the
> natural numbers, he is either just making a statement about language
> conventions, or he has an idea of the natural numbers that is
> independent of the axioms and is saying the axioms pick out the right
> set of natural numbers.
>
> Brent
No I'm actually saying that peano's axiom define the abstract rules
which permits to know if a number is a natural number or not. A number
is a natural number if it satisfies peano's axiom... so by definition
the set created by the numbers satisfying these rules is the set of
all natural numbers. So if you change the rules, you change the set
hence the new set(s) created by your new rules (axiom) is(are) not the
same set(s) than the one denoted by peano's axioms hence it is not N
and can't be by definition. The mathematical object you define with
your new rules is not the same.
And please note that modulo arithmetic is not the problem here. Torgny
is not talking about that, he said BIGGEST+1 is not in the set N, but
BIGGEST+1 is a natural number (Question1: What is a natural number ?,
Question2: How can a natural number not be in the set of **all**
natural numbers ?). With your version with modulo(BIGGEST), BIGGEST+1
is in the previously defined set, it is '0'. And in your version
BIGGEST+1 doesn't satisfy that it is strictly bigger than BIGGEST, but
in Torgny version it does.
Regards,
Quentin
>
> >
>
--
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RE: The seven step-Mathematical preliminaries
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> Date: Tue, 9 Jun 2009 15:22:10 -0700 > From: meekerdb@...> To: everything-list@...> Subject: Re: The seven step-Mathematical preliminaries > > > Jesse Mazer wrote: >> >> >>> Date: Tue, 9 Jun 2009 12:54:16 -0700 >>> From: meekerdb@...>>> To: everything-list@...>>> Subject: Re: The seven step-Mathematical preliminaries >>> >> >>> You don't justify definitions. How would you justify Peano's axioms >> as being >>> the "right" ones? You are just confirming my point that you are >> begging the >>> question by assuming there is a set called "the natural numbers" >> that exists >>> independently of it's definition and it satisfies Peano's axioms. >> >> What do you mean by "exists" in this context? What would it mean to >> have a well-defined, non-contradictory definition of some mathematical >> objects, and yet for those mathematical objects not to "exist"? > > A good question. But if one talks about some mathematical object, like > the natural numbers, having properties that are unprovable from their > defining set of axioms then it seems that one has assumed some kind of > existence apart from the particular definition.
Isn't this based on the idea that there should be an objective truth about every well-formed proposition about the natural numbers even if the Peano axioms cannot decide the truth about all propositions? I think that the statements that cannot be proved are disproved would all be ones of the type "for all numbers with property X, Y is true" or "there exists a number (or some finite group of numbers) with property X" (i.e. propositions using either the 'for all' or 'there exists' universal quantifiers in logic, with variables representing specific numbers or groups of numbers). So to believe these statements are objectively true basically means there would be a unique way to "extend" our judgment of the truth-values of propositions from the judgments already given by the Peano axioms, in such a way that if we could flip through all the infinite propositions judged true by the Peano axioms, we would *not* find an example of a proposition like "for this specific number N with property X, Y is false" (which would disprove the 'for all' proposition above), and likewise we would not find that for every possible number (or group of numbers) N, the Peano axioms proved a proposition like "number N does not have property X" (which would disprove the 'there exists' proposition above).
We can't actual flip through an infinite number of propositions in a finite time of course, but if we had a "hypercomputer" that could do so (which is equivalent to the notion of a hypercomputer that can decide in finite time if any given Turing program halts or not), then I think we'd have a well-defined notion of how to program it to decide the truth of every "for all" or "there exists" proposition in a way that's compatible with the propositions already proved by the Peano axioms. If I'm right about that, it would lead naturally to the idea of something like a "unique consistent extension" of the Peano axioms (not a real technical term, I just made up this phrase, but unless there's an error in my reasoning I imagine mathematicians have some analogous notion...maybe Bruno knows?) which assigns truth values to all the well-formed propositions that are undecidable by the Peano axioms themselves. So this would be a natural way of understanding the idea of truths "about the natural numbers" that are not decidable by the Peano axioms.
