Re: The seven step-Mathematical preliminaries
Jesse Mazer skrev:
Date: Fri, 5 Jun 2009 08:33:47 +0200
From: tor...@dsv.su.se
To: everything-list@googlegroups.com
Subject: Re: The seven step-Mathematical preliminaries
Brian Tenneson skrev:
How can BIGGEST+1 be a natural number but not belong to the set of all
natural numbers?
One way to represent natural number as sets is:
0 = {}
1 = {0} = {{}}
2 = {0, 1} = 1 union {1} = {{}, {{}}}
3 = {0, 1, 2} = 2 union {2} = ...
. . .
n+1 = {0, 1, 2, ..., n} = n union {n}
. . .
Here you can then define that a is less then b if and only if a belongs
to b.
With this notation you get the set N of all natural numbers as {0,
1, 2,
...}. But the remarkable thing is that N is exactly the same as
BIGGEST+1. BIGGEST+1 is a set with the same structure as all the other
natural numbers, so it is then a natural number. But BIGGEST+1 is not a
member of N, the set of all natural numbers.
Here you're just contradicting yourself. If you say BIGGEST+1 is then
a natural number, that just proves that the set N was not in fact the
set of all natural numbers. The alternative would be to say
BIGGEST+1 is *not* a natural number, but then you need to provide a
definition of natural number that would explain why this is the case.
It depends upon how you define natural number. If you define it by: n
is a natural number if and only if n belongs to N, the set of all
natural numbers, then of course BIGGEST+1 is *not* a natural number. In
that case you have to call BIGGEST+1 something else, maybe unnatural
number.
--
Torgny Tholerus
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RE: The seven step-Mathematical preliminaries
Date: Sat, 6 Jun 2009 16:48:21 +0200
From: tor...@dsv.su.se
To: everything-list@googlegroups.com
Subject: Re: The seven step-Mathematical preliminaries
Jesse Mazer skrev:
Date: Fri, 5 Jun 2009 08:33:47 +0200
From: tor...@dsv.su.se
To: everything-list@googlegroups.com
Subject: Re: The seven step-Mathematical preliminaries
Brian Tenneson skrev:
How can BIGGEST+1 be a natural number but not belong to the set of all
natural numbers?
One way to represent natural number as sets is:
0 = {}
1 = {0} = {{}}
2 = {0, 1} = 1 union {1} = {{}, {{}}}
3 = {0, 1, 2} = 2 union {2} = ...
. . .
n+1 = {0, 1, 2, ..., n} = n union {n}
. . .
Here you can then define that a is less then b if and only if a belongs
to b.
With this notation you get the set N of all natural numbers as {0,
1, 2,
...}. But the remarkable thing is that N is exactly the same as
BIGGEST+1. BIGGEST+1 is a set with the same structure as all the other
natural numbers, so it is then a natural number. But BIGGEST+1 is not a
member of N, the set of all natural numbers.
Here you're just contradicting yourself. If you say BIGGEST+1 is then
a natural number, that just proves that the set N was not in fact the
set of all natural numbers. The alternative would be to say
BIGGEST+1 is *not* a natural number, but then you need to provide a
definition of natural number that would explain why this is the case.
It depends upon how you define natural number. If you define it by: n
is a natural number if and only if n belongs to N, the set of all
natural numbers, then of course BIGGEST+1 is *not* a natural number. In
that case you have to call BIGGEST+1 something else, maybe unnatural
number.
OK, but then you need to define what you mean by N, the set of all natural
numbers. Specifically you need to say what number is BIGGEST. Is it
arbitrary? Can I set BIGGEST = 3, for example? Or do you have some
philosophical ideas related to what BIGGEST is, like the number of particles in
the universe or the largest number any human can conceptualize?
Also, any comment on my point about there being an infinite number of possible
propositions about even a finite set, or about my question about whether you
have any philosophical/logical argument for saying all sets must be finite, as
opposed to it just being a sort of aesthetic preference on your part? Do you
think there is anything illogical or incoherent about defining a set in terms
of a rule that takes any input and decides whether it's a member of the set or
not, such that there may be no upper limit on the number of possible inputs
that the rule would define as being members? (such as would be the case for the
rule 'n is a natural number if n=1 or if n is equal to some other natural
number+1')
Jesse
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Re: The seven step-Mathematical preliminaries
Torgny Tholerus wrote:
Jesse Mazer skrev:
Date: Fri, 5 Jun 2009 08:33:47 +0200
From: tor...@dsv.su.se
To: everything-list@googlegroups.com
Subject: Re: The seven step-Mathematical preliminaries
Brian Tenneson skrev:
How can BIGGEST+1 be a natural number but not belong to the set of all
natural numbers?
One way to represent natural number as sets is:
0 = {}
1 = {0} = {{}}
2 = {0, 1} = 1 union {1} = {{}, {{}}}
3 = {0, 1, 2} = 2 union {2} = ...
. . .
n+1 = {0, 1, 2, ..., n} = n union {n}
. . .
Here you can then define that a is less then b if and only if a belongs
to b.
