An Ultrafinite Set Theory
RussellE
I found was a remark by Zermelo:
"The 'ultrafinite antinomies of set theory',
which the scientific reactionaries and
anti-mathematicians eagerly and delightedly
call on in their campaign ..."
I get the impression Zermelo didn't like
ultrafinitists.
There were some articles about Essenin-Volpin's
set theory as well as finite abelian groups.
I couldn't find an actual ultrafinite set thory.
So, I decided to come up with my own set theory.
I looked at the axioms of ZFC, but many of these
axioms are obviously inconsistent with any fixed
finite theory.
The axiom of pairing states if A and B are sets,
there exists a set with A and B as elements.
This allows the creation of arbitrarily large sets.
Given the sets: {0} and {1}
{{0}, {1}}
{{0}, {{0}, {1}}
etc.
Similarly, the powerset axiom assumes sets can
grow without limit.
Many set theories use FOL which is based on predicate
calculus which is based on propositional calculus. This
set theory will use propositional calculus.
There are four axioms:
1) The exists N urelements. Each urelement is
a Boolean variable.
2) A set is a N-tuple which assigns a truth value
to each urelement.
3) A function is N well formed Boolean expressions,
one for each urelement.
4) A proper class is a sequence of sets defined by an
initial set and a function.
Let N = 4. It is simple to show there are exactly
2^4 sets. There are 2^(2^4) possible Boolean
expressions with four variables. There can be no
more than 2^64 possible functions or 2^68 proper
classes. This theory is provably finite.
We can define simple mathematical objects with
this theory. Assuming N=4, the "natural" numbers
can be defined as the 16 possbile sets. Like
any set theory. we must define a method of
representing natural numbers. Assume we define
natural numbers as base 2 binary numbers.
Now, we can assign a hexadecimal digit to each set.
(d0,c0,b0,a0)=0, (d0,c0,b0,a1)=1, ... (d1,c1,b1,a1)=f.
A proper class can be represented by a sequence
of unique sets and a "first repeat" set. The first
repeat set uniquely determines the function that
generated the proper class.
Consider this sequence of sets:
0,1,2,3,4,5,6,7,8,9,a,b,c,d,e,f
This is not a proper class because there is
no first repeat set. We haven't defined the
"successor" of set f. There are 16 functions
that will generate this sequence of sets.
We can arbitrarily define the successor of set f.
Assume we define 0 to be the successor of f.
This is the proper class of natural numbers:
0,1,2,3,4,5,6,7,8,9,a,b,c,d,e,f,0
We now have the proper class of natural numbers
and a unique successor function. We can also
define addition and multiplication, but there
is a problem.
Most theories assume any two abitrarily large numbers
can added together. This isn't true in this set theory.
Addition is a binary operator. Since there are only
four variables, we must split the variables between
the two operands. There are functions to add 1-bit
with 3-bit numbers and functions to add two 2-bit numbers.
For example, there is a function to add any number
represented by variables A and B to any number
represented by variables C and D. To add 2+2:
(d1,c0,b1,a0) +> (d0,c1,b0,a0)
A similar function can be found for multiplication.
This simple theory shows the natural numbers can
be represented as a finite proper class, allows
the definition of a unique successor function,
and has functions capable of adding and multiplying
"small" natural numbers.
Does anyone see an obvious inconsistency in this theory?
One advantage of an ultrafinite theory is that it
should be straightfoward to prove the theory is
inconsistent.
I find it interesting this theory shows there can
be many possible successor functions. It is also
interesting to think it may be impossible to define
addition and multiplication for all pairs of natural
numbers.
Russell
- 2 many 2 count
MoeBlee
> Many set theories use FOL which is based on predicate
> calculus which is based on propositional calculus. This
> set theory will use propositional calculus.
>
> There are four axioms:
>
> 1) The exists N urelements. Each urelement is
> a Boolean variable.
You just used predicate language (not just propositional).
Also, you haven't stated your language and primitives.
What is 'N'? What are 'urelements'? What is meant by 'N urelements'?
What is a 'Boolean variable'?
Without definitions, we need to take those as primitives, in which
case your axiom may as well be stated:
There exist x burblements. Each burblement is a goolean bairable.
> 2) A set is a N-tuple which assigns a truth value
> to each urelement.
What is a 'set'? What is an 'N-tuple'? What is 'assigns'? What is a
'truth-value'.
Might as well be stated:
A fret is an x-shoople which fursigns a loosh crabble to each
burblement.
> 3) A function is N well formed Boolean expressions,
> one for each urelement.
Might as well be stated:
A zumption is x krell dormed megressions flum for each burblement.
> 4) A proper class is a sequence of sets defined by an
> initial set and a function.
Might as well be stated:
A slopper trass is a peaquince of frets bemined by an orifal fret and
a zumption.
MoeBlee
RussellE
> On Feb 24, 7:52 pm, RussellE <reaste...@gmail.com> wrote:
>
> > Many set theories use FOL which is based on predicate
> > calculus which is based on propositional calculus. This
> > set theory will use propositional calculus.
>
> > There are four axioms:
>
> > 1) The exists N urelements. Each urelement is
> > a Boolean variable.
>
> You just used predicate language (not just propositional).
I used a quantifier? I guess I did use the word "each".
http://en.wikipedia.org/wiki/Predicate_logic
> Also, you haven't stated your language and primitives.
>
> What is 'N'?
N is some number.
http://en.wikipedia.org/wiki/Number
> What are 'urelements'?
An urelement is an object that can be a member
of a set, but is not itself a set.
http://en.wikipedia.org/wiki/Urelement
> What is meant by 'N urelements'?
This theory has a fixed number, N, of urelements.
> What is a 'Boolean variable'?
Its this thing a guy named George Boole invented.
http://en.wikipedia.org/wiki/Boolean_algebra_(logic)
> Without definitions, we need to take those as primitives, in which
> case your axiom may as well be stated:
>
> There exist x burblements. Each burblement is a goolean bairable.
These axioms define four "primitives": urelement, set, function, and
proper class.
> > 2) A set is a N-tuple which assigns a truth value
> > to each urelement.
>
> What is a 'set'?
A set is an ordered list of assignments for
the Boolean variables defined by axiom 1.
Many set theories don't define set. Set is a primitive.
Of course, we can call them "fret" if you want.
> A fret is an x-shoople which fursigns a loosh crabble to each
> burblement.
William Elliot
> On Feb 25, 9:03�am, MoeBlee <jazzm...@hotmail.com> wrote:
>> On Feb 24, 7:52�pm, RussellE <reaste...@gmail.com> wrote:
>>
>>> Many set theories use FOL which is based on predicate
>>> calculus which is based on propositional calculus. This
>>> set theory will use propositional calculus.
>>
>>> There are four axioms:
>>
>>> 1) The exists N urelements.
>
>>> Each urelement is a Boolean variable.
That's predicate calculus.
forall x, (x urelement -> x Boolean_variable)
> N is some number.
You could have a FOL with n constants, u_1,.. u_N
which can be considered as urelements (in the metalanguage).
> An urelement is an object that can be a member
> of a set, but is not itself a set.
What's an object? What is "a member of".
> This theory has a fixed number, N, of urelements.
>
>> What is a 'Boolean variable'?
>
> Its this thing a guy named George Boole invented.
> http://en.wikipedia.org/wiki/Boolean_algebra_(logic)
>
> These axioms define four "primitives": urelement, set, function, and
> proper class.
>
>>> 2) A set is a N-tuple which assigns a truth value
>>> to each urelement.
>>
>> What is a 'set'?
>
> A set is an ordered list of assignments for
> the Boolean variables defined by axiom 1.
>
What's an ordered list? What are assignments?
> Many set theories don't define set. Set is a primitive.
You however prefered to defined set by undefined terms.
For the object language, first pick your primitives. Then make
definitions in terms of the primitives. After you've done that,
then you may indicate in the meta-language some intended sematics
for the primitives.
----
MoeBlee
> On Feb 25, 9:03 am, MoeBlee <jazzm...@hotmail.com> wrote:
>
> > On Feb 24, 7:52 pm, RussellE <reaste...@gmail.com> wrote:
>
> > > Many set theories use FOL which is based on predicate
> > > calculus which is based on propositional calculus. This
> > > set theory will use propositional calculus.
>
> > > There are four axioms:
>
> > > 1) The exists N urelements. Each urelement is
> > > a Boolean variable.
>
> > You just used predicate language (not just propositional).
>
> I used a quantifier? I guess I did use the word "each".http://en.wikipedia.org/wiki/Predicate_logic
You used 'there exists'.
> > Also, you haven't stated your language and primitives.
>
> > What is 'N'?
>
> N is some number.http://en.wikipedia.org/wiki/Number
> > What are 'urelements'?
>
> An urelement is an object that can be a member
> of a set, but is not itself a set.http://en.wikipedia.org/wiki/Urelement
>
> > What is meant by 'N urelements'?
>
> This theory has a fixed number, N, of urelements.
>
> > What is a 'Boolean variable'?
>
> Its this thing a guy named George Boole invented.http://en.wikipedia.org/wiki/Boolean_algebra_(logic)
Sorry, I thought I could make my point with you without bludgeoning
you over the head with the obvious. I'm not asking what are the
ordinary mathematical definitions of your terminology, but rather
whether you are taking this terminology as primitive or defined per
YOUR SYSTEM.
> > Without definitions, we need to take those as primitives, in which
> > case your axiom may as well be stated:
Oh, sorry, I did bludgeon you with it after all, and you still didn't
get it.
> > There exist x burblements. Each burblement is a goolean bairable.
>
> These axioms define four "primitives": urelement, set, function, and
> proper class.
Axioms don't define primitives (except possibly in an informal sense
of 'define').
So your four primitives are 'urelement, 'set', 'function', 'proper
class'? Any others?
> > > 2) A set is a N-tuple which assigns a truth value
> > > to each urelement.
>
> > What is a 'set'?
>
> A set is an ordered list of assignments for
> the Boolean variables defined by axiom 1.
Please, what is 'ordered', 'list', 'assignments', 'and defined by
axiom' in YOUR SYSTEM?
Don't answer that, please, since it's a rhetorical question.
It's apparent that you're clueless as to how axiomatic systems work.
> Many set theories don't define set. Set is a primitive.
> Of course, we can call them "fret" if you want.
You are virtually completely uninformed about how 'set' may be defined
in certain ordinary set theories (which does not contradict that also
we may take the basic intuitive notion of set to be undefined).
You have no system or theory you've presented at all. What you have is
merely a bunch of mathematical terminology thrown together.
If you wish to have an intelligible system, you'd do well to specify:
the logical system (are you using classical predicate logic with
identity?)
the entire list of non-logical primitives
the axioms written only with the primitives (or, written with symbols
properly defined from primitives)
MoeBlee
RussellE
> On Thu, 25 Feb 2010, RussellE wrote:
> You could have a FOL with n constants, u_1,.. u_N
> which can be considered as urelements (in the metalanguage).
I was trying to come up with a type of "circuit board" system
with a finite number of inputs and a finite number of outputs.
> For the object language, first pick your primitives. Then make
> definitions in terms of the primitives. After you've done that,
> then you may indicate in the meta-language some intended sematics
> for the primitives.
OK. I can simplify the theory and better define my primitives.
Some definitions.
Urelement - a mathematical object that can be an element
of a set. An urelement can not be a set in this theory.
Set - a collection of urelements. In this theory, sets can
only have distinct urelements as members. A set can be empty.
Urelement variable - Define X to be the urelement variable
for urelement x. If a set has x as a member then X is true,
else X is false.
Function - a process to convert an input set into an output set.
A function has a well formed Boolean expression for each urelement.
This expression determines if the urelement is present in the output
set.
If the expression for urelement x is true, then x is a member of the
output set.
A Boolean expression can consist of parenthesis, OR, AND, NOT, the
constants True or False, and/or urelement variables.
Proper class - a sequence of sets defined by repeatedly applying a
function to an initial set.
Now, I will try to steal axioms from ZFC.
http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
1) Axiom of extensionality: Two sets are equal if they have the same
elements.
This theory doesn't need the Axiom of foundation. Sets can't be an
element
of a set by definition.
The axiom schema of specification defines the existence of certain
sets.
I will define the empty set and use the empty set to derive all other
sets.
2) Axiom of the empty set: The empty set exists.
3) Axiom of replacement: S is a set if a function exists to convert
the empty set into S.
As an example, assume x is the only urelement.
X is the urelement variable. Define the function:
X_out = ~X_in
Apply this function to the empty set to get {x}.
The empty set and {x} are the only sets if x
is the only urelement.
The axioms of pairing, union, and collection are a problem.
These axioms assume sets can be elements. They also
assume we can perform operations on more than one
set at a time.
I deliberately defined functions to be unary operators.
A function is defined over all urelements. Maybe this
is too restrictive. I can't define binary operators
like pairing or union because I can't have two input sets.
The only way I can specify more than one set is
as a proper class.
The axiom of infinity is also a problem.
I want to replace AoI with something like
"there are a finite number of urelements".
Unfortunately, I haven't defined finite yet.
The only solution I can come up with is to
specify the existence of certain urelements.
4) Axiom of Finiteness: The urelements a,b,c, and d exist.
I think I can derive the powerset axiom from these axioms.
The powerset of the urelements is a proper class.
And I don't need a well ordering axiom. A proper class
is ordered by definition.
It would be nice if I could come up with a general method
for binary operators. Maybe I could define proper classes
in such a way as to allow n-ary functions.
Any ideas are welcome.
William Elliot
> On Feb 25, 11:42�pm, William Elliot <ma...@rdrop.remove.com> wrote:
>> On Thu, 25 Feb 2010, RussellE wrote:
>
>> You could have a FOL with n constants, u_1,.. u_N
>> which can be considered as urelements (in the metalanguage).
>
> I was trying to come up with a type of "circuit board" system
> with a finite number of inputs and a finite number of outputs.
>
>> For the object language, first pick your primitives. �Then make
>> definitions in terms of the primitives. �After you've done that,
>> then you may indicate in the meta-language some intended sematics
>> for the primitives.
>
> OK. I can simplify the theory and better define my primitives.
>
> Some definitions.
>
> Urelement - a mathematical object that can be an element
> of a set. An urelement can not be a set in this theory.
>
> Set - a collection of urelements. In this theory, sets can
> only have distinct urelements as members. A set can be empty.
>
> Urelement variable - Define X to be the urelement variable
> for urelement x. If a set has x as a member then X is true,
> else X is false.
>
The urelement variable X is true because x is in the set {x}.
> Function - a process to convert an input set into an output set.
> A function has a well formed Boolean expression for each urelement.
> This expression determines if the urelement is present in the output
> set.
All those definitions are in the metalanguage.
> If the expression for urelement x is true, then x is a member of the
> output set.
>
Vague.
> A Boolean expression can consist of parenthesis, OR, AND, NOT, the
> constants True or False, and/or urelement variables.