Of course even if the notion of a "unique consistent extension" of certain types of axiomatic systems is well-defined, it would only make sense for axiomatic systems that are consistent in the first place. I guess in judging the question of the consistency of the Peano axioms, we must rely on some sort of ill-defined notion of our "understanding" of how the axioms should represent true statements about things like counting discrete objects. For example, we understand that the order you count a group of discrete objects doesn't affect the total number, which is a convincing argument for believing that A + B = B + A regardless of what numbers you choose for A and B. Likewise, we understand that multiplying A * B can be thought of in terms of a square array of discrete objects with the horizontal side having A objects and the vertical side having B objects, and we can see that just by rotating this you get a square array with B on the horizontal side and A on the vertical side, so if we believe that just mentally rotating an array of discrete objects won't change the number in the array that's a good argument for believing A * B = B * A. So thinking along these lines, as long as we don't believe that true statements about counting collections of discrete objects could ever lead to logical contradictions, we should believe the same for the Peano axioms.
Jesse
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Re: The seven step-Mathematical preliminaries
Jesse Mazer wrote:
>
>
> > Date: Tue, 9 Jun 2009 15:22:10 -0700
> > From: meekerdb@...
> > To: everything-list@...
> > Subject: Re: The seven step-Mathematical preliminaries
> >
> >
> > Jesse Mazer wrote:
> >>
> >>
> >>> Date: Tue, 9 Jun 2009 12:54:16 -0700
> >>> From: meekerdb@...
> >>> To: everything-list@...
> >>> Subject: Re: The seven step-Mathematical preliminaries
> >>>
> >>
> >>> You don't justify definitions. How would you justify Peano's axioms
> >> as being
> >>> the "right" ones? You are just confirming my point that you are
> >> begging the
> >>> question by assuming there is a set called "the natural numbers"
> >> that exists
> >>> independently of it's definition and it satisfies Peano's axioms.
> >>
> >> What do you mean by "exists" in this context? What would it mean to
> >> have a well-defined, non-contradictory definition of some mathematical
> >> objects, and yet for those mathematical objects not to "exist"?
> >
> > A good question. But if one talks about some mathematical object, like
> > the natural numbers, having properties that are unprovable from their
> > defining set of axioms then it seems that one has assumed some kind of
> > existence apart from the particular definition.
>
> Isn't this based on the idea that there should be an objective truth
> about every well-formed proposition about the natural numbers even if
> the Peano axioms cannot decide the truth about all propositions? I
> think that the statements that cannot be proved are disproved would
> all be ones of the type "for all numbers with property X, Y is true"
> or "there exists a number (or some finite group of numbers) with
> property X" (i.e. propositions using either the 'for all' or 'there
> exists' universal quantifiers in logic, with variables representing
> specific numbers or groups of numbers). So to believe these statements
> are objectively true basically means there would be a unique way to
> "extend" our judgment of the truth-values of propositions from the
> judgments already given by the Peano axioms, in such a way that if we
> could flip through all the infinite propositions judged true by the
> Peano axioms, we would *not* find an example of a proposition like
> "for this specific number N with property X, Y is false" (which would
> disprove the 'for all' proposition above), and likewise we would not
> find that for every possible number (or group of numbers) N, the Peano
> axioms proved a proposition like "number N does not have property X"
> (which would disprove the 'there exists' proposition above).
>
> We can't actual flip through an infinite number of propositions in a
> finite time of course, but if we had a "hypercomputer" that could do
> so (which is equivalent to the notion of a hypercomputer that can
> decide in finite time if any given Turing program halts or not), then
> I think we'd have a well-defined notion of how to program it to decide
> the truth of every "for all" or "there exists" proposition in a way
> that's compatible with the propositions already proved by the Peano
> axioms. If I'm right about that, it would lead naturally to the idea
> of something like a "unique consistent extension" of the Peano axioms
> (not a real technical term, I just made up this phrase, but unless
> there's an error in my reasoning I imagine mathematicians have some
> analogous notion...maybe Bruno knows?) which assigns truth values to
> all the well-formed propositions that are undecidable by the Peano
> axioms themselves. So this would be a natural way of understanding the
> idea of truths "about the natural numbers" that are not decidable by
> the Peano axioms.