With this notation you get the set N of all natural numbers as {0,
1, 2,
...}. But the remarkable thing is that N is exactly the same as
BIGGEST+1. BIGGEST+1 is a set with the same structure as all the other
natural numbers, so it is then a natural number. But BIGGEST+1 is not a
member of N, the set of all natural numbers.
Here you're just contradicting yourself. If you say BIGGEST+1 is then
a natural number, that just proves that the set N was not in fact the
set of all natural numbers. The alternative would be to say
BIGGEST+1 is *not* a natural number, but then you need to provide a
definition of natural number that would explain why this is the case.
It depends upon how you define natural number. If you define it by: n
is a natural number if and only if n belongs to N, the set of all
natural numbers, then of course BIGGEST+1 is *not* a natural number. In
that case you have to call BIGGEST+1 something else, maybe unnatural
number.
I wonder if anyone has tried work with a theory of finite numbers: where
BIGGEST+1=BIGGEST or BIGGEST+1=-BIGGEST as in some computers?
Brent
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Re: The seven step-Mathematical preliminaries
I wonder if anyone has tried work with a theory of finite numbers: where BIGGEST+1=BIGGEST or BIGGEST+1=-BIGGEST as in some computers? There is a group of faculty who address this problem directly in my department. But any general-purpose computer can emulate true, unlimited natural numbers (which is what people often do, rather than relying on bounded ints). The only real limitations that make computer not-equal-to Turing machine are memory and the limited patience of humans. This is one reason why people spend more time researching P vs. NP than artificially-imposed limits. When you add bounds to numbers it requires additional proof obligations, which makes it more difficult to prove things. And you can't directly prove anything about numbers that exist outside the bounds under which you're working. Anna --~--~-~--~~~---~--~~ 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 2
Bruno, Before I leave on holiday, I am following your advice to make my own table of symbols. Let me ask first whether the smaller rectangles have a different reference from the larger ones as seen in your example below? - Original Message - From: Bruno Marchal To: everything-list@googlegroups.com Sent: Wednesday, June 03, 2009 1:15 PM Subject: Re: The seven step-Mathematical preliminaries 2 ∅ ∪ A = ∅ ∪ B = A ∪ ∅ = B ∪ ∅ = N ∩ ∅ = B ∩ ∅ = ∅ ∩ B = ∅ ∩ ∅ = ∅ ∪ ∅ = --- 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 -~--~~~~--~~--~--~--- --~--~-~--~~~---~--~~ 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 skrev: Date: Sat, 6 Jun 2009 16:48:21 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: Here you're just contradicting yourself. If you say BIGGEST+1 is then a natural number, that just proves that the set N was not in fact the set of all natural numbers. The alternative would be to say BIGGEST+1 is *not* a natural number, but then you need to provide a definition of natural number that would explain why this is the case. It depends upon how you define natural number. If you define it by: n is a natural number if and only if n belongs to N, the set of all natural numbers, then of course BIGGEST+1 is *not* a natural number. In that case you have to call BIGGEST+1 something else, maybe unnatural number. OK, but then you need to define what you mean by N, the set of all natural numbers. Specifically you need to say what number is BIGGEST. Is it arbitrary? Can I set BIGGEST = 3, for example? Or do you have some philosophical ideas related to what BIGGEST is, like the number of particles in the universe or the largest number any human can conceptualize? It is rather the last, the largest number any human can conceptualize. More natural numbers are not needed. Also, any comment on my point about there being an infinite number of possible propositions about even a finite set, There is not an infinite number of possible proposition. You can only create a finite number of proposition with finite length during your lifetime. Just like the number of natural numbers are unlimited but finite, so are the possible propositions unlimited but finte. or about my question about whether you have any philosophical/logical argument for saying all sets must be finite, 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. Mostly that set is all objects in the universe, and if you stay inside the universe, there is no problems. But as soon you go outside universe, you must be carefull with what substitutions you do. If you have all quantified with all object inside the universe, you can not substitute it with an object outside the universe, because that object was not included in the original statement. as opposed to it just being a sort of aesthetic preference on your part? Do you think there is anything illogical or incoherent about defining a set in terms of a rule that takes any input and decides whether it's a member of the set or not, such that there may be no upper limit on the number of possible inputs that the rule would define as being members? (such as would be the case for the rule 'n is a natural number if n=1 or if n is equal to some other natural number+1') In the last sentence you have an implicite all: The full sentence would be: For all n in the universe hold that n is a natural number if n=1 or if n is equal to some other natural number+1. And you may now be able to understand, that if the number of objects in the universe is finite, then this sentence will just define a finite set. -- 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