>
> Proper class - a sequence of sets defined by repeatedly applying a
> function to an initial set.
>
Consider recursive sets.
> Now, I will try to steal axioms from ZFC.
> http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
>
You'll need to modify ZFC for if u is an object (urelements),
then by ZFC, u = empty set because for all x, x not in u
and for all x, x not in empty set.
Thus for all x, (x in u iff x in empty) and u = empty set.
> As an example, assume x is the only urelement.
> X is the urelement variable. Define the function:
>
> X_out = ~X_in
>
X_out and X_in are undefined.
> Apply this function to the empty set to get {x}.
> The empty set and {x} are the only sets if x
> is the only urelement.
>
Doesn't make any sense.
> I deliberately defined functions to be unary operators.
> A function is defined over all urelements. Maybe this
> is too restrictive. I can't define binary operators
> like pairing or union because I can't have two input sets.
> The only way I can specify more than one set is
> as a proper class.
>
> The axiom of infinity is also a problem.
> I want to replace AoI with something like
> "there are a finite number of urelements".
Both statements can be true.
> Unfortunately, I haven't defined finite yet.
>
That and a lot of other stuff and a bunch of vaguery.
> The only solution I can come up with is to
> specify the existence of certain urelements.
>
You're solving a problem?
> 4) Axiom of Finiteness: The urelements a,b,c, and d exist.
>
That doesn't say there are finite many objects (urelements),
only that there are four objects, a, b, c, and d.
> I think I can derive the powerset axiom from these axioms.
Of course, it's part of the ZFC package.
> The powerset of the urelements is a proper class.
> And I don't need a well ordering axiom.
You get that with ZFC.
> A proper class is ordered by definition.
It is? Oh yes, sequences are ordered.
Calling a sequence a class is a misnomer.
> It would be nice if I could come up with a general method
> for binary operators. Maybe I could define proper classes
> in such a way as to allow n-ary functions.
>
> Any ideas are welcome.
>
Learn up on propositional and predicated calculus.
Take special note how a formal object language is constructedd
and the distinction between the object language and the metalanguage.
Clarify your ideas before trying to fomalize them.
You've got a set O of objects.
You've got functions from subsets of O to subsets of O.
You've got inputs and outputs of some mysterious things.
Virgil
<b4492d25-d033-4b65...@s25g2000prd.googlegroups.com>,
RussellE <reas...@gmail.com> wrote:
> Some definitions.
>
> Urelement - a mathematical object that can be an element
> of a set. An urelement can not be a set in this theory.
>
> Set - a collection of urelements. In this theory, sets can
> only have distinct urelements as members. A set can be empty.
According to the two definitions above, it is impossible for any set to
be a member of any set, which is going to lead to an excessively
constrictive set theory of little if any use to mathematics.
Typical of RussellE.
RussellE
> On Fri, 26 Feb 2010, RussellE wrote:
> > On Feb 25, 11:42�pm, William Elliot <ma...@rdrop.remove.com> wrote:
> >> On Thu, 25 Feb 2010, RussellE wrote:
>
> >> You could have a FOL with n constants, u_1,.. u_N
> >> which can be considered as urelements (in the metalanguage).
>
> > I was trying to come up with a type of "circuit board" system
> > with a finite number of inputs and a finite number of outputs.
>
> Consider automata.
The system I am trying to describe is very similar to cellular
automata.
> >> For the object language, first pick your primitives. �Then make
> >> definitions in terms of the primitives. �After you've done that,
> >> then you may indicate in the meta-language some intended sematics
> >> for the primitives.
>
> > OK. I can simplify the theory and better define my primitives.
>
> > Some definitions.
>
> > Urelement - a mathematical object that can be an element
> > of a set. An urelement can not be a set in this theory.
>
> > Set - a collection of urelements. In this theory, sets can
> > only have distinct urelements as members. A set can be empty.
>
> > Urelement variable - Define X to be the urelement variable
> > for urelement x. If a set has x as a member then X is true,
> > else X is false.
>
> The urelement variable X is true because x is in the set {x}.
Yes. I keep forgetting everyone isn't a computer programmer.
> > Function - a process to convert an input set into an output set.
> > A function has a well formed Boolean expression for each urelement.
> > This expression determines if the urelement is present in the output
> > set.
>
> All those definitions are in the metalanguage.
OK. I see now functions must be defined in the meta-language.
> > If the expression for urelement x is true, then x is a member of the
> > output set.
>
> Vague.
>
> > A Boolean expression can consist of parenthesis, OR, AND, NOT, the
> > constants True or False, and/or urelement variables.
>
> > Proper class - a sequence of sets defined by repeatedly applying a
> > function to an initial set.
>
> Consider recursive sets.
>
> > Now, I will try to steal axioms from ZFC.
> >http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
>
> You'll need to modify ZFC for if u is an object (urelements),
> then by ZFC, u = empty set because for all x, x not in u
> and for all x, x not in empty set.
> Thus for all x, (x in u iff x in empty) and u = empty set.
Yes. I think i can make some simple modifications to ZFC.
> > As an example, assume x is the only urelement.
> > X is the urelement variable. Define the function:
>
> > X_out = ~X_in
>
> X_out and X_in are undefined.
X_out determines if urelement x is in the output set.
X_in is true if x is a member of the input set.
> > I think I can derive the powerset axiom from these axioms.
>
> Of course, it's part of the ZFC package.
ZFC has a powerset axiom. We can't derive the powerset
of all sets exist from the other axioms.
> > The powerset of the urelements is a proper class.
> > And I don't need a well ordering axiom.
>
> You get that with ZFC.
Again, as an axiom.
> > A proper class is ordered by definition.
>
> It is? Oh yes, sequences are ordered.
> Calling a sequence a class is a misnomer.
Yes. I should call them proper sequences.
> > It would be nice if I could come up with a general method
> > for binary operators. Maybe I could define proper classes
> > in such a way as to allow n-ary functions.
>
> > Any ideas are welcome.
>
> Learn up on propositional and predicated calculus.
> Take special note how a formal object language is constructedd
> and the distinction between the object language and the metalanguage.
>
> Clarify your ideas before trying to fomalize them.
>
> You've got a set O of objects.
> You've got functions from subsets of O to subsets of O.
> You've got inputs and outputs of some mysterious things
Is there a version of ZFC where sets can only have urelements as
members?
It would still have the axiom of extensionality. It doesn't need
foundation.
It could have a union axiom. It would need a different schema of
specification.
The schema would have to differentiate between sets and elements.
William Elliot
>>>> On Thu, 25 Feb 2010, RussellE wrote:
>>
>>> I was trying to come up with a type of "circuit board" system
>>> with a finite number of inputs and a finite number of outputs.
Certainly this has been done before.
>>> OK. I can simplify the theory and better define my primitives.
>>
>>> Some definitions.
>>
>>> Urelement - a mathematical object that can be an element
>>> of a set. An urelement can not be a set in this theory.
>>
>>> Set - a collection of urelements. In this theory, sets can
>>> only have distinct urelements as members. A set can be empty.
>>
>>> Urelement variable - Define X to be the urelement variable
>>> for urelement x. If a set has x as a member then X is true,
>>> else X is false.
>>
>> The urelement variable X is true because x is in the set {x}.
>
> Yes. I keep forgetting everyone isn't a computer programmer.
>
Likely you're ignorant of the fact that programmers know less math
learning programing than they'd know if they didn't study programming.
Yes, programmers need to unlearn what math they thought they learned.
>>> Function - a process to convert an input set into an output set.
>>> A function has a well formed Boolean expression for each urelement.
>>> This expression determines if the urelement is present in the output
>>> set.
>>
>> All those definitions are in the metalanguage.
>
> OK. I see now functions must be defined in the meta-language.
>
The choice is yours depending upon how you construct the object language.
An object language has nothing to do with object programming. It's the
formal language defined by it's syntax, devoid of all possible semmatics.
>>> If the expression for urelement x is true, then x is a member of the
>>> output set.
>>
>> Vague.
>>
>>> A Boolean expression can consist of parenthesis, OR, AND, NOT, the
>>> constants True or False, and/or urelement variables.
>>
>>> Proper class - a sequence of sets defined by repeatedly applying a
>>> function to an initial set.
>>
>> Consider recursive sets.
>>
>>> Now, I will try to steal axioms from ZFC.
>>> http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
>>
>> You'll need to modify ZFC for if u is an object (urelements),
>> then by ZFC, u = empty set because for all x, x not in u
>> and for all x, x not in empty set.
>> Thus for all x, (x in u iff x in empty) and u = empty set.
>
> Yes. I think i can make some simple modifications to ZFC.
>
>>> As an example, assume x is the only urelement.
>>> X is the urelement variable. Define the function:
>>
>>> X_out = ~X_in
>>
>> X_out and X_in are undefined.
>
> X_out determines if urelement x is in the output set.
> X_in is true if x is a member of the input set.
>
Egads. You've got undefined and unmentioned things possibly like
automata or Turning machines. Oh what the heck, they're nothing
but functions. If f is a function from X into Y, f:X -> Y
then X is the domain of f, the imput set and
f(X) = { f(x) | x in X } the output set.
Of course, f(X) subset Y.
Calling your X as a urelement variable is incorrect.
A urelement variable is a variable, x,y,z or whatever, that
ranges over all urelements.
If f is a function, operator ect., then f_in = domain f
and f_out = f(domain f). To say that an urelement x is in
the imput set of f, write "x in domain f".
>>> I think I can derive the powerset axiom from these axioms.
>>
>> Of course, it's part of the ZFC package.
>
> ZFC has a powerset axiom. We can't derive the powerset
> of all sets exist from the other axioms.
>
>>> The powerset of the urelements is a proper class.
>>> And I don't need a well ordering axiom.
>>
>> You get that with ZFC.
>
> Again, as an axiom.
>
>>> A proper class is ordered by definition.
>>
>> It is? Oh yes, sequences are ordered.
>> Calling a sequence a class is a misnomer.
>
> Yes. I should call them proper sequences.
>
What's proper about them? Proper means a subset of that's
less than the set itself.
Given a set of A and a function f, then the sequence is
f^n(A) where f^n is defined inductively, f^1(A) = A and
for all n in N (the positive integers),
f^(n+1)(A) = f(f^n(A)) = ff^n(A).
You know about induction, don't you?
>>> It would be nice if I could come up with a general method
>>> for binary operators. Maybe I could define proper classes
>>> in such a way as to allow n-ary functions.
>>
A binary operator usually is a function from XxX into X
where XxX = { (x,y) | x,y in X } and (x,y) is the ordered pair
with x first and y second.
>>> Any ideas are welcome.
>>
>> Learn up on propositional and predicated calculus.
>> Take special note how a formal object language is constructedd
>> and the distinction between the object language and the metalanguage.
>>
>> Clarify your ideas before trying to fomalize them.
>>
>> You've got a set O of objects.
>> You've got functions from subsets of O to subsets of O.
>> You've got inputs and outputs of some mysterious things
>
> Is there a version of ZFC where sets can only have urelements as
> members?
Look up New Foundations NF, web site that IIRC talks about
urelements. Let me know if theirs are the same as yours.
> It would still have the axiom of extensionality. It doesn't need
> foundation.
Foundations and AxC can be removed from ZFC if you chose.
Foundations gets rid of a lot of nonsense sets.
> It could have a union axiom. It would need a different schema of
> specification.
> The schema would have to differentiate between sets and elements.
>
Perhaps you'd want a FOL with two types of variables.
Another approach would be to use a FOL with equality FOL_=,
a primative constant U and the axiom
for all a,b not in U,
for all x, (x in a iff x in b) implies a = b
instead of the axiom
for all a,b, (for all x, (x in a iff x in b) implies a = b).
Barb Knox
William Elliot <ma...@rdrop.remove.com> wrote:
> On Fri, 26 Feb 2010, RussellE wrote:
[SNIP]
> > Yes. I keep forgetting everyone isn't a computer programmer.
>
> Likely you're ignorant of the fact that programmers know less math
> learning programing than they'd know if they didn't study programming.
>
> Yes, programmers need to unlearn what math they thought they learned.
[SNIP]
Similar folk wisdom exists within programming itself:
"You can learn Lisp in a week if you don't know Fortran, or in two weeks
if you do know Fortran."[1]
I'd like to know if there have been any experimental studies that
actually demonstrated (or refuted) the Programming v Mathematics
interference. I don't count the fact that most programmers are
pig-ignorant about mathematics as evidence for such interference, since
most everyone is pig-ignorant about mathematics.[2]
[1] I can't find the source for this quote; it's presumably from the
1960s. Do you know the source?
[2] By "mathematics" I mean actual mathematics with proofs, not
arithmetic or cook-book algebra.
[Added comp.programming, comp.lang.lisp]
--
---------------------------
| BBB b \ Barbara at LivingHistory stop co stop uk
| B B aa rrr b |
| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum videtur.
| BBB aa a r bbb |
-----------------------------
staticd
> In article <20100227020331.V50...@agora.rdrop.com>,
I would say that computer scientists don't use math as often as they
should.... Engineers know how to apply all the math they have to
learn... :|O
Chris F Clark
> I'd like to know if there have been any experimental studies that
> actually demonstrated (or refuted) the Programming v Mathematics
> interference. I don't count the fact that most programmers are
> pig-ignorant about mathematics as evidence for such interference, since
> most everyone is pig-ignorant about mathematics.[2]
...
> [2] By "mathematics" I mean actual mathematics with proofs, not
> arithmetic or cook-book algebra.
I don't have those studies, but I would have to agree with asking the
question. While it is true that one has limits on what one learns if
life, as most of us spend a finite amount of time in school, and time
spent learning one thing is not time spent on another, i.e. we can
apply the pigeon-hole principle, it is not clear that we do not spend
as much time on any given topic as we would have otherwise if we had
not done other things. In other words, it is not clear that the time
spent learning programming negatively impacts the time spent learning
math, nor that it impacts the learning of math.
If we want, to formalize this more, it is not true:
For all (person), person an element of people,
learn(person, programming) implies learn(person, math) <
learn(person, math) given not learn(person, programming)
While it is quite possibly true:
There exits (person), person an element of people,
learn(person, programming) implies learn(person, math) <
learn(person, math) given not learn(person, programming)
And, the experiment would try to establish:
Sum over all (person), person an element of people,
learn(person, programming) implies learn(person, math) <
learn(person, math) given not learn(person, programming)
That is that people who learn programming do so at the expense of
mathematical learning.
And, now I will explain, using a simple example, why I think the sum
probably doesn't hold, acknowledging full well the limits annecdotal
reasoning.