I think Godel's imcompleteness theorem already implies that there must
be non-unique extensions, (e.g. maybe you can add an axiom either that
there are infinitely many pairs of primes differing by two or the
negative of that). That would seem to be a reductio against the
existence of a hypercomputer that could decide these propositions by
inspection.
>
> Of course even if the notion of a "unique consistent extension" of
> certain types of axiomatic systems is well-defined, it would only make
> sense for axiomatic systems that are consistent in the first place. I
> guess in judging the question of the consistency of the Peano axioms,
> we must rely on some sort of ill-defined notion of our "understanding"
> of how the axioms should represent true statements about things like
> counting discrete objects. For example, we understand that the order
> you count a group of discrete objects doesn't affect the total number,
> which is a convincing argument for believing that A + B = B + A
> regardless of what numbers you choose for A and B. Likewise, we
> understand that multiplying A * B can be thought of in terms of a
> square array of discrete objects with the horizontal side having A
> objects and the vertical side having B objects, and we can see that
> just by rotating this you get a square array with B on the horizontal
> side and A on the vertical side, so if we believe that just mentally
> rotating an array of discrete objects won't change the number in the
> array that's a good argument for believing A * B = B * A. So thinking
> along these lines, as long as we don't believe that true statements
> about counting collections of discrete objects could ever lead to
> logical contradictions, we should believe the same for the Peano axioms.
>
> Jesse
So we believe in the consistency of Peano's arithmetic because we have a
physical model.
Brent
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RE: The seven step-Mathematical preliminaries
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> Date: Tue, 9 Jun 2009 17:20:39 -0700 > From: meekerdb@...> To: everything-list@...> Subject: Re: The seven step-Mathematical preliminaries > > > Jesse Mazer wrote: >> >> >>> Date: Tue, 9 Jun 2009 15:22:10 -0700 >>> From: meekerdb@...>>> To: everything-list@...>>> Subject: Re: The seven step-Mathematical preliminaries >>> >>> >>> Jesse Mazer wrote: >>>> >>>> >>>>> Date: Tue, 9 Jun 2009 12:54:16 -0700 >>>>> From: meekerdb@...>>>>> To: everything-list@...>>>>> Subject: Re: The seven step-Mathematical preliminaries >>>>> >>>> >>>>> You don't justify definitions. How would you justify Peano's axioms >>>> as being >>>>> the "right" ones? You are just confirming my point that you are >>>> begging the >>>>> question by assuming there is a set called "the natural numbers" >>>> that exists >>>>> independently of it's definition and it satisfies Peano's axioms. >>>> >>>> What do you mean by "exists" in this context? What would it mean to >>>> have a well-defined, non-contradictory definition of some mathematical >>>> objects, and yet for those mathematical objects not to "exist"? >>> >>> A good question. But if one talks about some mathematical object, like >>> the natural numbers, having properties that are unprovable from their >>> defining set of axioms then it seems that one has assumed some kind of >>> existence apart from the particular definition. >> >> Isn't this based on the idea that there should be an objective truth >> about every well-formed proposition about the natural numbers even if >> the Peano axioms cannot decide the truth about all propositions? I >> think that the statements that cannot be proved are disproved would >> all be ones of the type "for all numbers with property X, Y is true" >> or "there exists a number (or some finite group of numbers) with >> property X" (i.e. propositions using either the 'for all' or 'there >> exists' universal quantifiers in logic, with variables representing >> specific numbers or groups of numbers). So to believe these statements >> are objectively true basically means there would be a unique way to >> "extend" our judgment of the truth-values of propositions from the >> judgments already given by the Peano axioms, in such a way that if we >> could flip through all the infinite propositions judged true by the >> Peano axioms, we would *not* find an example of a proposition like >> "for this specific number N with property X, Y is false" (which would >> disprove the 'for all' proposition above), and likewise we would not >> find that for every possible number (or group of numbers) N, the Peano >> axioms proved a proposition like "number N does not have property X" >> (which would disprove the 'there exists' proposition above). >> >> We can't actual flip through an infinite number of propositions in a >> finite time of course, but if we had a "hypercomputer" that could do >> so (which is equivalent to the notion of a hypercomputer that can >> decide in finite time if any given Turing program halts or not), then >> I think we'd have a well-defined notion of how to program it to decide >> the truth of every "for all" or "there exists" proposition in a way >> that's compatible with the propositions already proved by the Peano >> axioms. If I'm right about that, it would lead naturally to the idea >> of something like a "unique consistent extension" of the Peano axioms >> (not a real technical term, I just made up this phrase, but unless >> there's an error in my reasoning I imagine mathematicians have some >> analogous notion...maybe Bruno knows?) which assigns truth values to >> all the well-formed propositions that are undecidable by the Peano >> axioms themselves. So this would be a natural way of understanding the >> idea of truths "about the natural numbers" that are not decidable by >> the Peano axioms. > > I think Godel's imcompleteness theorem already implies that there must > be non-unique extensions, (e.g. maybe you can add an axiom either that > there are infinitely many pairs of primes differing by two or the > negative of that). That would seem to be a reductio against the > existence of a hypercomputer that could decide these propositions by > inspection.