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 Jesse Mazer skrev: Date: Sat, 6 Jun 2009 16:48:21 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: Here you're just contradicting yourself. If you say BIGGEST+1 is then a natural number, that just proves that the set N was not in fact the set of all natural numbers. The alternative would be to say BIGGEST+1 is *not* a natural number, but then you need to provide a definition of natural number that would explain why this is the case. It depends upon how you define natural number. If you define it by: n is a natural number if and only if n belongs to N, the set of all natural numbers, then of course BIGGEST+1 is *not* a natural number. In that case you have to call BIGGEST+1 something else, maybe unnatural number. OK, but then you need to define what you mean by N, the set of all natural numbers. Specifically you need to say what number is BIGGEST. Is it arbitrary? Can I set BIGGEST = 3, for example? Or do you have some philosophical ideas related to what BIGGEST is, like the number of particles in the universe or the largest number any human can conceptualize? It is rather the last, the largest number any human can conceptualize. More natural numbers are not needed. Why humans, specifically? What if an alien could conceptualize a larger number? For that matter, since you deny any special role to consciousness, why should it have anything to do with the conceptualizations of beings with brains? A volume of space isn't normally said to conceptualize the number of atoms contained in that volume, but why should that number be any less real than the largest number that's been conceptualized by a biological brain? Also, any comment on my point about there being an infinite number of possible propositions about even a finite set, There is not an infinite number of possible proposition. You can only create a finite number of proposition with finite length during your lifetime. Just like the number of natural numbers are unlimited but finite, so are the possible propositions unlimited but finte. But you said earlier that as long as we admit only a finite collection of numbers, we can prove the consistency of mathematics involving only those numbers. Well, how can we prove that? If we only show that all the propositions we have generated to date are consistent, how do we know the next proposition we generate won't involve an inconsistency? Presumably you are implicitly suggesting there should be some upper limit on the number of propositions about the numbers as well as on the numbers themselves, but if you define this limit in terms of how many a human could generate in their lifetime, we get back to problems like what if some other being (genetically engineered humans, say) would have a longer lifetime, or what if we built a computer that generated propositions much faster than a human could and checked their consistency automatically, etc. or about my question about whether you have any philosophical/logical argument for saying all sets must be finite, 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? One of the first thing kids learn about number is that if you count some collection of objects, it doesn't matter what order you count them in, the final number you get will be the same regardless of the order (i.e. it doesn't matter which you point to when you say 1 and which you point to when you say 2, as long as you point to each object exactly once). Also, am I understanding correctly in thinking you don't believe there can be truths about numbers independent of what humans actually know about them (i.e. there is no truth about the sum of two very large numbers unless some human has actually calculated that sum at one point)? If in fact you don't believe there are truths about numbers independent of human thoughts about them, why do you think there can be truths about the physical universe which humans don't know about? For example, is there a truth about the surface topography of some planet that
Re: The seven step-Mathematical preliminaries 2
Marty, Bruno, Before I leave on holiday, I am following your advice to make my own table of symbols. Let me ask first whether the smaller rectangles have a different reference from the larger ones as seen in your example below? We do have problem of symbols, with the mail. I don't see any rectangle in the message below! Take it easy and . We will go very slowly. It will also be the exam periods. There is no rush ... Have a good holiday Bruno - Original Message - From: Bruno Marchal To: everything-list@googlegroups.com Sent: Wednesday, June 03, 2009 1:15 PM Subject: Re: The seven step-Mathematical preliminaries 2 ∅ ∪ A = ∅ ∪ B = A ∪ ∅ = B ∪ ∅ = N ∩ ∅ = B ∩ ∅ = ∅ ∩ B = ∅ ∩ ∅ = ∅ ∪ ∅ = --- 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 -~--~~~~--~~--~--~--- http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ 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: 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 Jesse Mazer skrev: Date: Sat, 6 Jun 2009 16:48:21 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: Here you're just contradicting yourself. If you say BIGGEST+1 is then a natural number, that just proves that the set N was not in fact the set of all natural numbers. The alternative would be to say BIGGEST+1 is *not* a natural number, but then you need to provide a definition of natural number that would explain why this is the case. It depends upon how you define natural number. If you define it by: n is a natural number if and only if n belongs to N, the set of all natural numbers, then of course BIGGEST+1 is *not* a natural number. In that case you have to call BIGGEST+1 something else, maybe unnatural number. OK, but then you need to define what you mean by N, the set of all natural numbers. Specifically you need to say what number is BIGGEST. Is it arbitrary? Can I set BIGGEST = 3, for example? Or do you have some philosophical ideas related to what BIGGEST is, like the number of particles in the universe or the largest number any human can conceptualize? It is rather the last, the largest number any human can conceptualize. More natural numbers are not needed. Why humans, specifically? What if an alien could conceptualize a larger number? For that matter, since you deny any special role to consciousness, why should it have anything to do with the conceptualizations of beings with brains? A volume of space isn't normally said to conceptualize the number of atoms contained in that volume, but why should that number be any less real than the largest number that's been conceptualized by a biological brain? Also, any comment on my point about there being an infinite number of possible propositions about even a finite set, There is not an infinite number of possible