I will take myself as the example. When I was around fifth or sixth
grade, I decided I would be a scientist as my career after seeing a
movie (whose title I can no longer remember, but it was in the style
of "Those Magnificent Men and Their Flying Machines") where people
were attempting to invent perpetual motion machines, and deciding I
would do that. By junior high and the study of algebra, I permutted
that goal to being a mathematician. I loved symbolically manipulating
equations. By High School, I took one of the first "Computer Math"
classes and discovered programming, which morphed my goal even more,
and I wrote my first (toy) compiler in High School. Writing
compilers, a subset of the programming field became my specialty and
my career through most of my adult life and I've used that talent to
do other things in the programming field. I would like to imagine
that if I had not had the specialty I did, I would have successfully
had a career as a mathematician doing some form of symbolic algebra or
set theory, but that kind of work is not that different in principle
from the work I do. It still deals with abstract and very structured
items. It does not in general involve complex calculations (i.e. what
I think of as engineering math), often using those trigonometic
functions. I would have been a failure at that.
Therein lies my point. I have one degree and it is officially a
degree in mathematics, and I know a certain amount of calculus and
differential equations. However, it was only when we got to "advanced
calculus" where I could do epsilon-delta proofs that I did well in
those subjects. Similarly, I was fortunately taught topology from a
point-set perspective, where I could map everything to union and
intersection operations, and thus was able to comprehend it. The point
being, that I essentially reached my own limits mathematically in the
course work I took.
Moreover, the courses I took in programming, grounded me in similar
rigor to the parts of mathematics that I do well in. Moreover, some
of the course, such as automata, allowed me to practice doing proofs
and gave me areas of mathematics to know more deeply. And, in
particular, it was those courses, that allowed me to take the course
in symbolic logic taught by the philosophy department, where I could
do even more proofs in areas I was learning. Sadly, the course in
which one actually recreated Goedel's incompleteness proof wasn't
offered. I did however manage to recreate the proof of Post's
correspondence theorem in one class. So, in the areas I do well, I
was able to study "mathematics" well outside the bounds of the
department.
Even the person whose attempt to devise an ultrafinite set theory
illustrates this point. Here is a programmer and one of his basic
tools of trade is Boolean Algebra, which is essentially a mathematical
formalism. That mathematics he was taught because he studied
programming not inspite of it.
Given the above annecdotal evidence, I find it suspect to believe that
people who learn programming, learn less mathematics than they would
have otherwise. I think it is more likely that the reverse is true.
The only difference I can see in my own life, is that had I continued
in mathematics, my career option would have required more schooling
and it wasn't clear if that would have been possible since I stopped
school because I was broke and had no means of support (i.e. I needed
to take a paying full-time job) rather than any other reason. I then
continued my own education by reading the relevant papers in the
fields that interested me and eventually did publish a paper and later
ran a workshop in my own field. While I am confident I could have
found a school somewhere that would have offered me an assistanceship
to further my education, I don't think I would have done anything more
mathematically significant than the work I ended up doing in my chosen
career.
Hope this helps,
-Chris
******************************************************************************
Chris Clark email: christoph...@compiler-resources.com
Compiler Resources, Inc. Web Site: http://world.std.com/~compres
23 Bailey Rd voice: (508) 435-5016
Berlin, MA 01503 USA twitter: @intel_chris
------------------------------------------------------------------------------
Aatu Koskensilta
> If we want, to formalize this more, it is not true:
>
> For all (person), person an element of people,
> learn(person, programming) implies learn(person, math) <
> learn(person, math) given not learn(person, programming)
>
> While it is quite possibly true:
>
> There exits (person), person an element of people,
> learn(person, programming) implies learn(person, math) <
> learn(person, math) given not learn(person, programming)
>
> And, the experiment would try to establish:
>
> Sum over all (person), person an element of people,
> learn(person, programming) implies learn(person, math) <
> learn(person, math) given not learn(person, programming)
What's the point of these "formalizations"? I posit they're just formal
mumbo jumbo -- as perhaps befits sci.logic -- of no apparent interest or
use. (We needn't go into the purely technical deficiencies of these
proposed formalizations.) A truly deplorable feature of the more dreary
sort of analytical philosophy is the fetishistic bandying about of just
this kind of gratuitous formal bunk, apparently in the incorrect belief
that it somehow makes the reasoning more rigorous or exact.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Barb Knox
Chris F Clark <c...@shell01.TheWorld.com> wrote:
> Barb Knox <s...@sig.below> writes:
>
> > I'd like to know if there have been any experimental studies that
> > actually demonstrated (or refuted) the Programming v Mathematics
> > interference. I don't count the fact that most programmers are
> > pig-ignorant about mathematics as evidence for such interference, since
> > most everyone is pig-ignorant about mathematics.[2]
> ...
> > [2] By "mathematics" I mean actual mathematics with proofs, not
> > arithmetic or cook-book algebra.
>
> I don't have those studies, but I would have to agree with asking the
> question. While it is true that one has limits on what one learns if
> life, as most of us spend a finite amount of time in school, and time
> spent learning one thing is not time spent on another, i.e. we can
> apply the pigeon-hole principle, it is not clear that we do not spend
> as much time on any given topic as we would have otherwise if we had
> not done other things. In other words, it is not clear that the time
> spent learning programming negatively impacts the time spent learning
> math, nor that it impacts the learning of math.
My question is not about the amount of time spent or not spent learning
maths, but whether or not there is some *active* cognitive interference
such that learning programming first (which was not your experience)
makes it significantly more difficult to then learn actual maths.
The maxim "You can learn Lisp in a week if you don't know Fortran, or in
two weeks if you do know Fortran" deals with the phenomenon that having
a Fortran mindset (procedural, non-recursive, mostly mutable data, a
small fixed set of data types, etc.) actively interferes with acquiring
the Lisp mindset (largely functional, recursive, much immutable data,
roll-your-own data types, etc.)
Whether or not there is good evidence for the Fortran v Lisp
interference, my question is whether there is good evidence for a
similar programming v maths interference. I find it intuitively
plausible, especially if the previously learnied programming is of the
Fortran / C / etc. variety.
For example, such programming has almost exclusively a time-oriented
operational semantics ("if such-and-such is the case right before that
statement is executed then so-and-so will be the case right after").
This is not the sort of semantics that is common in mathematics, which
usually deals with "eternal truths".
--
Frederick Williams
> If we want, to formalize this more, [...]
Having read what I've snipped before I snipped it, I say let's not want.
RussellE
> In article <20100227020331.V50...@agora.rdrop.com>,
> William Elliot <ma...@rdrop.remove.com> wrote:
>
> > On Fri, 26 Feb 2010, RussellE wrote:
>
> [SNIP]
>
> > > Yes. I keep forgetting everyone isn't a computer programmer.
>
> > Likely you're ignorant of the fact that programmers know less math
> > learning programing than they'd know if they didn't study programming.
>
> > Yes, programmers need to unlearn what math they thought they learned.
>
> [SNIP]
>
> Similar folk wisdom exists within programming itself:
> "You can learn Lisp in a week if you don't know Fortran, or in two weeks
> if you do know Fortran."[1]
>
> I'd like to know if there have been any experimental studies that
> actually demonstrated (or refuted) the Programming v Mathematics
> interference. I don't count the fact that most programmers are
> pig-ignorant about mathematics as evidence for such interference, since
> most everyone is pig-ignorant about mathematics.[2]
>
> [1] I can't find the source for this quote; it's presumably from the
> 1960s. Do you know the source?
>
> [2] By "mathematics" I mean actual mathematics with proofs, not
> arithmetic or cook-book algebra.
I once helped a grad student validate his thesis. He derived a
formula for the translational friction of toroids in a viscous fluid.
Think of a donut sinking into a vat of jelly. I compared his
results with results our lab had derived.
The student didn't know programming, so I worte a program
to compute his results. His formula had a function I had never seen.
He told me it was the gamma function. When I asked him what
gamma function was, he looked at me like a I was an idiot.
He explained it was a generalization of the factorial function.
His face went blank when I asked him how to calculate it.
I don't think it ever occurred to him that someone might want to
compute the gamma function.Two weeks later he gave me a
paper on numerical methods for computing the gamma function.
Looking things up took a lot longer before the internet.
I learned a lot working with this mathematician.
Our first meeting was like trying to communicate with
someone from another planet. His work was based
on atmospheric models of rain drop formation.
Our lab was mostly interested in calculating the
diffusion coefficients of macro-molocules.
It wasn't obvious we even defined friction the same way.
We couldn't come up with a way to convert his units into our units.
This is such a common problem in hydrodynamics,
there is a standard convention to deal with it.
We computed the friction of an object divided by
the friction of a sphere of the same volume.
This gave us a dimensionless ratio we could compare.
I think "math vs programming" is mostly about language differences.
Two groups are solving the same problems for different reasons.
Each group develops methods to solve the problem that
answers the questions they are trying to answer.
For example, I defined functions as circuits.
There are a fixed number of inputs and a fixed number of outputs.
Each output is a circuit of inputs.
WE wants to define function using FOL just like ZFC does.
I am not trying to come up with a powerful set theory like ZFC.
I want a set theory that is provably finite and provably
consistent. I defined functions the way I did because
I know how to count circuits. Using my definition,
I can easily count the number of possible functions.
Defining functions using FOL makes them much
more understandable. We can usually determine
what a fuction does by looking at the FOL statement
that defines it.
Writing out a function using my definition gives us
something that looks like a computer circuit.
We get the same set of functions with either definition.
WE wants to use the full power of FOL. I want to count functions.
I knew Fortran when I learned Lisp.
At first, I tried to convert Lisp into something
"understandable" like Fortran. That didn't work at all.
So, there was some "unlearning" involved.
Of course, people who learned Lisp first will never
understand Fortran.
RussellE
Primitives:
Urelement - an element of a set. A set or proper class can not be an
urlelement.
Set - a collection of urelements.
Proper Class - a collection of sets.
1) Axiom of extensionality: Two sets are equal (are the same set) if
they have the same elements.
2) Axiom of singletons: If x is an urelement there exists a set, {x},
with x as its only element.
3) Axiom of union: If A and B are sets there exists a set with the
elements of both A and B.
4) Axiom of intersection: If A and B are sets there exists a set with
the elements common to both A and B.
5) Axiom of complement: If A is a set there exists a set of urelements
not in A.
6) Axiom of well ordering: The urelements are well ordered.
7) Axiom of finiteness: There is a largest and smallest urelement.
I probably don't need the axiom of complement.
It can be derived from the other axioms.
I included the axiom of intersection because I don't really
understand
how set theories like ZFC define intersection. Maybe intersection
can also be derived from the other axioms.
I don't need an axiom schema of specification.
The singleton axiom and union axiom are enough to create any set.
Many people have pointed out the difference between
"all sets are finite" which is consistent with ZFC-Inf,
and "there exists a largest element" which is not
consistent with the other axioms of ZFC.
The axiom of finiteness make this an UST.
The simplest way to represent natural numbers in this
system is to assume each natural number is an urelement.
This gives us the finite set of all natural numbers.
Many people have told me all known UST's are inconsistent.
Obviously, no UST will be consistent with axioms from other
set theories. No UST will be consistent with the axiom
"if n is a natural number then n+1 is a natural number".
My UST doesn't have this axiom.
This UST shows some of the properties I think all UST's must have.
For example, we can not define addition for every pair of natural
numbers.
Some natural numbers are just too big to be added together.
This is also true for multiplication and exponentiation (powerset).
Virgil
<3c6ba639-63b2-4381...@z1g2000prc.googlegroups.com>,
RussellE <reas...@gmail.com> wrote:
> Simpler is better. Here is a simple ultrafinite set theory (UST).
>
> Primitives:
>
> Urelement - an element of a set. A set or proper class can not be an
> urlelement.
> Set - a collection of urelements.
> Proper Class - a collection of sets.
These forbid a set being a member of a set, which means that such a set
theory would be of damn all use in any part of mathematics.
William Elliot
> Simpler is better. Here is a simple ultrafinite set theory (UST).
>
> Urelement - an element of a set. A set or proper class can not be an
> Urelement.
> Set - a collection of urelements.
> Proper Class - a collection of sets.
>
You can model that with U \/ P(U) \/ P(P(U))
where U is the set of urelements.
> 1) Axiom of extensionality: Two sets are equal (are the same set) if
> they have the same elements.
>
> 2) Axiom of singletons: If x is an urelement there exists a set, {x},
> with x as its only element.
>
> 3) Axiom of union: If A and B are sets there exists a set with the
> elements of both A and B.
>
> 4) Axiom of intersection: If A and B are sets there exists a set with
> the elements common to both A and B.
>
> 5) Axiom of complement: If A is a set there exists a set of urelements
> not in A.
>
> 6) Axiom of well ordering: The urelements are well ordered.
>
> 7) Axiom of finiteness: There is a largest and smallest urelement.
>
That doesn't make U finite. The ordinal number omega_0 + 1
has a smallest and largest element and isn't finite.
> The singleton axiom and union axiom are enough to create any set.
No. Even assuming U is finite, you can't construct an empty set.
> The simplest way to represent natural numbers in this
> system is to assume each natural number is an urelement.
> This gives us the finite set of all natural numbers.
No it doesn't. It shows that the natural
numbers cannot be represented in UST.
> Many people have told me all known UST's are inconsistent.
> Obviously, no UST will be consistent with axioms from other
> set theories. No UST will be consistent with the axiom
> "if n is a natural number then n+1 is a natural number".
> My UST doesn't have this axiom.
Of course it doesn't. You haven't even defined incrementation.
> Some natural numbers are just too big to be added together.
Most natural numbers are too big for computers to comprehend.
Patricia Shanahan
...
> The simplest way to represent natural numbers in this
> system is to assume each natural number is an urelement.
> This gives us the finite set of all natural numbers.
How do you define the term "natural numbers"?
Patricia
RussellE
> On Sun, 28 Feb 2010, RussellE wrote:
> > 7) Axiom of finiteness: There is a largest and smallest urelement.
>
> That doesn't make U finite. The ordinal number omega_0 + 1
> has a smallest and largest element and isn't finite.
Yes, I know. I am still having problems coming up with
an axiom of finiteness.
I could use my bijection proof. The axiom says if A and B
are sets and have a bijection there exists a bijection
between A-B and B-A.
This would eliminate sets having a bijection with a proper subset.
But, I would have to define bijection.
> > The singleton axiom and union axiom are enough to create any set.
>
> No. Even assuming U is finite, you can't construct an empty set.
I think I can derive that from intersection.
> > The simplest way to represent natural numbers in this
> > system is to assume each natural number is an urelement.
> > This gives us the finite set of all natural numbers.
>
> No it doesn't. It shows that the natural
> numbers cannot be represented in UST.
Which natural numbers?
This certainly isn't the same set of natural numbers
defined by ZFC. ZFC defines natural numbers
as the intersection of all inductive sets with
the empty set as a member.
> > Many people have told me all known UST's are inconsistent.
> > Obviously, no UST will be consistent with axioms from other
> > set theories. No UST will be consistent with the axiom
> > "if n is a natural number then n+1 is a natural number".
> > My UST doesn't have this axiom.
>
> Of course it doesn't. You haven't even defined incrementation.