I think I remember reading in one of Roger Penrose's books that there is a difference between an ordinary consistency condition (which just means that no two propositions explicitly contradict each other) and "omega-consistency"--see http://en.wikipedia.org/wiki/Omega-consistent_theory . I can't quite follow the details, but I'm guessing the condition means (or at least includes) something like the idea that if you have a statement of the form "there exists a number (or set of numbers) with property X" then there must actually be some other proposition describing a particular number (or set of numbers) does in fact have this property. The fact that you can add either a Godel statement or its negation to the Peano axioms without creating a contradiction (as long as the Peano axioms are not inconsistent) may not mean you can add either one and still have an omega-consistent theory; if that's true, would there be a unique omega-consistent way to set the truth value of all well-formed propositions about arithmetic which are undecidable by the Peano axioms? Again, Bruno might know...
>
> So we believe in the consistency of Peano's arithmetic because we have a
> physical model.
Well, I would say we generalize our understanding from a physical model, but once we have that understanding it's sort of generalized and doesn't depend on checking that things work for each specific number of discrete objects. For example, would you agree with the intuition that if we have a square array of idealized marbles, then simply mentally rotating it so we count them in a different order shouldn't change the total number of marbles in the array, so we can be confident that A*B = B*A for arbitrary numbers of marbles on the vertical and horizontal sides of the square?
Jesse
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Re: The seven step-Mathematical preliminaries
Jesse Mazer skrev:
>
>
> > Date: Tue, 9 Jun 2009 18:38:23 +0200
> > From: torgny@...
> > To: everything-list@...
> > Subject: Re: The seven step-Mathematical preliminaries
> >
> > For you to be able to use the word "all", you must define the "domain"
> > of that word. If you do not define the domain, then it will be
> > impossible for me and all other humans to understand what you are
> > talking about.
>
> OK, so how do you say I should define this type of "universe"? Unless
> you are demanding that I actually give you a list which spells out
> every symbol-string that qualifies as a member, can't I simply provide
> an abstract *rule* that would allow someone to determine in principle
> if a particular symbol-string they are given qualifies? Or do you have
> a third alternative besides spelling out every member or giving an
> abstract rule?
You have to spell out every member. Because in a *rule* you are
(implicitely) using this type of "universe", and you will then get a
circular definition. When you say that *every* number have a successor,
you are presupposing that you already know what *every* means.
--
Torgny Tholerus
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Re: The seven step-Mathematical preliminaries
Well if you have problem with word definition, use google then
http://www.google.be/search?source=ig&hl=fr&rlz=&=&q=define%3A+every&btnG=Recherche+Google&meta=lr%3D2009/6/10 Torgny Tholerus < torgny@...>:
>
> Jesse Mazer skrev:
>>
>>
>> > Date: Tue, 9 Jun 2009 18:38:23 +0200
>> > From: torgny@...
>> > To: everything-list@...
>> > Subject: Re: The seven step-Mathematical preliminaries
>> >
>> > For you to be able to use the word "all", you must define the "domain"
>> > of that word. If you do not define the domain, then it will be
>> > impossible for me and all other humans to understand what you are
>> > talking about.