proposition. You can only create a finite number of proposition with finite length during your lifetime. Just like the number of natural numbers are unlimited but finite, so are the possible propositions unlimited but finte. But you said earlier that as long as we admit only a finite collection of numbers, we can prove the consistency of mathematics involving only those numbers. Well, how can we prove that? If we only show that all the propositions we have generated to date are consistent, how do we know the next proposition we generate won't involve an inconsistency? Presumably you are implicitly suggesting there should be some upper limit on the number of propositions about the numbers as well as on the numbers themselves, but if you define this limit in terms of how many a human could generate in their lifetime, we get back to problems like what if some other being (genetically engineered humans, say) would have a longer lifetime, or what if we built a computer that generated propositions much faster than a human could and checked their consistency automatically, etc. or about my question about whether you have any philosophical/logical argument for saying all sets must be finite, 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? One of the first thing kids learn about number is that if you count some collection of objects, it doesn't matter what order you count them in, the final number you get will be the same regardless of the order (i.e. it doesn't matter which you point to when you say 1 and which you point to when you say 2, as long as you point to each object exactly once). Also, am I understanding correctly in thinking you don't believe there can be truths about numbers independent of what humans actually know about them (i.e. there is no truth about the sum of two very large numbers unless some human has actually calculated that sum at one point)? If in fact you don't believe there are truths about numbers independent of human thoughts about them, why do you think there can be truths about the physical universe which
RE: The seven step-Mathematical preliminaries 2
If it helps, here's a screenshot of how the symbols are supposed to look: http://img34.imageshack.us/img34/3345/picture2uzk.png From: marc...@ulb.ac.be To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries 2 Date: Sat, 6 Jun 2009 22:36:01 +0200 Marty, Bruno, Before I leave on holiday, I am following your advice to make my own table of symbols. Let me ask first whether the smaller rectangles have a different reference from the larger ones as seen in your example below? We do have problem of symbols, with the mail. I don't see any rectangle in the message below! Take it easy and . We will go very slowly. It will also be the exam periods. There is no rush ... Have a good holiday Bruno - Original Message -From: Bruno MarchalTo: everything-l...@googlegroups.comsent: Wednesday, June 03, 2009 1:15 PMSubject: Re: The seven step-Mathematical preliminaries 2 ∅ ∪ A =∅ ∪ B =A ∪ ∅ =B ∪ ∅ =N ∩ ∅ =B ∩ ∅ =∅ ∩ B =∅ ∩ ∅ =∅ ∪ ∅ = ---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 -~--~~~~--~~--~--~--- http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ 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 2
(I'll be here till Tuesday.) Evidently, the symbol you are using for such that is being shown on my screen as a small rectangle. In the copy below, I see two rectangles before the A=, two before the B=, two after the A, two after the B. The UNION symbol (inverted U) shows up but is followed by a rectangle in the next two examples and preceded by a rectangle in the last three. In checking a table of logic notaion, I find that the relation such that is designated by a reversed capital E. Is this the symbol you are using? m.a. - Original Message - From: Bruno Marchal To: everything-list@googlegroups.com Sent: Saturday, June 06, 2009 4:36 PM Subject: Re: The seven step-Mathematical preliminaries 2 We do have problem of symbols, with the mail. I don't see any rectangle in the message below! Take it easy and . We will go very slowly. It will also be the exam periods. There is no rush ... Have a good holiday Bruno - Original Message - From: Bruno Marchal To: everything-list@googlegroups.com Sent: Wednesday, June 03, 2009 1:15 PM Subject: Re: The seven step-Mathematical preliminaries 2 ∅ ∪ A = ∅ ∪ B = A ∪ ∅ = B ∪ ∅ = N ∩ ∅ = B ∩ ∅ = ∅ ∩ B = ∅ ∩ ∅ = ∅ ∪ ∅ = --- 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 -~--~~~~--~~--~--~--- http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ 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 2
Bruno,
I've encountered some difficulty with the examples below. You say
that in extension describes exhaustion or quasi-exhaustion. And you give
the example: B = {3, 6, 9, 12, ... 99}.
Then you define in intension with exactly the same type of set:
Example: Let A be the set {2, 4, 6, 8, 10, ... 100}.
Can you see the cause of my confusion? Incidentally, may I suggest
you use smaller than rather than more little than. Your English is
generally too good to include that kind of error. marty a.
- Original Message -
From: Bruno Marchal
To: everything-list@googlegroups.com
Sent: Wednesday, June 03, 2009 1:15 PM
Subject: Re: The seven step-Mathematical preliminaries 2
=== Intension and extension
In the case of finite and little set we have seen that we can define them
by exhaustion. This means we can give an explicit complete description of all
element of the set.
Example. A = {0, 1, 2, 77, 98, 5}
When the set is still finite and too big, or if we are lazy, we can sometimes
define the set by quasi exhaustion. This means we describe enough elements of
the set in a manner which, by requiring some good will and some imagination, we
can estimate having define the set.
Example. B = {3, 6, 9, 12, ... 99}. We can understand in this case that we
meant the set of multiple of the number three, below 100.
A fortiori, when a set in not finite, that is, when the set is infinite, we
have to use either quasi-exhaustion, or we have to use some sentence or phrase
or proposition describing the elements of the set.
Definition.
I will say that a set is defined IN EXTENSIO, or simply, in extension, when
it is defined in exhaustion or quasi-exhaustion.