I have a well ordering axiom. What else do I need?
> > Some natural numbers are just too big to be added together.
>
> Most natural numbers are too big for computers to comprehend.
Actually, this is true. At least, it is true for the natural
numbers defined by ZFC.
William Elliot
> On Feb 28, 9:11�pm, William Elliot <ma...@rdrop.remove.com> wrote:
>> On Sun, 28 Feb 2010, RussellE wrote:
>
>>> 7) Axiom of finiteness: There is a largest and smallest urelement.
>>
>> That doesn't make U finite. The ordinal number omega_0 + 1
>> has a smallest and largest element and isn't finite.
>
> Yes, I know. I am still having problems coming up with
> an axiom of finiteness.
>
You could include in the language, k constant symbols u1,.. uk,
define U = { u1,.. uk } and state that if u is an urelement,
then u in U.
> I could use my bijection proof. The axiom says if A and B
> are sets and have a bijection there exists a bijection
> between A-B and B-A.
>
> This would eliminate sets having a bijection with a proper subset.
> But, I would have to define bijection.
>
>>> The singleton axiom and union axiom are enough to create any set.
>> No. Even assuming U is finite, you can't construct an empty set.
> I think I can derive that from intersection.
>
You can't if there's only one urelement.
>>> The simplest way to represent natural numbers in this
>>> system is to assume each natural number is an urelement.
>>> This gives us the finite set of all natural numbers.
>>
>> No it doesn't. It shows that the natural
>> numbers cannot be represented in UST.
>
> Which natural numbers?
Most of them.
> This certainly isn't the same set of natural numbers
> defined by ZFC. ZFC defines natural numbers
> as the intersection of all inductive sets with
> the empty set as a member.
>
It also excludes the positive integers of Piano's axiom.
Your natural numbers are unnatural. If it doesn't smell
like a dog nor bark or look like a dog, then it isn't a dog.
>>> Many people have told me all known UST's are inconsistent.
>>> Obviously, no UST will be consistent with axioms from other
>>> set theories. No UST will be consistent with the axiom
>>> "if n is a natural number then n+1 is a natural number".
>>> My UST doesn't have this axiom.
>>
>> Of course it doesn't. You haven't even defined incrementation.
>
> I have a well ordering axiom. What else do I need?
A definition of n + 1 as the successor urelement.
>>> Some natural numbers are just too big to be added together.
>>
>> Most natural numbers are too big for computers to comprehend.
>
> Actually, this is true. At least, it is true for the natural
> numbers defined by ZFC.
>
Some numbers are too small for a computer to comprehend
and others are too precise for a computer to comprehend.
In fact it's worse than computers not being able to comprehend most
numbers. All they can ever hope to do is to comprehend almost no numbers.
----
RussellE
I define a natural number to be an urelement.
The set of all natural numbers is the set of all urelements.
This isn't the same definition as Peano's axoims or ZFC.
My natural numbers serve the same purpose as natural numbers
in these other systems. Natural numbers have an order.
I have a well ordering axiom.
Herman Jurjus
[...]
> My question is not about the amount of time spent or not spent learning
> maths, but whether or not there is some *active* cognitive interference
> such that learning programming first (which was not your experience)
> makes it significantly more difficult to then learn actual maths.
Wasn't the original question about the other way around: after learning
maths, people supposedly had to 'unlearn' maths before they could learn
to program?
> The maxim "You can learn Lisp in a week if you don't know Fortran, or in
> two weeks if you do know Fortran" deals with the phenomenon that having
> a Fortran mindset (procedural, non-recursive, mostly mutable data, a
> small fixed set of data types, etc.) actively interferes with acquiring
> the Lisp mindset (largely functional, recursive, much immutable data,
> roll-your-own data types, etc.)
>
> Whether or not there is good evidence for the Fortran v Lisp
> interference, my question is whether there is good evidence for a
> similar programming v maths interference. I find it intuitively
> plausible, especially if the previously learnied programming is of the
> Fortran / C / etc. variety.
My experience is opposite: people who are 'mathematically challenged'
can sometimes become mathematically minded -by learning to program- (and
doing a lot of it).
Moreover, programming is about the only activity that I have seen
achieving this miracle.
> For example, such programming has almost exclusively a time-oriented
> operational semantics ("if such-and-such is the case right before that
> statement is executed then so-and-so will be the case right after").
> This is not the sort of semantics that is common in mathematics, which
> usually deals with "eternal truths".
Programming also forces people to become aware of the difference between
(vague) intuitions and (exact) definitions.
From where I'm standing, this 'dealing with eternal truths', as you
call it, is no more than a personal preference in reasoning style
(static versus dynamic), and is not the demarcation line between
mathematical and a-mathematical.
--
Cheers,
Herman Jurjus
Virgil
<45b1afa9-21f1-45a9...@k18g2000prf.googlegroups.com>,
RussellE <reas...@gmail.com> wrote:
> On Feb 28, 9:47�pm, Patricia Shanahan <p...@acm.org> wrote:
> > RussellE wrote:
> >
> > ...> The simplest way to represent natural numbers in this
> > > system is to assume each natural number is an urelement.
> > > This gives us the finite set of all natural numbers.
> >
> > ...
> >
> > How do you define the term "natural numbers"?
>
>
> I define a natural number to be an urelement.
And are all your urelements natural numbers? If not, you so called
definition is unusable.
> The set of all natural numbers is the set of all urelements.
> This isn't the same definition as Peano's axoims or ZFC.
> My natural numbers serve the same purpose as natural numbers
> in these other systems. Natural numbers have an order.
> I have a well ordering axiom.
But you do not have any arithmetic.
Nick Keighley
> I knew Fortran when I learned Lisp.
> At first, I tried to convert Lisp into something
> "understandable" like Fortran. That didn't work at all.
> So, there was some "unlearning" involved.
> Of course, people who learned Lisp first will never
> understand Fortran.
of course they will! Fortran is just Lisp with some funny syntax (and
some very odd restrictions). Everything is just Lisp with funny
syntax.
[which would you rather do, write a Fortran compiler in Lisp or a Lisp
compiler in Fortran?]
pete
>
> Barb Knox wrote:
> [...]
> > My question is not about the amount of time spent
> > or not spent learning
> > maths, but whether or not there is some *active*
> > cognitive interference
> > such that learning programming first (which was not your experience)
> > makes it significantly more difficult to then learn actual maths.
>
> Wasn't the original question about the other way around:
> after learning maths,
> people supposedly had to 'unlearn' maths before they could learn
> to program?
I don't know what the original question was,
but out of the two the best programmers that I ever worked with:
one of them had a masters degree in math
and the other was an engineer.
--
pete
William Elliot
>>> RussellE wrote:
>>>
>>> ...> The simplest way to represent natural numbers in this
>>>> system is to assume each natural number is an urelement.
>>>> This gives us the finite set of all natural numbers.
>>>
>>
>> I define a natural number to be an urelement.
>
> And are all your urelements natural numbers? If not, you so called
> definition is unusable.
>
urelements the natural numbers is a misnomer. He needs another term
such as natural computer numbers, bounded natural numbers,
initial segment of natural numbers, minimus numbers.
>> The set of all natural numbers is the set of all urelements.
>> This isn't the same definition as Peano's axioms or ZFC.
>> My natural numbers serve the same purpose as natural numbers
>> in these other systems. Natural numbers have an order.
>> I have a well ordering axiom.
>
The natural numbers are also well ordered.
> But you do not have any arithmetic.
>
It will be worse than the arithmetic of the extended naturals.
Patricia Shanahan
In that case, I suggest you pick a different term, to avoid confusing
yourself and others.
If you had a combination of zero element and successor operation that
satisfied the Peano axioms, you could use the normal definitions of
natural number arithmetic and any theorem about natural numbers that has
been proved from the Peano axioms. That is how ZFC gets its arithmetic,
using the empty set as zero and the set containing only x as the
successor of x. Obviously, you cannot do that given the fact that your
numbers do not satisfy the Peano axioms.
If you go on using the term "natural numbers" you may fool yourself into
assuming that something is already defined or proved because it has been
defined or proved for systems satisfying the Peano axioms. If you want
arithmetic in your system, you will need to go back to the drawing board
to define it, and prove each theorem you want using only your
definitions, axioms, and any theorems you have already proved.
I suggest "Easterly numbers" as a placeholder. Similarly, you could use
"Easterly arithmetic" for the corresponding system of arithmetic
definitions and theorems. As you develop your definitions you may be
able to prove your numbers are isomorphic to some previously defined
system, and adopt the name of that system for them.
Patricia
Brian
> in certain ordinary set theories (which does not contradict that also
> we may take the basic intuitive notion of set to be undefined).
> MoeBlee
I have not seen this before. How is the word set defined in "certain
ordinary set theories"?
cr88192
"pete" <pfi...@mindspring.com> wrote in message
news:4B8BAE...@mindspring.com...
programming uses lots of math, granted, but the types of math used, and the
way they are used, are not so much similar.
one can think they understand stuff, making heavy use of vectors and
quaternions, for example, but then find themselves very much defeated in
math classes due to far more subtle enemies, such as solving for variables
in polynomials, ...
math can be malevolent in this respect...
more subtly, one can be faced with terrors such as set theory, which proceed
in being not so easy to grasp, ...
some people claim set theory is very simple and easy, but I don't find it
this way...
one is left to try to understand how something can look like a collection of
discrete items at one moment, a continuous volume the next, and at the same
time behave as if it were predicate logic. this entity is difficult to make
sense of (how does one imagine such a beast?...).
then again, even in programming land, both pure functional programming and
SSA also remain just outside of grasp (impure FP languages, such as Lisp,
make sense enough, but this did not extend to attempts to use Haskell...).
(granted, this leaves me with conventional languages, plain tree-structured
IR's and/or RPN, ... but, all this seems to work well enough...).
once, long ago, my math skills were impressive (vs others my age), but in
time, it seems now my math skills suck. it would seem about like
calculus-level math is beyond me (granted, I am fairly good at using math to
implement stuff, but this is hardly the same thing...).
once, long ago, my coding skills were sort of a novelty, but anymore, it is
difficult to find anyone to even carry on a basic conversation with.
life goes on, as others gain more but I seem to stay just the same.
I have found there is a friend and an enemy: standards...
one can make something novel, and maybe more people will take interest, but
at costs:
there is little compatibility, little chance of reuse, or ability to "mix
and match" parts.
one can also implement things according to standards, and suddenly find that
now hardly anyone cares anymore. one can mix and match parts, but who cares,
given as to them it seems it has all been done before...
one may often also be left implementing piles of largely non-functional
boilerplate, as is my experience would seem to be of the Java class
library... much of the code seems to do little beyond redirecting things
elsewhere, and many things, apart from minor complications requiring
alterations, would be a matter of copying and pasting.
and, why? to maybe escape both GPL and the Apache liscense (my compiler
subproject is now free of GPL, but still has some Apache code, notably, the
class libraries).
so, no one cares, and it is unlikely useful to anyone besides myself (and
possibly, even this can be debated...).
but, hell, maybe I am just boring and/or stupid, I don't know...
Chris F Clark
> Chris F Clark <c...@shell01.TheWorld.com> writes:
>
>> If we want, to formalize this more, it is not true:
>
> What's the point of these "formalizations"? I posit they're just formal
> mumbo jumbo -- as perhaps befits sci.logic -- of no apparent interest or
> use. (We needn't go into the purely technical deficiencies of these
> proposed formalizations.) A truly deplorable feature of the more dreary
> sort of analytical philosophy is the fetishistic bandying about of just
> this kind of gratuitous formal bunk, apparently in the incorrect belief
> that it somehow makes the reasoning more rigorous or exact.
The purpose of formalization is to make things more clear by
expressing things in a language that is less ambiguous. I take it
from your reaction and the reaction of another poster, that that goal
was not achieved, at least not in communication. (It was acheived in
the sense that it helped me clarify what I wanted to address, but it
may have been superfluous after that point.)
Note, I take this clarity as being one of the central points of the
mathematics that interests me. If one can draw the analogy is a
precise formal system and apply the theorems and axioms of a system to
derive a valid result, then one has successfully reasoned in a more
careful way. the key point being that the formal system must model the
attributes of the question one is trying to answer.
In any casre, given that I did not argue from the formalizations, I
will grant that they were extraneous and not germane. I apologize if
they were puffery.
Chris F Clark
> My question is not about the amount of time spent or not spent learning
> maths, but whether or not there is some *active* cognitive interference
> such that learning programming first (which was not your experience)
> makes it significantly more difficult to then learn actual maths.
I think that question can be asked more generally, does learn any
subset of maths or programming interfere with learning other branches,
and again I have no studies to draw on.
However, I will posit that different parts of mathematics draw on
different mental skills. The mathematics that many engineering fields
draw on is very computational, one finds the appropriate equation and
applies the correct numbers to it to calculate a precise result.
Behind that mathematics is often the ability to visualize curves and
translate them into relevant equations. That is a separate skill.
Other parts of mathematics depend upon symbolic manipulation, which is
yet a separate skill--and the only one I possess to any significant
degree. BTW, these are just three representative skills that come to
my mind, they are neither conclusive nor exclusive.
However, one of the beauties of mathemtatics is that for many problems
there are solution methods in each of those skills. We can model a
computational problem as the abstract intersection of two curves and
use that to form a closed-form solution to the problem. We can model
curves as entities we can manipulate symbolically through axioms and
theorems. We can perform symbolic manipulations in a rote calculation
like manner and achieve correct results.
It is not clear that learning one skill precludes learning the others.
In fact, learning one skill can be a crutch to help learn one of the
others, allowing one to understand that answer as one practices the
non-yet-learned skill. However, at some level those skills are the
application of mathematics and not the mathematics itself--and I will
get to that in a second.
But, first, I want to induldge in an illustrative example of what I
mean. When I first took calculus, limits did not make much sense, and
the computational and geometric visualization problems were all
difficult and not aiding my learning. My poor background in trig did
not help, of course. However, eventually, I was introduced to
epsilon-delta proofs and that simplified matter for me tremendously.
Even better was the work of sequences and series. Here my programming
background proved useful, as I could see induction on these series and
taking the limit of the induction as things I could model in familiar
programming terms, arrays and loops. In that sense, my learning of
calculus was backward, all of the appeal to "familiar" geometric
objects did not help my comprehension. My intuition about such things
is relatively poor and my calculation skills far worse. Only dealing
with calculus as a symbolic system helped it make sense to me.
Now, are there flaws in this. Yes, my programming skills mean I tend
to reason far more by case analysis than I should. However, I think
that was a tendancy, reasoning from the specific to the general, which
I had before I learned to program. However, because my first class
requiring proofs, geometry, roughly corresponded to my first learning
programming, I cannot be certain either way.