>>
>> OK, so how do you say I should define this type of "universe"? Unless
>> you are demanding that I actually give you a list which spells out
>> every symbol-string that qualifies as a member, can't I simply provide
>> an abstract *rule* that would allow someone to determine in principle
>> if a particular symbol-string they are given qualifies? Or do you have
>> a third alternative besides spelling out every member or giving an
>> abstract rule?
>
> You have to spell out every member. Because in a *rule* you are
> (implicitely) using this type of "universe", and you will then get a
> circular definition. When you say that *every* number have a successor,
> you are presupposing that you already know what *every* means.
>
> --
> Torgny Tholerus
>
> >
>
--
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Re: The seven step-Mathematical preliminaries
Torgny,
your par. 1:
I like your including "all universes" into "UNIVERSE" if you talk about it. WE, here in this universe think about them. No contact, no lead, just our mental efforts. It all occurs in our prceived reality by thinking about more.
your par.2:
domain is tricky. I like to write about 'totality' vs 'models i.e. the identified cuts of it for our interest (other lists, other topics) and a smart fellow (NZ) replied: "your 'totality' IS a model. You identified it as 'all' (we can imagine)" - which is not "all that can, or cannot exist". Possible, or impossible in our present views.
BTW to 'understand' what somebody talks about is also tricky: we can only translate the 3rd pers. communication into our 1st pers. mindset so what we understand is not (necessarily) what the other said. Or wanted to say. Mindset is individual, no two persons can match in genetic origin (DNA, input of lineage, circumstances in gestational development, plus plus plus), AND the accumulated (personal) experience-material as applied to the individual life-history and emotional responses.
"Duo si faciunt idem, non est idem" valid in ideation as well.
I once wrote a sci-fi with an intelligent alien society where the communication consisted of direct transfer of ideas.
There was NO discussion.
Respectfully
John Mikes
On Tue, Jun 9, 2009 at 12:38 PM, Torgny Tholerus <torgny@...> wrote:
Jesse Mazer skrev: > > > > Date: Sat, 6 Jun 2009 21:17:03 +0200 > > From: torgny@...> > To: everything-list@...
> > Subject: Re: The seven step-Mathematical preliminaries > > > > My philosophical argument is about the mening of the word "all". To be
> > able to use that word, you must associate it with a value set.
>
> What's a "value set"? And why do you say we "must" associate it in
> this way? Do you have a philosophical argument for this "must", or is
> it just an edict that reflects your personal aesthetic preferences?
>
> > Mostly that set is "all objects in the universe", and if you stay
> inside the
> > universe, there is no problems.
>
> *I* certainly don't define numbers in terms of any specific mapping
> between numbers and objects in the universe, it seems like a rather
> strange notion--shall we have arguments over whether the number 113485
> should be associated with this specific shoelace or this specific
> kangaroo?
When I talk about "universe" here, I do not mean our physical universe.
What I mean is something that can be called "everything". It includes
all objects in our physical universe, as well as all symbols and all
words and all numbers and all sets and all other universes. It includes
everything you can use the word "all" about.
For you to be able to use the word "all", you must define the "domain"
of that word. If you do not define the domain, then it will be
impossible for me and all other humans to understand what you are
talking about.
--
Torgny Tholerus
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Re: The seven step-Mathematical preliminaries
On 10 Jun 2009, at 01:50, Jesse Mazer wrote:
Isn't this based on the idea that there should be an objective truth about every well-formed proposition about the natural numbers even if the Peano axioms cannot decide the truth about all propositions? I think that the statements that cannot be proved are disproved would all be ones of the type "for all numbers with property X, Y is true" or "there exists a number (or some finite group of numbers) with property X" (i.e. propositions using either the 'for all' or 'there exists' universal quantifiers in logic, with variables representing specific numbers or groups of numbers). So to believe these statements are objectively true basically means there would be a unique way to "extend" our judgment of the truth-values of propositions from the judgments already given by the Peano axioms, in such a way that if we could flip through all the infinite propositions judged true by the Peano axioms, we would *not* find an example of a proposition like "for this specific number N with property X, Y is false" (which would disprove the 'for all' proposition above), and likewise we would not find that for every possible number (or group of numbers) N, the Peano axioms proved a proposition like "number N does not have property X" (which would disprove the 'there exists' proposition above).