I will say that a set is defined IN INTENSIO, or simply in intension, with a
s, when it is defined by a sentence explaining the typical attribute of the
elements.
Example: Let A be the set {2, 4, 6, 8, 10, ... 100}. We can easily define A
in intension: A = the set of numbers which are even and more little than 100.
mathematician will condense this by the following:
A = {x such that x is even and little than 100} = {x ⎮ x is even x 100}.
⎮ is a special character, abbreviating such that, and I hope it goes
through the mail. If not I will use such that, or s.t., or things like that.
The expression {x ⎮ x is even} is literally read as: the set of object x,
(or number x if we are in a context where we talk about number) such that x is
even.
Exercise 1: Could you define in intension the following infinite set C =
{101, 103, 105, ...}
C = ?
Exercise 2: I will say that a natural number is a multiple of 4 if it can be
written as 4*y, for some y. For example 0 is a multiple of 4, (0 = 4*0), but
also 28, 400, 404, ... Could you define in extension the following set D = {x
⎮ x 10x is a multiple of 4}.
A last notational, but important symbol. Sets have elements. For example the
set A = {1, 2, 3} has three elements 1, 2 and 3. For saying that 3 is an
element of A in an a short way, we usually write 3 ∈ A. this is read as 3
belongs to A, or 3 is in A. Now 4 does not belong to A. To write this in a
short way, we will write 4 ∉ A, or we will write ¬ (4 ∈ A) or sometimes just
NOT(4 ∈ A). It is read: 4 does not belong to A, or: it is not the case that 4
belongs to A.
Having those notions and notations at our disposition we can speed up on the
notion of union and intersection.
The intersection of the sets A and B is the (new) set of those elements which
belongs to both A and B. Put in another way:
The intersection of the sets A with the set B is the set of those elements
which belongs to A and which belongs to B.
This new set, obtained from A and B is written A ∩ B, or A inter. B (in case
the special character doesn't go through).
With our notations we can write or define the intersection A ∩ B directly
A ∩ B = {x ⎮ x ∈ A and x ∈ B}.
Example {3, 4, 5, 6, 8} ∩ {5, 6, 7, 9} = {5, 6}
Similarly, we can directly define the union of two sets A and B, written A ∪
B in the following way:
A ∪ B = {x ⎮ x ∈ A or x ∈ B}.Here we use the usual logical or. p or q
is suppose to be true if p is true or q is true (or both are true). It is not
the exclusive or.
Example {3, 4, 5, 6, 8} ∪ {5, 6, 7, 9} = {3, 4, 5, 6, 7, 8}.
Exercice 3.
Let N = {0, 1, 2, 3, ...}
Let A = {x ⎮ x 10}
Let B = {x ⎮ x is even}
Describe in extension (that is: exhaustion or quasi-exhaustion) the following
sets:
N ∪ A =
N ∪ B =
A ∪ B =
B ∪ A =
N ∩ A =
B ∩ A =
N ∩ B =
A ∩ B =
Exercice 4
Is it true that A ∩ B = B ∩ A, whatever A and B are?
Is it true that A ∪ B = B ∪ A, whatever A and B are?
Now, I could give you exercise so that you would be lead to discoveries, but
I prefer to be as simple and approachable as possible, and my goal is not even
to give you the
Re: The seven step-Mathematical preliminaries
2009/6/6 Torgny Tholerus tor...@dsv.su.se: Jesse Mazer skrev: Date: Sat, 6 Jun 2009 16:48:21 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: Here you're just contradicting yourself. If you say BIGGEST+1 is then a natural number, that just proves that the set N was not in fact the set of all natural numbers. The alternative would be to say BIGGEST+1 is *not* a natural number, but then you need to provide a definition of natural number that would explain why this is the case. It depends upon how you define natural number. If you define it by: n is a natural number if and only if n belongs to N, the set of all natural numbers, then of course BIGGEST+1 is *not* a natural number. In that case you have to call BIGGEST+1 something else, maybe unnatural number. OK, but then you need to define what you mean by N, the set of all natural numbers. Specifically you need to say what number is BIGGEST. Is it arbitrary? Can I set BIGGEST = 3, for example? Or do you have some philosophical ideas related to what BIGGEST is, like the number of particles in the universe or the largest number any human can conceptualize? It is rather the last, the largest number any human can conceptualize. More natural numbers are not needed. What is the last number human can invent ? Your theory can't explain why addition works... If N is limited, then addition can and will (in human lifetime) create number which are still finite and not in N. N can be defined solelly as the successor function, you don't need anything else. You just have to assert that the function is true always. Also, any comment on my point about there being an infinite number of possible propositions about even a finite set, There is not an infinite number of possible proposition. Prove it please. You can only create a finite number of proposition with finite length during your lifetime. What is a lifetime . What is truth ? Either you CAN*** define a limit or you ***CAN'T***. Just like the number of natural numbers are unlimited but finite, so are the possible propositions unlimited but finte. EVERY*** ***MEMBER*** of the set ***N*** is FINITE* or about my question about whether you have any philosophical/logical argument for saying all sets must be finite, 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. Mostly that set is all objects in the universe, and if you stay inside the universe, there is no problems. But as soon you go outside universe, you must be carefull with what substitutions you do. If you have all quantified with all object