Finally, I want to talk a bit about the distinction in my mind between
applying mathematics and doing mathematics. The application of
mathematics being about using models of mathematics to solve problems.
there is something mathematical is recognizing the corresponce of a
problem to a mathematical model, but after that the work is of a
different (and to me a non-mathematical) nature. Doing mathematics on
the other hand is coming up with appropriate formal systems in the
first place. There is something entirely different in realizing that
a theorem holds true and being able to express that in a sound way,
than that which we do when we apply the theorem to a problem. I have
no idea whether my training in programming has helped or hindered my
having moments of mathematical insight. All I can say for certain, is
that it has given me subject matter over which I can have those
insights.
Regards,
Pascal J. Bourguignon
> Aatu Koskensilta <aatu.kos...@uta.fi> writes:
>
>> Chris F Clark <c...@shell01.TheWorld.com> writes:
>>
>>> If we want, to formalize this more, it is not true:
>>
>> What's the point of these "formalizations"? I posit they're just formal
>> mumbo jumbo -- as perhaps befits sci.logic -- of no apparent interest or
>> use. (We needn't go into the purely technical deficiencies of these
>> proposed formalizations.) A truly deplorable feature of the more dreary
>> sort of analytical philosophy is the fetishistic bandying about of just
>> this kind of gratuitous formal bunk, apparently in the incorrect belief
>> that it somehow makes the reasoning more rigorous or exact.
>
> The purpose of formalization is to make things more clear by
> expressing things in a language that is less ambiguous. I take it
> from your reaction and the reaction of another poster, that that goal
> was not achieved, at least not in communication. (It was acheived in
> the sense that it helped me clarify what I wanted to address, but it
> may have been superfluous after that point.)
>
> Note, I take this clarity as being one of the central points of the
> mathematics that interests me. If one can draw the analogy is a
> precise formal system and apply the theorems and axioms of a system to
> derive a valid result, then one has successfully reasoned in a more
> careful way. the key point being that the formal system must model the
> attributes of the question one is trying to answer.
This is I think the main point that is criticized. There's in general
no difficuly in "formalizing" a static description. However, finding
the right inference rules, axioms and theorems, matching the model is
the hard part and often overlooked. This is science and of course,
science is hard. Having just a descriptive formalism is not very
useful, without the right inference rules, if the informal descriptive
language is already good enough.
> In any casre, given that I did not argue from the formalizations, I
> will grant that they were extraneous and not germane. I apologize if
> they were puffery.
--
__Pascal Bourguignon__
Pascal J. Bourguignon
> Aatu Koskensilta <aatu.kos...@uta.fi> writes:
>
>> Chris F Clark <c...@shell01.TheWorld.com> writes:
>>
>>> If we want, to formalize this more, it is not true:
>>
>> What's the point of these "formalizations"? I posit they're just formal
>> mumbo jumbo -- as perhaps befits sci.logic -- of no apparent interest or
>> use. (We needn't go into the purely technical deficiencies of these
>> proposed formalizations.) A truly deplorable feature of the more dreary
>> sort of analytical philosophy is the fetishistic bandying about of just
>> this kind of gratuitous formal bunk, apparently in the incorrect belief
>> that it somehow makes the reasoning more rigorous or exact.
>
> The purpose of formalization is to make things more clear by
> expressing things in a language that is less ambiguous. I take it
> from your reaction and the reaction of another poster, that that goal
> was not achieved, at least not in communication. (It was acheived in
> the sense that it helped me clarify what I wanted to address, but it
> may have been superfluous after that point.)
>
> Note, I take this clarity as being one of the central points of the
> mathematics that interests me. If one can draw the analogy is a
> precise formal system and apply the theorems and axioms of a system to
> derive a valid result, then one has successfully reasoned in a more
> careful way. the key point being that the formal system must model the
> attributes of the question one is trying to answer.
This is I think the main point that is criticized. There's in general
no difficuly in "formalizing" a static description. However, finding
the right inference rules, axioms and theorems, matching the model is
the hard part and often overlooked. This is science and of course,
science is hard. Having just a descriptive formalism is not very
useful, without the right inference rules, if the informal descriptive
language is already good enough.
> In any casre, given that I did not argue from the formalizations, I
> will grant that they were extraneous and not germane. I apologize if
> they were puffery.
--
__Pascal Bourguignon__
aminer
Barb Knox wrote:
> However, I will posit that different parts of mathematics draw on
> different mental skills. The mathematics that many engineering fields
> draw on is very computational, one finds the appropriate equation and
> applies the correct numbers to it to calculate a precise result.
>Finally, I want to talk a bit about the distinction in my mind between
> applying mathematics and doing mathematics.
Sorry Sir, I don't agree...
When you are doing mathematics - topology etc.. -
the reasonning process about abstract objects use logic and ressemble
a graph...
That's the same for engineers in operational research , when
they reason about reality and mathematics models , they use 'logic'
in there reasonning process and that also ressemble a graph...
That's the same process...
Regards,
Amine.
> ******************************************************************************
> Chris Clark email: christopher.f.cl...@compiler-resources.com
> Compiler Resources, Inc. Web Site:http://world.std.com/~compres
> 23 Bailey Rd voice: (508) 435-5016
> Berlin, MA 01503 USA twitter: @intel_chris
> ------------------------------------------------------------------------------
aminer
I wrote:
> > However, I will posit that different parts of mathematics draw on
> > different mental skills. The mathematics that many engineering fields
> > draw on is very computational, one finds the appropriate equation and
> > applies the correct numbers to it to calculate a precise result.
> [...]
> >Finally, I want to talk a bit about the distinction in my mind between
> > applying mathematics and doing mathematics.
>
> Sorry Sir, I don't agree...
>
> When you are doing mathematics - topology etc.. -
> the reasonning process about abstract objects use logic and ressemble
> a graph...
>
> That's the same for engineers in operational research , when
> they reason about reality and mathematics models , they use 'logic'
> in there reasonning process and that also ressemble a graph...
>
> That's the same process...
>
> Regards,
> Amine.
Even when we are reasonning on lock-free algorithms etc.
this reasonning on abstract objects is also complex
and we are using 'logic' (even if it's not yet write on the paper!).
Take also for example the capacity planning example:
http://pages.videotron.com/aminer/parallelhashlist/queue.htm
I am also using a complex reasonning process on abstract mathematical
objects and on the reality , and i am using also 'logic' !
The reasonning process is the same as Topology etc..
Think about it !
Regards,
Amine Moulay Ramdane.
MoeBlee
> Simpler is better. Here is a simple ultrafinite set theory (UST).
>
> Primitives:
>
> Urelement - an element of a set. A set or proper class can not be an
> urlelement.
> Set - a collection of urelements.
> Proper Class - a collection of sets.
If they're primitives, then what is the part following the dash
symbol?
Are those definitions or axioms or combination above? Are the
primitives 'collection' and 'element'? Or what?
PLEASE look up how primitives, defintitions, and axioms work!
> 1) Axiom of extensionality: Two sets are equal (are the same set) if
> they have the same elements.
>
> 2) Axiom of singletons: If x is an urelement there exists a set, {x},
> with x as its only element.
>
> 3) Axiom of union: If A and B are sets there exists a set with the
> elements of both A and B.
>
> 4) Axiom of intersection: If A and B are sets there exists a set with
> the elements common to both A and B.
Okay, all Z set theory so far.
> 5) Axiom of complement: If A is a set there exists a set of urelements
> not in A.
Okay, you're own axiom.
> 6) Axiom of well ordering: The urelements are well ordered.
Assuming the ordinary definiton of 'well ordered', I guess.
> 7) Axiom of finiteness: There is a largest and smallest urelement.
WHAT 'large' and 'small'? According to WHAT relation?
What is the purpose of your theory? Do you think it makes ordinary set
theory otiose? If you think that, then please show how to derive
ordinary mathematics for the sciences from your axioms.
> I probably don't need the axiom of complement.
> It can be derived from the other axioms.
> I included the axiom of intersection because I don't really
> understand
> how set theories like ZFC define intersection.
Why don't you just READ how it's done?
> Maybe intersection
> can also be derived from the other axioms.
Yes. You can read about it in virtually any textbook on set theory.
> I don't need an axiom schema of specification.
> The singleton axiom and union axiom are enough to create any set.
Depends on what you think are "enough" sets.
MoeBlee
MoeBlee
> The set of all natural numbers is the set of all urelements.
Forgive me for skimming at this point, but you proved there is such a
set or you took it as an axiom that there is such a set?
MoeBlee
aminer
Question:
Do you think that i have to prove all the mathematics before using
it ?
Of course not !
Take a look at
http://pages.videotron.com/aminer/parallelhashlist/queue.htm
I am using 'logic' in my process of reasonning about
mathematical models and the reality
and when i say for example:
"Since we have mostly two kind of transactions Find() and Add(),
the service in the network side will be hyperexponentially
distributed"
It's like a theorem that i am using !
aminer
Hello,
As an example:
Take a look at lockfree_mpmc:
We can state something like this, when we are reasonning
about lockfree algorithms in a multithreaded envirenement:
See lockfree_mpmc: http://pages.videotron.com/aminer/
"IF two variables(like head and tail) resides in the same cache line
and at least one of them is write to by a thread, THEN there is a
false sharing..."
It's like a theorem that i am applying, it's why when you look at
lockfree_mpmc the head and tail resides in different cache lines.
The same for deadlock , we can apply theorems to avoid deadlock.
etc.
It's like mathematics that use logic...
And the mathematics itself was invented by using logic ...
Regards,
Amine Moulay Ramdane.
Jesse F. Hughes
>> 6) Axiom of well ordering: The urelements are well ordered.
>
> Assuming the ordinary definiton of 'well ordered', I guess.
>
>> 7) Axiom of finiteness: There is a largest and smallest urelement.
>
> WHAT 'large' and 'small'? According to WHAT relation?
I'm with you on most of your criticisms, but I don't get this one.
Clearly, the relation mentioned in (7) is the well-ordering mentioned
in (6).
--
"[Y]ou never understood the real role of mathematicians. The
position is one of great responsibility and power. [...] You people
have no concept of what it means to be an actual mathematician versus
pretending to be one, dreaming you understand." -- James S. Harris
Transfer Principle
> In article
> <3c6ba639-63b2-4381-b060-c50a35101...@z1g2000prc.googlegroups.com>,
> RussellE <reaste...@gmail.com> wrote:
> > Simpler is better. Here is a simple ultrafinite set theory (UST).
> > Primitives:
> > Urelement - an element of a set. A set or proper class can not be an
> > urlelement.
> > Set - a collection of urelements.
> > Proper Class - a collection of sets.
When RE first posted this theory last week, I noticed the thread,
but I decided not to post until RE explained his theory more. But
now I feel like responding to Virgil's objection here:
> These forbid a set being a member of a set, which means that such a set
> theory would be of damn all use in any part of mathematics.
This reminds me of the discussion from the Sixth Grade Math thread
almost a month ago. In sixth grade math -- indeed, in all K-12 and
possibly even undergrad level math -- one thinks about sets as
containing elements like natural numbers, real numbers, complex
numbers, ordered pairs, functions -- but not other sets. A student
_never_ thinks about sets as having other sets as members until
learning about standard set theories such as ZFC. Only then does
one learn that not only can sets have other sets as elements, but
_every_ element of _every_ set is another set.
But RE makes it clear that he doesn't want sets to have other sets
as elements. To see why, we notice what RE has written:
> > Here is a simple ultrafinite set theory (UST).
So RE states his goal here -- he wants his theory to be an
_ultrafinite_ set theory. So among another things, we expect this
theory UST to have a _finite_ model, M.
Now we go back to the original post:
> > The axiom of pairing states if A and B are sets,
> > there exists a set with A and B as elements.
> > This allows the creation of arbitrarily large sets.
> > Given the sets: {0} and {1}
> > {{0}, {1}}
> > {{0}, {{0}, {1}}
> > etc.
Technically speaking, the Axiom of Pairing doesn't allow the
creation of arbitrarily large sets (indeed, one can't even
create a set with just _three_ elements, let alone arbitrarily
many elements). But RE is on the right track, for any model
of the theory with at least these ZFC axioms:
Axiom of Extensionality
Axiom of Empty Set
Axiom of Pairing
must be infinite. Even weakening Pairing so that it only gives
_singleton_ sets produces (as Patricia Shanahan later alludes
to in one of her posts) all of the Zermelo naturals, of which
there are infinitely many.
So the ultrafinitist requires, in some way, to restrict set
formation somehow. In a theory with Extensionality and Empty
Set, there _must_ exist some set x such that {x} is not a set,
lest infinitely many sets will exist.
But Virgil insists that a set theory in which no set may be a
member of any other set "would be of damn all use in any part
of mathematics," despite the fact that in all of the math one
learns from kindergarten to lower-level undergrad, there is no
mention of sets containing other sets as elements. (In other
words, K-12 and lower-level undergrad math sets contain as
elements natural numbers, real numbers, etc., and these numbers
are treated in these classes like _urelements_ -- which is sort
of like what RE is trying to accomplish!)
Virgil, like many standard set theorists, is trying to impose
ZFC (at least the ZFC idea of sets being elements of other sets)
on everyone, whereas RE is trying to show that this idea is
not necessary (or in his case, desirable).
And never the twain shall meet? I like RE's idea of having sets
have only urelements as elements and being elements only of the
proper classes. If we allow sets to have other sets as elements,
then there must be some way to limit which sets are allowed to
have sets as elements in order to accomplish RE's goal of
having a finite universe. But is there any way to do so in a
theory that's acceptable to Virgil and other anti-"cranks"?
(In other words, instead of saying if x is a set then so is {x},
we must require x to satisfy some suitable property in order for
{x} to be a set.)
I might look back at some of the old zuhair threads, since this
sounds like a trick that zuhair often used in his theories.
RussellE
> RussellE wrote:
> >> How do you define the term "natural numbers"?
>
> > I define a natural number to be an urelement.
> > The set of all natural numbers is the set of all urelements.
> > This isn't the same definition as Peano's axoims or ZFC.
> > My natural numbers serve the same purpose as natural numbers
> > in these other systems. Natural numbers have an order.
> > I have a well ordering axiom.
>
> In that case, I suggest you pick a different term, to avoid confusing
> yourself and others.
Nope. I am calling my numbers the "natural numbers".
Mostly to annoy Moeblee.
> If you had a combination of zero element and successor operation that
> satisfied the Peano axioms, you could use the normal definitions of
> natural number arithmetic and any theorem about natural numbers that has
> been proved from the Peano axioms. That is how ZFC gets its arithmetic,
Natural numbers were around long before Peano and set theories came
along.
Set theories are an attempt to formalize our intuitive notions of
"natural number".
The ancient Greeks had intuitive notions of natural numbers.
They weren't the same as our intuitive notions.
They didn't consider zero to be a number. Many ancient Greeks
wouldn't
have considered one to be a natural number, either.
Many did assume natural numbers could grow without bound.
But, there were dissenters even 2500 years ago.
I have yet to see a convincing refutation of Zeno's paradoxes.
Aristotle couldn't come up believable argument.