We can't actual flip through an infinite number of propositions in a finite time of course, but if we had a "hypercomputer" that could do so (which is equivalent to the notion of a hypercomputer that can decide in finite time if any given Turing program halts or not),
Such an hypercomputer is just what Turing called an "oracle". And the haslting oracle is very low in the hierarchy of possible oracles. And Turing results is that even a transfinite ladder of more and more powerful oracles that you can add on Peano Arithmetic, will not give you a complete theory. Hypercomputing by constructive extension of PA, with more and more powerful oracles does not help to overcome incompleteness, unless you add non constructive ordinal extension of "hypercomputation". This is the obeject of the study of the degrees of unsolvability, originated by Emil Post. Arithmetical truth is big. No notion of hypercomputing can really help. Yet, the notion of arithmetical truth is well understood by everybody, and is easily definable (as opposed to effectively decidable or computable) in usual set theory. That is why logician have no problem with the notion of standrd model of Peano Arithmetic, for example.
then I think we'd have a well-defined notion of how to program it to decide the truth of every "for all" or "there exists" proposition in a way that's compatible with the propositions already proved by the Peano axioms.
Hypercomputation will not help. Unless you go to the higher non constructive transfinite. But of course, in that case you are using a theory much more powerful than peano Arithmetic and its extension by constructive ordinal. You have to already believe in the notion of truth on numbers to do that.
If I'm right about that, it would lead naturally to the idea of something like a "unique consistent extension" of the Peano axioms (not a real technical term, I just made up this phrase, but unless there's an error in my reasoning I imagine mathematicians have some analogous notion...maybe Bruno knows?)
Just go to set theory. Arithmetical truth, or standard model of PA, can play that role. It is not effective (constructive) but it is well defined. Mathematicians used such notions everyday. If you belive in the excluded middle principle on closed arithmetical sentences, you are using implictly such notions.
which assigns truth values to all the well-formed propositions that are undecidable by the Peano axioms themselves.
You can do that in set theory. Of course, this is not an effective way to do it, but we know, by Godel, that completeness can never be given in any effective way. Set theory can define the standard model (your "unique extension") of PA, but it is not a constructive object. In set theory, few object are constructive.
So this would be a natural way of understanding the idea of truths "about the natural numbers" that are not decidable by the Peano axioms.
After Godel, truth, even on numbers can be well defined, in richer theory, but have to be non effective, non mechanical. It is not a reason to doubt about the truth of the arithmetical propositions. On the contrary it shows that such truth kicks back and refiute all effective definition we could belive in about that whole truth.
Of course even if the notion of a "unique consistent extension" of certain types of axiomatic systems is well-defined, it would only make sense for axiomatic systems that are consistent in the first place. I guess in judging the question of the consistency of the Peano axioms, we must rely on some sort of ill-defined notion of our "understanding" of how the axioms should represent true statements about things like counting discrete objects.
We have to rely on our intuition of numbers and sets. It is ill-defined, but today we know there is nothing better, and there there will never anything better. That is why it is difficult to refute an ultra-intuitionist, or why it is difficult to refute a zombie. Notion like "natural numbers" or "consciousness" just cannot be defined so as to be comprehensible by someone who does not already grasp those notions. That is the main reason of the failure of logicism.
For example, we understand that the order you count a group of discrete objects doesn't affect the total number, which is a convincing argument for believing that A + B = B + A regardless of what numbers you choose for A and B.
A + B = B + A, for all A and B, is already not provable in the Robinson arithmetic, but you can prove in Peano Arithmetic. You need the schema of axioms of induction.
Likewise, we understand that multiplying A * B can be thought of in terms of a square array of discrete objects with the horizontal side having A objects and the vertical side having B objects, and we can see that just by rotating this you get a square array with B on the horizontal side and A on the vertical side, so if we believe that just mentally rotating an array of discrete objects won't change the number in the array that's a good argument for believing A * B = B * A. So thinking along these lines, as long as we don't believe that true statements about counting collections of discrete objects could ever lead to logical contradictions, we should believe the same for the Peano axioms.