inside the universe, you can not substitute it with an object outside the universe, because that object was not included in the original statement. as opposed to it just being a sort of aesthetic preference on your part? Do you think there is anything illogical or incoherent about defining a set in terms of a rule that takes any input and decides whether it's a member of the set or not, such that there may be no upper limit on the number of possible inputs that the rule would define as being members? (such as would be the case for the rule 'n is a natural number if n=1 or if n is equal to some other natural number+1') In the last sentence you have an implicite all: The full sentence would be: For all n in the universe hold that n is a natural number if n=1 or if n is equal to some other natural number+1. And you may now be able to understand, that if the number of objects in the universe is finite, then this sentence will just define a finite set. -- Torgny Tholerus I will read the rest (and others) email later unfortunatelly. -- 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 2
On 06 Jun 2009, at 23:54, m.a. wrote: (I'll be here till Tuesday.) Evidently, the symbol you are using for such that is being shown on my screen as a small rectangle. In the copy below, I see two rectangles before the A=, two before the B=, two after the A, two after the B. The UNION symbol (inverted U) shows up but is followed by a rectangle in the next two examples and preceded by a rectangle in the last three. In checking a table of logic notaion, I find that the relation such that is designated by a reversed capital E. Is this the symbol you are using? m.a. Yes, we have a problem. There should be no rectangles at all. We have to switch on english abbreviations. This explains the difficulty you did have with the union ... You could look on the archive, from here, http://www.mail-archive.com/everything-list@googlegroups.com/msg16531.html the symbols are correct on my computer, but we will think on easier mail symbols. Tell me if you see different symbols in the archive. Best, Bruno - Original Message - From: Bruno Marchal To: everything-list@googlegroups.com Sent: Saturday, June 06, 2009 4:36 PM Subject: Re: The seven step-Mathematical preliminaries 2 We do have problem of symbols, with the mail. I don't see any rectangle in the message below! Take it easy and . We will go very slowly. It will also be the exam periods. There is no rush ... Have a good holiday Bruno - Original Message - From: Bruno Marchal To: everything-list@googlegroups.com Sent: Wednesday, June 03, 2009 1:15 PM Subject: Re: The seven step-Mathematical preliminaries 2 ∅ ∪ A = ∅ ∪ B = A ∪ ∅ = B ∪ ∅ = N ∩ ∅ = B ∩ ∅ = ∅ ∩ B = ∅ ∩ ∅ = ∅ ∪ ∅ = --- 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 -~--~~~~--~~--~--~--- http://iridia.ulb.ac.be/~marchal/ http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ 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 2
I've encountered some difficulty with the examples below.
You say that in extension describes exhaustion or quasi-
exhaustion. And you give the example: B = {3, 6, 9, 12, ... 99}.
Then you define in intension with exactly the same type
of set: Example: Let A be the set {2, 4, 6, 8, 10, ... 100}.
I give A in extension there, but just to define it in intension after.
It is always the same set there. But I show its definition in
extension, to show the definition in intension after. You have to read
the to sentences.
Can you see the cause of my confusion?
It is always the same set. I give it in extension, and then in
intension.
Incidentally, may I suggest you use smaller than rather than
more little than. Your English is generally too good to include
that kind of error. marty a.
Well sure. Sometimes the correct expression just slip out from my
mind. smaller than is much better! Thanks for helping,
Bruno
- Original Message -
From: Bruno Marchal
To: everything-list@googlegroups.com
Sent: Wednesday, June 03, 2009 1:15 PM
Subject: Re: The seven step-Mathematical preliminaries 2
=== Intension and extension
In the case of finite and little set we have seen that we can
define them by exhaustion. This means we can give an explicit
complete description of all element of the set.
Example. A = {0, 1, 2, 77, 98, 5}
When the set is still finite and too big, or if we are lazy, we can
sometimes define the set by quasi exhaustion. This means we describe
enough elements of the set in a manner which, by requiring some good
will and some imagination, we can estimate having define the set.
Example. B = {3, 6, 9, 12, ... 99}. We can understand in this case
that we meant the set of multiple of the number three, below 100.
A fortiori, when a set in not finite, that is, when the set is
infinite, we have to use either quasi-exhaustion, or we have to use
some sentence or phrase or proposition describing the elements of
the set.
Definition.
I will say that a set is defined IN EXTENSIO, or simply, in
extension, when it is defined in exhaustion or quasi-exhaustion.
I will say that a set is defined IN INTENSIO, or simply in
intension, with a s, when it is defined by a sentence explaining
the typical attribute of the elements.
Example: Let A be the set {2, 4, 6, 8, 10, ... 100}. We can easily
define A in intension: A = the set of numbers which are even and
more little than 100. mathematician will condense this by the
following:
A = {x such that x is even and little than 100} = {x ⎮ x is even
x 100}. ⎮ is a special character, abbreviating such that, and
I hope it goes through the mail. If not I will use such that, or
s.t., or things like that.