> using the empty set as zero and the set containing only x as the
> successor of x.
The empty set is not zero in my set theory.
Peano's axioms don't actually define the successor function.
PA lists a bunch of properties the successor function must have.
Typically, the successor of n is given as "n+1".
Unfortunately, Peano's axioms don't define addition.
You have to define arithmetic to define addition.
And you can't define arithmetic without first defining natural
numbers.
Any system that defines natural numbers using arithematic is circular.
> Obviously, you cannot do that given the fact that your
> numbers do not satisfy the Peano axioms.
Nor are they meant to. Why would you expect the natural numbers
in an ultrafinite set theory to satisfy the axioms of a different
set theory, like Peano's axioms?
> If you go on using the term "natural numbers" you may fool yourself into
> assuming that something is already defined or proved because it has been
> defined or proved for systems satisfying the Peano axioms. If you want
> arithmetic in your system, you will need to go back to the drawing board
> to define it, and prove each theorem you want using only your
> definitions, axioms, and any theorems you have already proved.
>
> I suggest "Easterly numbers" as a placeholder.
I will call my numbers Easterly numbers if you call your numbers
Peano numbers.
> Similarly, you could use
> "Easterly arithmetic" for the corresponding system of arithmetic
> definitions and theorems.
I don't have any problem calling my arithmetic "Easterly" arithmetic.
We do talk about "Peano" arithmetic. And my arithematic is very
different from Peano arithematic.
But, my numbers have just as much right to be called
"natural numbers" as Peano's numbers.
Patricia Shanahan
> On Mar 1, 4:14 am, Patricia Shanahan <p...@acm.org> wrote:
>> RussellE wrote:
>
>>>> How do you define the term "natural numbers"?
>>> I define a natural number to be an urelement.
>>> The set of all natural numbers is the set of all urelements.
>>> This isn't the same definition as Peano's axoims or ZFC.
>>> My natural numbers serve the same purpose as natural numbers
>>> in these other systems. Natural numbers have an order.
>>> I have a well ordering axiom.
>> In that case, I suggest you pick a different term, to avoid confusing
>> yourself and others.
>
> Nope. I am calling my numbers the "natural numbers".
> Mostly to annoy Moeblee.
I was treating clear communication as desirable, and annoying others as
a bad thing. Given your priorities, it probably does not matter what you
call your numbers.
Patricia
Transfer Principle
> On Feb 28, 6:34 pm, RussellE <reaste...@gmail.com> wrote:
> > 5) Axiom of complement: If A is a set there exists a set of urelements
> > not in A.
> > 6) Axiom of well ordering: The urelements are well ordered.
> Assuming the ordinary definiton of 'well ordered', I guess.
> > 7) Axiom of finiteness: There is a largest and smallest urelement.
> WHAT 'large' and 'small'? According to WHAT relation?
> What is the purpose of your theory? Do you think it makes ordinary set
> theory otiose? If you think that, then please show how to derive
> ordinary mathematics for the sciences from your axioms.
Again with "the sciences." We already know how anti-"cranks"
doubt that so-called "crank" theories can provide enough math
for the sciences, while the "cranks" believe that standard
theory doesn't provide the right math for the sciences. (Of
course, "sciences" as used by both "cranks" and anti-"cranks"
usually refers to _physics_.)
So the question we ask is, can one derive enough math for
physics from an ultrafinitist theory like RE's? As mentioned
numerous times in other threads (especially AP threads), as
long as we believe that there are only finitely many particles
in the universe, and space and time can be quantized (using
Planck units, for example), then an ultrafinitist theory
should be sufficient for math for the sciences.
If we take the AP upper bound of 10^500, then we can have a
model of RE's theory with 11 urelements. Then this should give
us 2^11=2048 sets and 2^2048 classes. We note that since
2^2048 > 10^616, so we've covered. Since the largest of the
RSA numbers is around 2^2048, this theory should be sufficient
for math for the science of cryptography, at least.
If the standard anti-"cranks" feel that science requires the
existence of numbers larger than this, then let N be what
they feel is the largest natural number required by science,
then give a model of RE's theory in which there exist exactly
ceil(lg(lg(N))) urelements. (Of course, lg here is the base-2
logarithm function.) The standard theorists are hard-pressed
to argue that N exceeds 10^500, especially 10^10^500. But even
if they argue that science requires numbers on the order of
10^^10 (ten tetrated to the tenth), Moser's Number, or even
Graham's Number, these numbers are still _finite_, and so we
could still use these limits and still have a theory that
axiomatizes physics without need of an Axiom of Infinity.
> > I don't need an axiom schema of specification.
> > The singleton axiom and union axiom are enough to create any set.
> Depends on what you think are "enough" sets.
Obviously, RE thinks that finitely many sets are "enough." I
think that one should be able to have a theory like RE's that
is finitely axiomatized without any schemata -- since the
desired model of such a theory is _finite_. If there exist a
fixed finite number (say n) urelements, then we should be able
to prove the existence of 2^n sets and 2^2^n classes yet
without need of any Separation Schema. (Certainly only finitely
many instances of such a schema should be required.)
I'm fully aware of the possibility that MoeBlee might call me a
"liar" for saying this, but I don't care. This post, once again,
reveals the disdain of the standard anti-"cranks" for any
ultrafinitist theory -- Yessenin-Volpin notwithstanding. Of
course, standard theorists have never asked Y-V to "derive
ordinary mathematics for the sciences" using only the naturals
that he considers to exist (i.e., naturals n such that he will
answer "yes" to "Is n a natural number?").
Notice that RE mentions Y-V in the OP of this thread. I now
consider Y-V to be one of those figures, like Abraham Robinson,
whom both "cranks" and anti-"cranks" claim as being a member of
their side of the debate. So the standard theorists will defend
Y-V and Robinson as being non-"cranks," yet a Usenet poster who
espouses ideas similar or identical to either mathematician
regarding infinity is called a "crank." And the "cranks" will
appeal to Y-V and Robinson when arguing against Infinity with
the standard anti-"cranks."
RussellE
> In article
> > The set of all natural numbers is the set of all urelements.
> > This isn't the same definition as Peano's axoims or ZFC.
> > My natural numbers serve the same purpose as natural numbers
> > in these other systems. Natural numbers have an order.
> > I have a well ordering axiom.
>
> But you do not have any arithmetic.
If you mean Peano arithmetic can't be derived in my UST,
you are correct. This is a good thing.
Since UST's can't derive Peano arithmetic, they are not
subject to Godel's Incompleteness theorem.
An UST might be able to prove its own consistency.
Assume this is the set of all urelements:
{0,1,2,3}
+ 0 1 2 3
0 0 1 2 3
1 1 2 3 x
2 2 3 x x
3 3 x x x
There are 16 possible ordered pairs of natural numbers
Addition can be defined for 10 of them,
Addition is undefined for the other six pairs in this UST.
The purpose of a set theory is to examine the impact
of certain assumptions. Peano arithematic assumes
addition can be defined for any ordered pair of natural numbers.
My UST doesn't assume this.
I can completely define addition for "small" natural numbers.
Addition is completely defined for the numbers 0 and 1
in the UST above.
I don't have to assume arithematic is defined for all natural
numbers because Peano arithematic makes this assumption.
Aatu Koskensilta
> I like RE's idea of having sets have only urelements as elements and
> being elements only of the proper classes.
You'll find learning about third-order arithmetic a blast, then.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
aminer
Hello again,
Sorry, i don't speak very well english(so be cool),
but i love to clarify my ideas...
Doing something interresting with a minimum of knowledge,
is it not also intelligence ?
Do you think that i know all the mathematics ?
Do you think that i know a lot about hardware ?
...
BUT EVEN if i don't know a lot about mathematics and hardware etc.
i love to construct something usefull with a MINIMUM of objects,
and that's also intelligence !
And - like in a graph - i love also to find the shortest path to
knowledge
and understanding !
Regards,
Amine Moulay Ramdane.
> > Amine Moulay Ramdane.- Hide quoted text -
>
> - Show quoted text -
RussellE
> On Sun, 28 Feb 2010, RussellE wrote:
> > On Feb 28, 9:11�pm, William Elliot <ma...@rdrop.remove.com> wrote:
> >> On Sun, 28 Feb 2010, RussellE wrote:
> >>> 7) Axiom of finiteness: There is a largest and smallest urelement.
>
> >> That doesn't make U finite. The ordinal number omega_0 + 1
> >> has a smallest and largest element and isn't finite.
>
> > Yes, I know. I am still having problems coming up with
> > an axiom of finiteness.
>
> You could include in the language, k constant symbols u1,.. uk,
> define U = { u1,.. uk } and state that if u is an urelement,
> then u in U.
This seems to be the simplest solution.
It would be nice to have something more "elegant".
> > I could use my bijection proof. The axiom says if A and B
> > are sets and have a bijection there exists a bijection
> > between A-B and B-A.
>
> > This would eliminate sets having a bijection with a proper subset.
> > But, I would have to define bijection.
>
> >>> The singleton axiom and union axiom are enough to create any set.
> >> No. Even assuming U is finite, you can't construct an empty set.
> > I think I can derive that from intersection.
>
> You can't if there's only one urelement.
Yes. I am not sure this is a problem.
I could add an empty set axiom.
I want to minimize the number of axioms.
If I remember correctly, if an axiomatic set theory is
consistent, it is still consistent when we negate an axiom.
I am not sure how I can deal with an anti-empty set axiom.
I use to think anti-foundational set theories were strange.
Lately, I have been considering anti-union theories and
anti-comprehension theories. There exists two sets with
the same elements that are not equal.
> >>> The simplest way to represent natural numbers in this
> >>> system is to assume each natural number is an urelement.
> >>> This gives us the finite set of all natural numbers.
>
> >> No it doesn't. It shows that the natural
> >> numbers cannot be represented in UST.
>
> > Which natural numbers?
>
> Most of them.
>
> > This certainly isn't the same set of natural numbers
> > defined by ZFC. ZFC defines natural numbers
> > as the intersection of all inductive sets with
> > the empty set as a member.
>
> It also excludes the positive integers of Piano's axiom.
Of course. It's not an UST if it doesn't exclude these.
> Your natural numbers are unnatural. If it doesn't smell
> like a dog nor bark or look like a dog, then it isn't a dog.
Are you saying my UST is "counter-intuitive"?
I find it amusing that a finite set theory is "counter-intuitive".
We all have pre-conceived intuitions about "natual numbers".
I don't think natural numbers can grow without limit.
I want my set theory to formalized my pre-conceived notion
of natural numbers.
> >>> Many people have told me all known UST's are inconsistent.
> >>> Obviously, no UST will be consistent with axioms from other
> >>> set theories. No UST will be consistent with the axiom
> >>> "if n is a natural number then n+1 is a natural number".
> >>> My UST doesn't have this axiom.
>
> >> Of course it doesn't. You haven't even defined incrementation.
>
> > I have a well ordering axiom. What else do I need?
>
> A definition of n + 1 as the successor urelement.
How does ZFC define the successor function?
Is there a "successor" axiom?
"n+1" is meaningless for certain n in my UST.
> >>> Some natural numbers are just too big to be added together.
>
> >> Most natural numbers are too big for computers to comprehend.
>
> > Actually, this is true. At least, it is true for the natural
> > numbers defined by ZFC.
>
> Some numbers are too small for a computer to comprehend
> and others are too precise for a computer to comprehend.
>
> In fact it's worse than computers not being able to comprehend most
> numbers. All they can ever hope to do is to comprehend almost no numbers.
Some people think the universe is a computer.
If so, there are numbers too big and too small
for the universe to comprehend.
You can almost derive the uncertainty principle from this.
Position and momentum can't be computed beyond a certain precision.
It could be worse. If physicists come up with a set theory it will be
something like "there is a probability 1=1, a probability 1=2, ..."
aminer
Hello,
And logic - and intelligence - dictate that wisdom is a good thing
for humanity !
Hence, don't forget about wisdom !
And don't forget to read my poems:
http://pages.videotron.com/aminer/poems.html
:)
Regards,
Amine Moulay Ramdane.
> > - Show quoted text -- Hide quoted text -
MoeBlee
> MoeBlee <jazzm...@hotmail.com> writes:
> >> 6) Axiom of well ordering: The urelements are well ordered.
>
> > Assuming the ordinary definiton of 'well ordered', I guess.
>
> >> 7) Axiom of finiteness: There is a largest and smallest urelement.
>
> > WHAT 'large' and 'small'? According to WHAT relation?
>
> I'm with you on most of your criticisms, but I don't get this one.
>
> Clearly, the relation mentioned in (7) is the well-ordering mentioned
> in (6).
Perhaps I misunderstand, but I take it that his well ordering axiom
says that the set of urelements (by the way, where's the proof that
there is such a set? Maybe I skimmmed past it?) has a well ordering.
That doesn't specify any PARTICULAR well ordering; it doesn't specify
any particular relation that well orders the set of urelements.
Rather, it just says that there exists some well ordering, perhaps
many well orderings of the set of urelements.
MoeBlee
MoeBlee
> On Mar 1, 1:48 pm, MoeBlee <jazzm...@hotmail.com> wrote:
>
> > On Feb 28, 6:34 pm, RussellE <reaste...@gmail.com> wrote:
> > > 5) Axiom of complement: If A is a set there exists a set of urelements
> > > not in A.
> > Okay, you're [sic] own axiom.
> > > 6) Axiom of well ordering: The urelements are well ordered.
> > Assuming the ordinary definiton of 'well ordered', I guess.
> > > 7) Axiom of finiteness: There is a largest and smallest urelement.
> > WHAT 'large' and 'small'? According to WHAT relation?
> > What is the purpose of your theory? Do you think it makes ordinary set
> > theory otiose? If you think that, then please show how to derive
> > ordinary mathematics for the sciences from your axioms.
>
> Again with "the sciences."
It's like saying "Again with the numbers" or "again with the
theorems".
The question of mathematics for the sciences is a central question of
this subject matter. So I mention it in certain contexts. So what? Sue
me.
> as
> long as we believe that there are only finitely many particles
> in the universe, and space and time can be quantized (using
> Planck units, for example), then an ultrafinitist theory
> should be sufficient for math for the sciences.
Great, then let's see the derivation of it from RussellE's axioms
(assuming he ever cleans up his work to present actual primitives and
axioms).
> If there exist a
> fixed finite number (say n) urelements, then we should be able
> to prove the existence of 2^n sets and 2^2^n classes yet
> without need of any Separation Schema. (Certainly only finitely
> many instances of such a schema should be required.)
>
> I'm fully aware of the possibility that MoeBlee might call me a
> "liar" for saying this, but I don't care.
Why should I call you a 'liar' for it? What I called you a liar for
were the actual plain LIES you wrote about my postings.
> This post, once again,
> reveals the disdain of the standard anti-"cranks" for any
> ultrafinitist theory
Now you ARE lying again. I didn't express any disdain for
ultrafinitism. I just asked how RussellE proposes to prove a certain
amount of mathematics for the sciences IF he even proposes to do that.
