Re: The seven step-Mathematical preliminaries
Jesse Mazer skrev: Date: Sat, 6 Jun 2009 21:17:03 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com 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 --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
2009/6/9 Torgny Tholerus tor...@dsv.su.se: Jesse Mazer skrev: Date: Sat, 6 Jun 2009 21:17:03 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com 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, 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. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
Quentin Anciaux wrote:
2009/6/9 Torgny Tholerus tor...@dsv.su.se:
Jesse Mazer skrev:
Date: Sat, 6 Jun 2009 21:17:03 +0200
From: tor...@dsv.su.se
To: everything-list@googlegroups.com
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
--~--~-~--~~~---~--~~
You received this message because you are subscribed to the Google Groups
Everything List group.
To post to this group, send email to everything-list@googlegroups.com
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Re: The seven step-Mathematical preliminaries
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 meeke...@dslextreme.com:
Quentin Anciaux wrote:
2009/6/9 Torgny Tholerus tor...@dsv.su.se:
Jesse Mazer skrev:
Date: Sat, 6 Jun 2009 21:17:03 +0200
From: tor...@dsv.su.se
To: everything-list@googlegroups.com
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.
--~--~-~--~~~---~--~~
You received this message because you are subscribed to the Google Groups
Everything List group.
To post to this group, send email to everything-list@googlegroups.com
To unsubscribe from this group, send email to
everything-list+unsubscr...@googlegroups.com
For more options, visit this group at
http://groups.google.com/group/everything-list?hl=en
-~--~~~~--~~--~--~---
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 meeke...@dslextreme.com:
Quentin Anciaux wrote:
2009/6/9 Torgny Tholerus tor...@dsv.su.se:
Jesse Mazer skrev:
Date: Sat, 6 Jun 2009 21:17:03 +0200
From: tor...@dsv.su.se
To: everything-list@googlegroups.com
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
--~--~-~--~~~---~--~~
You received this message because you are subscribed to the Google Groups
Everything List group.
To post to this group, send email to everything-list@googlegroups.com
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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 allco...@gmail.com:
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 meeke...@dslextreme.com:
Quentin Anciaux wrote:
2009/6/9 Torgny Tholerus tor...@dsv.su.se:
Jesse Mazer skrev:
Date: Sat, 6 Jun 2009 21:17:03 +0200
From: tor...@dsv.su.se
To: everything-list@googlegroups.com
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.
--~--~-~--~~~---~--~~
You received this message because you are subscribed to the Google Groups
Everything List group.
To post to this group, send email to everything-list@googlegroups.com
To unsubscribe from this group, send email to
everything-list+unsubscr...@googlegroups.com
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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 --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
RE: The seven step-Mathematical preliminaries
Date: Tue, 9 Jun 2009 18:38:23 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: Date: Sat, 6 Jun 2009 21:17:03 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com 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 --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
2009/6/9 Brent Meeker meeke...@dslextreme.com: 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_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. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
2009/6/9 Quentin Anciaux allco...@gmail.com: 2009/6/9 Brent Meeker meeke...@dslextreme.com: 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. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
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 --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
RE: The seven step-Mathematical preliminaries
Date: Tue, 9 Jun 2009 12:54:16 -0700 From: meeke...@dslextreme.com To: everything-list@googlegroups.com 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? --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
Jesse Mazer wrote: Date: Tue, 9 Jun 2009 12:54:16 -0700 From: meeke...@dslextreme.com To: everything-list@googlegroups.com 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 --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: The seven step-Mathematical preliminaries
2009/6/10 Brent Meeker meeke...@dslextreme.com: Jesse Mazer wrote: Date: Tue, 9 Jun 2009 12:54:16 -0700 From: meeke...@dslextreme.com To: everything-list@googlegroups.com 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 -- All those moments will be lost in time, like tears in rain. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
RE: The seven step-Mathematical preliminaries
Date: Tue, 9 Jun 2009 15:22:10 -0700 From: meeke...@dslextreme.com To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer wrote: Date: Tue, 9 Jun 2009 12:54:16 -0700 From: meeke...@dslextreme.com To: everything-list@googlegroups.com 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 --~--~-~--~~~---~--~~ You received this message because you are subscribed to the
Re: The seven step-Mathematical preliminaries
Jesse Mazer wrote: Date: Tue, 9 Jun 2009 15:22:10 -0700 From: meeke...@dslextreme.com To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer wrote: Date: Tue, 9 Jun 2009 12:54:16 -0700 From: meeke...@dslextreme.com To: everything-list@googlegroups.com 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
RE: The seven step-Mathematical preliminaries
Date: Tue, 9 Jun 2009 17:20:39 -0700 From: meeke...@dslextreme.com To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer wrote: Date: Tue, 9 Jun 2009 15:22:10 -0700 From: meeke...@dslextreme.com To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer wrote: Date: Tue, 9 Jun 2009 12:54:16 -0700 From: meeke...@dslextreme.com To: everything-list@googlegroups.com 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