The expression {x ⎮ x is even} is literally read as: the set of
object x, (or number x if we are in a context where we talk about
number) such that x is even.
Exercise 1: Could you define in intension the following infinite set
C = {101, 103, 105, ...}
C = ?
Exercise 2: I will say that a natural number is a multiple of 4 if
it can be written as 4*y, for some y. For example 0 is a multiple of
4, (0 = 4*0), but also 28, 400, 404, ... Could you define in
extension the following set D = {x ⎮ x 10x is a multiple of
4}.
A last notational, but important symbol. Sets have elements. For
example the set A = {1, 2, 3} has three elements 1, 2 and 3. For
saying that 3 is an element of A in an a short way, we usually write
3 ∈ A. this is read as 3 belongs to A, or 3 is in A. Now 4
does not belong to A. To write this in a short way, we will write 4
∉ A, or we will write ¬ (4 ∈ A) or sometimes just NOT(4 ∈ A).
It is read: 4 does not belong to A, or: it is not the case that 4
belongs to A.
Having those notions and notations at our disposition we can speed
up on the notion of union and intersection.
The intersection of the sets A and B is the (new) set of those
elements which belongs to both A and B. Put in another way:
The intersection of the sets A with the set B is the set of those
elements which belongs to A and which belongs to B.
This new set, obtained from A and B is written A ∩ B, or A inter. B
(in case the special character doesn't go through).
With our notations we can write or define the intersection A ∩ B
directly
A ∩ B = {x ⎮ x ∈ A and x ∈ B}.
Example {3, 4, 5, 6, 8} ∩ {5, 6, 7, 9} = {5, 6}
Similarly, we can directly define the union of two sets A and B,
written A ∪ B in the following way:
A ∪ B = {x ⎮ x ∈ A or x ∈ B}.Here we use the usual
logical or. p or q is suppose to be true if p is true or q is true
(or both are true). It is not the exclusive or.
Example {3, 4, 5, 6, 8} ∪ {5, 6, 7, 9} = {3, 4, 5, 6, 7, 8}.
Exercice 3.
Let N = {0, 1, 2, 3, ...}
Let A = {x ⎮ x 10}
Let B = {x ⎮ x is
Re: The seven step-Mathematical preliminaries 2
On this date, you made the following correction: You cannot write D = 4*x
..., But you wrote D= 4*x in the exercise just above it. I don't get
the distinction between your use of the equation and mine.
- Original Message -
From: Bruno Marchal
Exercise 2: I will say that a natural number is a multiple of 4 if it can
be written as 4*y, for some y. For example 0 is a multiple of 4, (0 = 4*0), but
also 28, 400, 404, ... Could you define in extension the following set D = {x
⎮ x 10x is a multiple of 4}.D=4*x where x = 0 (but also 1,2,3...10)
You cannot write D = 4*x ..., given that D is a set, and 4*x is a (unknown)
number (a multiple of four when x is a natural number).
Read carefully the problem. I gave the set in intension, and the exercise
consisted in writing the set in extension. Let us translate in english the
definition of the set D = {x ⎮ x 10x is a multiple of 4}: it means that
D is the set of numbers, x, such that x is little than 10, and x is a multiple
of four. So D = {0, 4, 8}.
SEE BELOW
Example: the set of multiple of 4 is {0, 4, 8, 12, 16, 20, 24, 28, 32, 36,
...}, all have the shape 4*x, with x = to 0, 1, 2, 3, ...
The set of multiple of 5 is {0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55,
...}
Etc.
A ∩ B = {x ⎮ x ∈ A and x ∈ B}.
Example {3, 4, 5, 6, 8} ∩ {5, 6, 7, 9} = {5, 6}
Similarly, we can directly define the union of two sets A and B, written
A ∪ B in the following way:
A ∪ B = {x ⎮ x ∈ A or x ∈ B}.Here we use the usual logical or. p or
q is suppose to be true if p is true or q is true (or both are true). It is not
the exclusive or.
Example {3, 4, 5, 6, 8} ∪ {5, 6, 7, 9} = {3, 4, 5, 6, 7, 8}. Question:
In the example above, 5,6 were the intersection because they were the (only)
two numbers BOTH groups had in common. But in this example, 7 is only in the
second group yet it is included in the answer. Please explain.
In the example above (that is {3, 4, 5, 6, 8} ∩ {5, 6, 7, 9} = {5, 6}) we
were taking the INTERSECTION of the two sets.
But after that, may be too quickly (and I should have made a title perhaps) I
was introducing the UNION of the two sets.
If you read carefully the definition in intension, you should see that the
intersection of A and B is defined with an and. The definition of union is
defined with a or. Do you see that? It is just above in the quote.
I hope that your computer can distinguish A ∩ B (A intersection B) and A ∪ B
(A union B).
In the union of two sets, you put all the elements of the two sets together.
In the intersection of two sets, you take only those elements which belongs to
the two sets.
It seems you have not seen the difference between intersection and union.
This has indeed been the case. My usual math disabilities have been
exacerbated by the confusion of symbols due to E-mail limitations. The
profusion of little rectangles replacing the UNION symbol make the formulae
difficult to follow.