Stop lying about my words again.
> standard theorists have never asked Y-V to "derive
> ordinary mathematics for the sciences" using only the naturals
> that he considers to exist (i.e., naturals n such that he will
> answer "yes" to "Is n a natural number?").
Sure I might very well ask. I just haven't opined about Yessenin-
Volpin. But if I were to do so, and if the context came up, I might
very well ask.
> Notice that RE mentions Y-V in the OP of this thread. I now
> consider Y-V to be one of those figures, like Abraham Robinson,
> whom both "cranks" and anti-"cranks" claim as being a member of
> their side of the debate. So the standard theorists will defend
> Y-V and Robinson as being non-"cranks,"
Since you've been talking about ME, let it be known that I've never
said whether Yessenin-Volpin is a crank or not a crank or a
pointillist or not a pointillist or a member of the Tea Party or not.
> yet a Usenet poster who
> espouses ideas similar or identical to either mathematician
> regarding infinity is called a "crank." And the "cranks" will
> appeal to Y-V and Robinson when arguing against Infinity with
> the standard anti-"cranks."
Just please stop lying about what I have and have not said.
MoeBlee
William Elliot
> RussellE wrote:
>> I define a natural number to be an urelement.
>> The set of all natural numbers is the set of all urelements.
>> This isn't the same definition as Peano's axoims or ZFC.
> In that case, I suggest you pick a different term, to avoid confusing
> yourself and others.
>
> I suggest "Easterly numbers" as a placeholder. Similarly, you could use
> "Easterly arithmetic" for the corresponding system of arithmetic
> definitions and theorems.
No. It has nothing to do with any Asian cultures, not even Zen.
Send the baby back home with it's father who can christen it as
natural computer numbers in honor of his family name.
RussellE
>
> > Simpler is better. Here is a simple ultrafinite set theory (UST).
>
> > Primitives:
>
> > Urelement - an element of a set. A set or proper class can not be an
> > urlelement.
> > Set - a collection of urelements.
> > Proper Class - a collection of sets.
>
> If they're primitives, then what is the part following the dash
> symbol?
> Are those definitions or axioms or combination above? Are the
> primitives 'collection' and 'element'? Or what?
OK. The primitives are element and collection.
urelement - Only objects defined to be urelements can be elements of a
set
set - A collection of elements.
proper class - a collection of sets.
> PLEASE look up how primitives, defintitions, and axioms work!
>
> > 1) Axiom of extensionality: Two sets are equal (are the same set) if
> > they have the same elements.
>
> > 2) Axiom of singletons: If x is an urelement there exists a set, {x},
> > with x as its only element.
>
> > 3) Axiom of union: If A and B are sets there exists a set with the
> > elements of both A and B.
>
> > 4) Axiom of intersection: If A and B are sets there exists a set with
> > the elements common to both A and B.
>
> Okay, all Z set theory so far.
OK
> > 5) Axiom of complement: If A is a set there exists a set of urelements
> > not in A.
>
> Okay, you're own axiom.
I am not sure I need this axiom.
> > 6) Axiom of well ordering: The urelements are well ordered.
>
> Assuming the ordinary definiton of 'well ordered', I guess.
You got me. I don't define well ordering.
I can't define well ordering the way ZFC does.
My theory doesn't have sets of ordered pairs.
I could define proper classes as ordered pairs.
Any suggestions for a well ordering axiom would be welcome.
> > 7) Axiom of finiteness: There is a largest and smallest urelement.
>
> WHAT 'large' and 'small'? According to WHAT relation?
Again, this is not a great axiom.
A better finitenes axiom would be:
7) Axiom of finitenes: The set U = {u_0, u_1, ..., u_k} exists.
u is an element of U implies u is an urelement.
> What is the purpose of your theory?
I want to show it is possible to have a consistent, finite set theory.
> Do you think it makes ordinary set
> theory otiose?
No. Why would you think that?
Are anti-foundational set theories otiose?
> If you think that, then please show how to derive
> ordinary mathematics for the sciences from your axioms.
What do you mean by "ordinary mathematics for the sciences"?
Can you derive E=MC^2 from ZFC?
If you mean Peano arithematic, my theory can't do that.
I can derive part of PA. I can show there is a set of "small"
natural numbers for which addition is completely defined.
> > I probably don't need the axiom of complement.
> > It can be derived from the other axioms.
> > I included the axiom of intersection because I don't really
> > understand
> > how set theories like ZFC define intersection.
>
> Why don't you just READ how it's done?
>
> > Maybe intersection
> > can also be derived from the other axioms.
>
> Yes. You can read about it in virtually any textbook on set theory.
ZFC doesn't have an axiom of intersection.
I assume intersection can be derived from the axiom schema of
specification.
My theory doesn't have an axiom schema of specification.
> > I don't need an axiom schema of specification.
> > The singleton axiom and union axiom are enough to create any set.
>
> Depends on what you think are "enough" sets.
Do we really need a continuum number of sets to do "science"?
Russell
- Zeno was right. Motion is impossible.
Virgil
<bbc61265-cb81-487d...@k36g2000prb.googlegroups.com>,
RussellE <reas...@gmail.com> wrote:
> > Assuming the ordinary definiton of 'well ordered', I guess.
>
> You got me. I don't define well ordering.
With your "axioms" you can't define any sort of ordering, so we see that
your set theory is disorderly.
William Elliot
> On Mar 1, 12:14�am, William Elliot <ma...@rdrop.remove.com> wrote:
>>>>> 7) Axiom of finiteness: There is a largest and smallest urelement.
>>
>>>> That doesn't make U finite. The ordinal number omega_0 + 1
>>>> has a smallest and largest element and isn't finite.
>>
>>> Yes, I know. I am still having problems coming up with
>>> an axiom of finiteness.
>>
>> You could include in the language, k constant symbols u1,.. uk,
>> define U = { u1,.. uk } and state that if u is an urelement,
>> then u in U.
>
> This seems to be the simplest solution.
> It would be nice to have something more "elegant".
>
Simple is elegant.
>>>>> The singleton axiom and union axiom are enough to create any set.
>>>> No. Even assuming U is finite, you can't construct an empty set.
>>> I think I can derive that from intersection.
>>
>> You can't if there's only one urelement.
>
> Yes. I am not sure this is a problem.
> I could add an empty set axiom.
> I want to minimize the number of axioms.
>
> If I remember correctly, if an axiomatic set theory is
> consistent, it is still consistent when we negate an axiom.
It is not. You can delete an axiom but to negate an axiom
you first have to prove that each axiom is independent of
the others.
> I am not sure how I can deal with an anti-empty set axiom.
>
Don't. If you don't need an empty set, then don't create one.
> I use to think anti-foundational set theories were strange.
> Lately, I have been considering anti-union theories and
> anti-comprehension theories. There exists two sets with
> the same elements that are not equal.
>
Read about fuzzy set theory.
>> It also excludes the positive integers of Piano's axiom.
>
> Of course. It's not an UST if it doesn't exclude these.
>
Then don't call them natural numbers.
That expression has been taken. Call them something else.
>> Your natural numbers are unnatural. �If it doesn't smell
>> like a dog nor bark or look like a dog, then it isn't a dog.
>
> Are you saying my UST is "counter-intuitive"?
No. I'm saying don't call it the natural numbers
as they aren't natural and natural numbers already
means something that can't be your numbers. Call
them something else, like natural computer numbers.
> I find it amusing that a finite set theory is "counter-intuitive".
>
What's finite set theory? The theory of finite sets?
> We all have pre-conceived intuitions about "natual numbers".
> I don't think natural numbers can grow without limit.
That's because you're limited by the lack of visualization of computers.
> I want my set theory to formalized my pre-conceived notion
> of natural numbers.
I myself, have consider bounded integers and
found the complexity too much to be of worth.
>>>>> Many people have told me all known UST's are inconsistent.
>>>>> Obviously, no UST will be consistent with axioms from other
>>>>> set theories. No UST will be consistent with the axiom
>>>>> "if n is a natural number then n+1 is a natural number".
>>>>> My UST doesn't have this axiom.
>>
>>>> Of course it doesn't. You haven't even defined incrementation.
>>
>>> I have a well ordering axiom. What else do I need?
>>
>> A definition of n + 1 as the successor urelement.
>
> How does ZFC define the successor function?
S(x) = x \/ {x}
> Is there a "successor" axiom?
It's one of Peano's axioms.
> "n+1" is meaningless for certain n in my UST.
>
Which ones? You could call some large urelement oo (ie infinity)
or overflow and instead of a + b and S(u) being undefined for certain
urelements, you could say a + b = oo and S(oo) = oo.
> Some people think the universe is a computer.
> If so, there are numbers too big and too small
> for the universe to comprehend.
>
Their universe excludes the mind.
> Position and momentum can't be computed beyond a certain precision.
>
Commonplace physics. The national debt can't be computer beyond a
certain percision also. What happens when you've a finite set
of integers and some physicists has need for a larger or more
precise number that what you provide?
> It could be worse. If physicists come up with a set theory it will be
> something like "there is a probability 1=1, a probability 1=2, ..."
>
Look into fuzzy set theory.
----
RussellE
Thanks.
> > I am not sure how I can deal with an anti-empty set axiom.
>
> Don't. If you don't need an empty set, then don't create one.
>
> > I use to think anti-foundational set theories were strange.
> > Lately, I have been considering anti-union theories and
> > anti-comprehension theories. There exists two sets with
> > the same elements that are not equal.
>
> Read about fuzzy set theory.
>
> >> It also excludes the positive integers of Piano's axiom.
>
> > Of course. It's not an UST if it doesn't exclude these.
>
> Then don't call them natural numbers.
> That expression has been taken. Call them something else.
> Call
> them something else, like natural computer numbers.
OK. We can call them natural computer numbers.
> > I want my set theory to formalized my pre-conceived notion
> > of natural numbers.
>
> I myself, have consider bounded integers and
> found the complexity too much to be of worth.
Most computer engineers agree with you.
> >>>>> Many people have told me all known UST's are inconsistent.
> >>>>> Obviously, no UST will be consistent with axioms from other
> >>>>> set theories. No UST will be consistent with the axiom
> >>>>> "if n is a natural number then n+1 is a natural number".
> >>>>> My UST doesn't have this axiom.
>
> >>>> Of course it doesn't. You haven't even defined incrementation.
>
> >>> I have a well ordering axiom. What else do I need?
>
> >> A definition of n + 1 as the successor urelement.
>
> > How does ZFC define the successor function?
>
> S(x) = x \/ {x}
My theory won't allow the union of an element and a set.
I can define successor as a "circuit".
Assume we have the set U = {a,b,c,d}.
We arbitrarily choose a singleton set like {a}.
Define the variable K_in to be true if a set has k as an element.
Define K_out to be true if the successor has k as an element.
A_out = B_in
B_out = C_in
C_out = D_in
D_out = A_in
This is a successor function for the elements of U.
It doesn't actually define the "first" element.
> > Is there a "successor" axiom?
>
> It's one of Peano's axioms.
>
> > "n+1" is meaningless for certain n in my UST.
>
> Which ones? You could call some large urelement oo (ie infinity)
> or overflow
Programmers use NaN. Not a number.
> and instead of a + b and S(u) being undefined for certain
> urelements, you could say a + b = oo and S(oo) = oo.
The simplest is to define "0" as a successor.
I can also define modulo arithmetic.
> > Some people think the universe is a computer.
> > If so, there are numbers too big and too small
> > for the universe to comprehend.
>
> Their universe excludes the mind.
>
> > Position and momentum can't be computed beyond a certain precision.
>
> Commonplace physics. The national debt can't be computer beyond a
> certain percision also. What happens when you've a finite set
> of integers and some physicists has need for a larger or more
> precise number that what you provide?
Add another urelement.
> > It could be worse. If physicists come up with a set theory it will be
> > something like "there is a probability 1=1, a probability 1=2, ..."
>
> Look into fuzzy set theory.
I never found much use for fuzzy logic.
It works well for some things, but, knowing something
is 80% true doesn't help in a lot of situations.
There are easier ways to calculate odds than fuzzy logic.
I like multi-valued logics. A tri-value logic with
"true", "false", and "don't know" is interesting.
William Elliot
>>> On Mar 1, 12:14�am, William Elliot <ma...@rdrop.remove.com> wrote:
>>>> It also excludes the positive integers of Piano's axiom.
>>> Of course. It's not an UST if it doesn't exclude these.
>>
>> Then don't call them natural numbers.
>> That expression has been taken. �Call them something else.
>> Call them something else, like natural computer numbers.
>
> OK. We can call them natural computer numbers.
>
>>> I want my set theory to formalized my pre-conceived notion
>>> of natural numbers.
>>
>> I myself, have consider bounded integers and
>> found the complexity too much to be of worth.
>
> Most computer engineers agree with you.
>
> I can define successor as a "circuit".
What's a circuit?
> Assume we have the set U = {a,b,c,d}.
> We arbitrarily choose a singleton set like {a}.
> Define the variable K_in to be true if a set has k as an element.
Crazy. What you doing? Your syntax doesn't make any sense.
For each urelement you're labeling a variable?
No. Doesn't even make basic programming sense.
You are defining a proposition over urelements.
I(k) when k in U and some set A with k in A.
> Define K_out to be true if the successor has k as an element.
>
That defies any sense whatever.
Are you sure you wrote what you intended to write?
> A_out = B_in
> B_out = C_in
> C_out = D_in
> D_out = A_in
>
Meaningless other than it seems to be a circuit, a cycle.
Sure you could give U a circular order like the integers
modulus |U| > 1.
> This is a successor function for the elements of U.
> It doesn't actually define the "first" element.
Since U is well ordered, isn't the successor
of k the successor of k in the well ordering?
>>> Is there a "successor" axiom?
>> It's one of Peano's axioms.
>>
>>> "n+1" is meaningless for certain n in my UST.
>>
>> Which ones? You could call some large urelement oo (ie infinity)
>> or overflow
>
> Programmers use NaN. Not a number.
>
>> and instead of a + b and S(u) being undefined for certain
>> urelements, you could say a + b = oo and S(oo) = oo.
>
> The simplest is to define "0" as a successor.
> I can also define modulo arithmetic.
>
Make up your mind.
>>> Position and momentum can't be computed beyond a certain precision.
>>
>> Commonplace physics. The national debt can't be computer beyond a
>> certain precision also. What happens when you've a finite set
>> of integers and some physicists has need for a larger or more
>> precise number that what you provide?
>
> Add another urelement.
>
How would the others using your hippy numbers know that it was done
and what it was. How would you convert all previous work to the new
augmented yippy numbers. Most of all, who decides when and if more
urelements need to be included, how many are needed and what new
urelement will be? I can see it now: Natural Computer Numbers 2011.3.10.