I would say it is a complete mystery why we believe in those axioms, but I have never meet someone who does not believe in it. Even Torgny is forced to believe in it so as to be able to assert that he does not believe in it. A real ultrafinitist cannot assert that he is ultrafinitist. A real ulrafinitist should just ask, what do you mean by natural numbers. And the only answer we can give him is "sorry pal, but this is stidied in primary school, and if you have not understand it (as opposed to some building of a sophisticated philosophical argument against them), there is nothing we can do for you. Of course Torgny know very well what we are talking about, like he knows very well what consciousness is.
You can see the UDA+AUDA as a reduction of the mind-body appearance to the mystery of numbers. The beauty of numbers, is that we can explain today in detail why, IF someone believe in numbers, then by work alone he can understand why the numbers are impossible to define. But it is a fact that they are very easy to grasp.
Bruno
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Re: The seven step-Mathematical preliminaries
On 10 Jun 2009, at 02:20, Brent Meeker wrote:
> I think Godel's imcompleteness theorem already implies that there must
> be non-unique extensions, (e.g. maybe you can add an axiom either that
> there are infinitely many pairs of primes differing by two or the
> negative of that). That would seem to be a reductio against the
> existence of a hypercomputer that could decide these propositions by
> inspection.
Not at all. Gödel's theorem implies that there must be non-unique
*consistent* extensions. But there is only one sound extension. The
unsound consistent extensions, somehow, does no more talk about
natural numbers.
Typical example: take the proposition that PA is inconsistant. By
Gödel's second incompletenss theorem, we have that PA+"PA is
inconsistent" is a consistent extension of PA. But it is not a sound
one. It affirms the existence of a number which is a Gödel number of a
proof of 0=1. But such a number is not a usual number at all.
An oracle for the whole arithmetical truth is well defined in set
theory, even if it is a non effective object.
Bruno
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Re: The seven step-Mathematical preliminaries
On 10 Jun 2009, at 04:14, Jesse Mazer wrote:
I think I remember reading in one of Roger Penrose's books that there is a difference between an ordinary consistency condition (which just means that no two propositions explicitly contradict each other) and "omega-consistency"--see http://en.wikipedia.org/wiki/Omega-consistent_theory . I can't quite follow the details, but I'm guessing the condition means (or at least includes) something like the idea that if you have a statement of the form "there exists a number (or set of numbers) with property X" then there must actually be some other proposition describing a particular number (or set of numbers) does in fact have this property. The fact that you can add either a Godel statement or its negation to the Peano axioms without creating a contradiction (as long as the Peano axioms are not inconsistent) may not mean you can add either one and still have an omega-consistent theory; if that's true, would there be a unique omega-consistent way to set the truth value of all well-formed propositions about arithmetic which are undecidable by the Peano axioms? Again, Bruno might know...
The notion of omega-consistency is a red-herring. The notion exists only for technical reason. Gödel did not succeed in proving the undecidability of its "Gödel-sentences" without using it, but this will be done succesfully by Rosser. Smullyan introduced better notion than omega-consistency, like his notion of stability, but personaly I prefer to use the (non effective, ok) notion of soundness, and use the notion of stability only latter in more advanced course. The notion of arithmetical soundness was not well seen at the time of Gödel, due to historical circumstances. That's all.
But the answer is "no". There are non unique omega-consistent extension of PA. Omega-consistency is just a bit more powerful than consistency for proviong undecidability, but Rosser has been able to replace omega-consistency by consistency in the proof of the existence of undecidable statements. Would Gödel have seen Rosser point before Rosser, the notion of omega-consistency could have not appeared at all.
I will probably come back on stability, consistency and soundness when we arrive at the AUDA part. This is not for tomorrow. I can give references, well see my URL.
Bruno
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Re: The seven step-Mathematical preliminaries
On 10 Jun 2009, at 02:20, Brent Meeker wrote:
> So we believe in the consistency of Peano's arithmetic because we
> have a
> physical model.
Why physical? And do we have a physical model? I would say we belive
in the consistency (and soundness) of PA because we have a model of
PA, the well known structure (N, 0, +, *).
If comp is true, there is no physical model at all. (But this is not
something on which I want to insist for now).
Bruno
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