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Re: The seven step-Mathematical preliminaries 2
m.a. wrote:
*Bruno,*
* I've encountered some difficulty with the examples below.
You say that in extension describes exhaustion or
quasi-exhaustion. And you give the example: **B = {3, 6, 9, 12, ...
99}.*
* Then you define in intension with exactly the same type
of set: Example: Let A be the set {2, 4, 6, 8, 10, ... 100}.*
No, that's not the intensional definition. This We can easily define A
in intension: A = the set of numbers which are even and more little
than 100. is the intensional definition.
Brent
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Re: The seven step-Mathematical preliminaries 2
Bruno, When I tried to copy the symbols from the URL cited below, I found that my email server was not able to reproproduce the intersection or the union symbol. See below: From: Bruno Marchal To: everything-list@googlegroups.com ∅ ∪ A = I see two rectangles and A ∅ ∪ B = I see two rectangles and B A ∪ ∅ = I see A and two rectangles B ∪ ∅ = I see B and two rectangles N ∩ ∅ = I see N Inverted U and a rectangle B ∩ ∅ = I see B Inverted U and a rectangle ∅ ∩ B = I see a rectangle an inverted U and B ∅ ∩ ∅ = I see a rectangle an inverted U and a rectangle ∅ ∪ ∅ = I see three rectangles - Original Message - From: Bruno Marchal You could look on the archive, from here, http://www.mail-archive.com/everything-list@googlegroups.com/msg16531.html --~--~-~--~~~---~--~~ 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: 2009/6/6 Torgny Tholerus tor...@dsv.su.se: Jesse Mazer skrev: Date: Sat, 6 Jun 2009 16:48:21 +0200 From: tor...@dsv.su.se To: everything-list@googlegroups.com Subject: Re: The seven step-Mathematical preliminaries Jesse Mazer skrev: Here you're just contradicting yourself. If you say BIGGEST+1 is then a natural number, that just proves that the set N was not in fact the set of all natural numbers. The alternative would be to say BIGGEST+1 is *not* a natural number, but then you need to provide a definition of natural number that would explain why this is the case. It depends upon how you define natural number. If you define it by: n is a natural number if and only if n belongs to N, the set of all natural numbers, then of course BIGGEST+1 is *not* a natural number. In that case you have to call BIGGEST+1 something else, maybe unnatural number. OK, but then you need to define what you mean by N, the set of all natural numbers. Specifically you need to say what number is BIGGEST. Is it arbitrary? Can I set BIGGEST = 3, for example? Or do you have some philosophical ideas related to what BIGGEST is, like the number of particles in the universe or the largest number any human can conceptualize? It is rather the last, the largest number any human can conceptualize. More natural numbers are not needed. What is the last number human can invent ? Your theory can't explain why addition works... If N is limited, then addition can and will (in human lifetime) create number which are still finite and not in N. It is very unlikely that anyone will get to the number 10^10^100 by addition. :-) Would agree that a any given time there is a largest number which has been conceived by a human being? N can be defined solelly as the successor function, you don't need anything else. You just have to assert that the function is true always. Also, any comment on my point about there being an infinite number of possible propositions about even a finite set, There is not an infinite number of possible proposition. Prove it please. That would seem to turn on the meaning of possible. Many (dare I say infinitely many) things are logically possible which are not nomologically possible (although the posters on this list seem to doubt that). 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 2
Okay, so is it true to say that things written in EXTENSION are never in
formula style but are translated into formulas when we put them into INTENSION
form? You can see that my difficulty with math arises from an inability to
master even the simplest definitions.marty a.
- Original Message -
From: Bruno Marchal
I've encountered some difficulty with the examples below. You
say that in extension describes exhaustion or quasi-exhaustion. And you
give the example: B = {3, 6, 9, 12, ... 99}.
Then you define in intension with exactly the same type of
set: Example: Let A be the set {2, 4, 6, 8, 10, ... 100}.
I give A in extension there, but just to define it in intension after. It is
always the same set there. But I show its definition in extension, to show the
definition in intension after. You have to read the to sentences.
Can you see the cause of my confusion?
It is always the same set. I give it in extension, and then in intension.
http://iridia.ulb.ac.be/~marchal/
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Re: The seven step-Mathematical preliminaries 2
m.a. wrote: *Okay, so is it true to say that things written in EXTENSION are never in formula style but are translated into formulas when we put them into INTENSION form? You can see that my difficulty with math arises from an inability to master even the simplest definitions. marty a.* It's not that technical. I could define the set of books on my shelf by giving a list of titles: The Comprehensible Cosmos, Set Theory and It's Philosophy, Overshoot, Quintessence. That would be a definition by extension. Or I could point to them in succession and say, That and that and that and that. which would be a definition by ostension. Or I could just say, The books on my shelf. which is a definition by intension. An intensional definition is a descriptive phrase with an implicit variable, which in logic you might write as: The set of things x such that x is a book and x is on my shelf. 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 -~--~~~~--~~--~--~---