>>> It could be worse. If physicists come up with a set theory it will be
>>> something like "there is a probability 1=1, a probability 1=2, ..."
>>
>> Look into fuzzy set theory.
>
> I never found much use for fuzzy logic.
> It works well for some things, but, knowing something
> is 80% true doesn't help in a lot of situations.
If you knew that the US dollar was about to collapsed with
an 80% assurance you'd keep 20% of your money in US dollars
and 80% in gold or in food, land and guns, depending upon
how much of an optimist you are.
> There are easier ways to calculate odds than fuzzy logic.
>
> I like multi-valued logics. A tri-value logic with
> "true", "false", and "don't know" is interesting.
>
You'd also like the tribe that counts one, two, many.
Most language count one, many; singular and plural.
Sanskrit counts, one, two, many; singular, dual and plural.
s + s = d; s+d = d+d = s+p = d+p = p+p = p
S(s) = d; S(d) = p; S(p) = p
Jesse F. Hughes
> On Mar 1, 1:48 pm, MoeBlee <jazzm...@hotmail.com> wrote:
>> On Feb 28, 6:34 pm, RussellE <reaste...@gmail.com> wrote:
>>
>> > Simpler is better. Here is a simple ultrafinite set theory (UST).
>>
>> > Primitives:
>>
>> > Urelement - an element of a set. A set or proper class can not be an
>> > urlelement.
>> > Set - a collection of urelements.
>> > Proper Class - a collection of sets.
>>
>> If they're primitives, then what is the part following the dash
>> symbol?
>
>> Are those definitions or axioms or combination above? Are the
>> primitives 'collection' and 'element'? Or what?
>
> OK. The primitives are element and collection.
It seems to me that you also have predicates for Urelement, Set and
Class.
>
> urelement - Only objects defined to be urelements can be elements of a
> set
> set - A collection of elements.
> proper class - a collection of sets.
So, it seems to me that you want the following axioms
(Ax)(Urelement(x) -> (Ay)~y e x)
(Ax)(Set(x) -> (Ay)(y e x -> Urelement(y))
(Ax)(Class(x) -> (Ay)(y e x -> Set(y))
(Ax)(Set(x) -> (~Urelement(x) & ~Class(x))
(Ax)(Class(x) -> ~Urelement(x))
>> > 6) Axiom of well ordering: The urelements are well ordered.
>>
>> Assuming the ordinary definiton of 'well ordered', I guess.
>
> You got me. I don't define well ordering.
> I can't define well ordering the way ZFC does.
> My theory doesn't have sets of ordered pairs.
> I could define proper classes as ordered pairs.
> Any suggestions for a well ordering axiom would be welcome.
Just add another relation < and axioms
(Ax)(Ay)(x < y -> (Urelement(x) & Urelement(y)))
and the usual axioms specifying that < is a well-ordering.
>> > 7) Axiom of finiteness: There is a largest and smallest urelement.
Simply add another axiom that <^op (i.e., >) is also a well-ordering.
Every well-ordering has a least element.
You'll have to check, of course, that you *can* write down the axioms
for well-ordering. I don't see any issues, but I haven't thought it
through.
--
"At some point in the future history of humanity, AP will eclipse even
Jesus." -- Archimedes Plutonium, 10/21/07
"I wrote those lines because I am not a megalomania [sic] but rather
very humble and down to earth." -- Archimedes Plutonium, 10/22/07
MoeBlee
> On Mar 1, 1:48 pm, MoeBlee <jazzm...@hotmail.com> wrote:
> OK. The primitives are element and collection.
>
> urelement - Only objects defined to be urelements can be elements of a
> set
> set - A collection of elements.
> proper class - a collection of sets.
That is quite confused.
The following is the best I can make sense of what you might be
driving at. (By the way, the reason yours is confused is not because
of the words 'set', etc., that you use but rather your table of the
words is mixed up. However, I've proposed new words since there is no
sense in confusing with the ordinary use of words in set theory):
Primitives:
1-place predicate - 'x is a ret'
1-place predicate - 'x is an urment'
2-place predicate - 'xey' ('x is an element of y')
Axiom:
x is a ret -> Ay(yex -> y is an urment)
Definition:
x is a rass <-> Ay(yex -> y is a ret)
(So you don't need 'collection'.)
Maybe you still want to revise what I did, but at least mine is clear.
> > > 6) Axiom of well ordering: The urelements are well ordered.
>
> > Assuming the ordinary definiton of 'well ordered', I guess.
>
> You got me. I don't define well ordering.
> I can't define well ordering the way ZFC does.
Sure you can. Definitions of these kinds of predicates (in a language
with 'e') don't depend on axioms.
> 7) Axiom of finitenes: The set U = {u_0, u_1, ..., u_k} exists.
Nope. What are '0', '1'? What does "..." mean?
You're basically question begging by using "..." in this way to mean
something like "finite" when what you need to do is DEFINE
'finite' (I'd call it 'r-nite') in your language.
> > What is the purpose of your theory?
>
> I want to show it is possible to have a consistent, finite set theory.
What do you MEAN by a 'finite theory'? You seem to have your own
definition of 'finite'.
(1) Do you mean (given the ordinary definition of 'finite'), a theory
that has a theorem that there exist only finite sets? Then that is
easy: The theory, in the language of set theory, whose sole axiom is
"There exist only finite sets" is a consistent theory.
(2) Do you mean (given the ordinary definition of 'finite'), a theory
that has only models with finite domains? Again, that is easy. The
theory, in the langauge of set theory, whose sole axiom is "Axy(x=y)"
is consistent and has models only with finite domains.
So what?
> > Do you think it makes ordinary set
> > theory otiose?
>
> No. Why would you think that?
I'm just asking?
> What do you mean by "ordinary mathematics for the sciences"?
I have no PRECISE definition. It is left open to reasonable
interpretation. But a typical minimum would be some calculus for
predicting the motions of objects, for caclulating probabilites and
for statistics.
> Can you derive E=MC^2 from ZFC?
As I understand, that is a statement of physics, not just of
mathematics. I'm referring to the mathematics uses for physics, not
the physics itself.
> If you mean Peano arithematic,
No, I don't. First order PA by itself, is, as far as I know, not
adequate for a theory for the sciences.
> > > I don't need an axiom schema of specification.
> > > The singleton axiom and union axiom are enough to create any set.
>
> > Depends on what you think are "enough" sets.
>
> Do we really need a continuum number of sets to do "science"?
I don't know. But that doesn't obviate the sense of my question as to
what you consider enough sets. Why do you need sets at all? Why do you
need a set theory? Why do you need any theory at all? What is your
purpose in having a theory (other than merely to have a consistent one
such that (1) or (2) from above) or in having a foundational theory?
MoeBlee
RussellE
1) Axiom of extensionality - two sets are equal if they have the same
elements.
2) Axiom of finiteness - the set U = {u0,u1,u2,...,uk) exists.
u is an element of U iff u is an urelement.
3) Axiom of well ordering - the urelements have the following order:
u0 < u1 < u2 < ... < uk
4) Axiom of singleton - if u is an urelement there exists a set with u
as the only element.
5) Axiom of union - if A and B are sets there exists a set with all
the elements of both A and B.
6) Axiom of intersection - if A and B are sets there exists a set with
all elements common to A and B.
7) Axiom of complement - if A is a set there exists a set of
urelements not in A.
Define every urelement except the largest to be a natural number.
The largest urelement is defined as NaN - not a number.
Any arithmetic operation with NaN as an operand equals NaN.
For example, NaN+1 = NaN.
Arithmetic can be completely defined.
Every natural number has a unique successor.
The successor of NaN is NaN.
MoeBlee
> Simpler is better
COHERENT would be nice too.
> 1) Axiom of extensionality - two sets are equal if they have the same
> elements.
I guess you're adopting identity theory also with this.
> 2) Axiom of finiteness - the set U = {u0,u1,u2,...,uk) exists.
> u is an element of U iff u is an urelement.
You skipped what I said about that.
> 3) Axiom of well ordering - the urelements have the following order:
> u0 < u1 < u2 < ... < uk
This is not an axiom in the language of your system.
PLEASE, you tell us you're giving us primitives and stuff, but then
you just skip right past the matter.
> 4) Axiom of singleton - if u is an urelement there exists a set with u
> as the only element.
Okay, but only assuming that you have identity theory (or first order
logic with identity) to work with to explicate the expression "the
only".
> 5) Axiom of union - if A and B are sets there exists a set with all
> the elements of both A and B.
Okay.
> 6) Axiom of intersection - if A and B are sets there exists a set with
> all elements common to A and B.
Okay.
> 7) Axiom of complement - if A is a set there exists a set of
> urelements not in A.
Okay.
> Define every urelement except the largest to be a natural number.
This depends on your Axiom 3, which is still just floating
mathematical verbiage.
> The largest urelement is defined as NaN - not a number.
> Any arithmetic operation with NaN as an operand equals NaN.
Freefloating mathematical verbiage.
> For example, NaN+1 = NaN.
Freefloating mathematical verbiage.
> Arithmetic can be completely defined.
Whatever that means.
> Every natural number has a unique successor.
> The successor of NaN is NaN.
To quote the scribe, "Huh?"
MoeBlee
RussellE
> On Mar 2, 12:48 am, RussellE <reaste...@gmail.com> wrote:
> > > What is the purpose of your theory?
>
> > I want to show it is possible to have a consistent, finite set theory.
>
> What do you MEAN by a 'finite theory'? You seem to have your own
> definition of 'finite'.
The title of the thread is "An Ultrafinite Set Theory".
I want a theory with a largest natural number.
This is more restrictive than just a finite theory.
> (1) Do you mean (given the ordinary definition of 'finite'), a theory
> that has a theorem that there exist only finite sets? Then that is
> easy: The theory, in the language of set theory, whose sole axiom is
> "There exist only finite sets" is a consistent theory.
>
> (2) Do you mean (given the ordinary definition of 'finite'), a theory
> that has only models with finite domains? Again, that is easy. The
> theory, in the langauge of set theory, whose sole axiom is "Axy(x=y)"
> is consistent and has models only with finite domains.
>
> So what?
I have often been told there are no "consistent" ultrafinite set
theories (UST).
I suspect people don't mean we can always derive a contradiction from
the axioms of a UST. I think they mean UST's aren't consistent with
their idea of arithematic.
> > > Do you think it makes ordinary set
> > > theory otiose?
>
> > No. Why would you think that?
>
> I'm just asking?
>
> > What do you mean by "ordinary mathematics for the sciences"?
>
> I have no PRECISE definition. It is left open to reasonable
> interpretation. But a typical minimum would be some calculus for
> predicting the motions of objects, for caclulating probabilites and
> for statistics.
The philosophy of science says truth can only be determined
by measurement and repeatable experiments.
In some ways, science is the antithesis of mathematics which
says truth can be derived by pure reasoning.
Could a UST predict the orbit of Mercury?
I think one could. Assume a set of 2^500 natural numbers.
Predicting the orbit of Mercury to within experimental error
should not be much harder than writnig a video game for a
really large computer monitor with a finite number of pixels.
> > Can you derive E=MC^2 from ZFC?
>
> As I understand, that is a statement of physics, not just of
> mathematics. I'm referring to the mathematics uses for physics, not
> the physics itself.
The first experimental evidence for the theory of relativity
was a deviation in the orbit of Mercury as calulated using
Newton's laws. How do you draw a line between math and science?
> > If you mean Peano arithematic,
>
> No, I don't. First order PA by itself, is, as far as I know, not
> adequate for a theory for the sciences.
What does "adequate" mean? For the most part,
science doesn't care about theory.
> > > > I don't need an axiom schema of specification.
> > > > The singleton axiom and union axiom are enough to create any set.
>
> > > Depends on what you think are "enough" sets.
>
> > Do we really need a continuum number of sets to do "science"?
>
> I don't know. But that doesn't obviate the sense of my question as to
> what you consider enough sets. Why do you need sets at all?
I don't need sets. I really want natural numbers.
> Why do you
> need a set theory?
To formalize intuitive notions of "natural numbers".
> Why do you need any theory at all?
So I can experiment on them. Before I started playing
with this UST, I might not have considered the idea
that arithmetic may not be defined for every pair of
natural numbers. What if some numbers are so big
they can't be added to another number. How would
this change our intutions about natural numbers.
Can I come up with a test for this in the real world?
Russell
- The universe is one dimensional
RussellE
> On Mar 2, 12:12 pm, RussellE <reaste...@gmail.com> wrote:
>
> > Simpler is better
>
> COHERENT would be nice too.
>
> > 1) Axiom of extensionality - two sets are equal if they have the same
> > elements.
>
> I guess you're adopting identity theory also with this.
>
> > 2) Axiom of finiteness - the set U = {u0,u1,u2,...,uk) exists.
> > u is an element of U iff u is an urelement.
> You skipped what I said about that.
You said I needed to define finite ("r-nite").
This may be difficult to do with one axiom considering
the contortions we have to go through to define finite in ZFC.
The simplest method does seem to be to define a set of urelements
(you called it a ret). The axiom of infinity defines a set.
I could have an axiom schema for urelements. Each urelement
is a new axiom. Then, I could require the theory to have
a "finite" number of axioms.
> > 3) Axiom of well ordering - the urelements have the following order:
> > u0 < u1 < u2 < ... < uk
>
> This is not an axiom in the language of your system.
Do I need something like a 1-place predicate?
> PLEASE, you tell us you're giving us primitives and stuff, but then
> you just skip right past the matter.
>
> > 4) Axiom of singleton - if u is an urelement there exists a set with u
> > as the only element.
>
> Okay, but only assuming that you have identity theory (or first order
> logic with identity) to work with to explicate the expression "the
> only".
>
> > 5) Axiom of union - if A and B are sets there exists a set with all
> > the elements of both A and B.
>
> Okay.
>
> > 6) Axiom of intersection - if A and B are sets there exists a set with
> > all elements common to A and B.
>
> Okay.
>
> > 7) Axiom of complement - if A is a set there exists a set of
> > urelements not in A.
>
> Okay.
>
> > Define every urelement except the largest to be a natural number.
>
> This depends on your Axiom 3, which is still just floating
> mathematical verbiage.
>
> > The largest urelement is defined as NaN - not a number.
> > Any arithmetic operation with NaN as an operand equals NaN.
>
> Freefloating mathematical verbiage.
I want to have two types of urelements. One type is a natural number.
The other type is "not a number". I think this makes sense for an UST.
The largest "natural number" is not a natural number.
> > For example, NaN+1 = NaN.
>
> Freefloating mathematical verbiage.
Arithmetic is defined for all urelements, not just natural numbers.
Assume addition is defined as a two place predicate.
The sum of NaN and any other urelement is NaN.
> > Arithmetic can be completely defined.
>
> Whatever that means.
It means I can define 2-place predicates that correspond
to addition, multiplication, etc.
MoeBlee
> On Mar 2, 10:03 am, MoeBlee <jazzm...@hotmail.com> wrote:
>
> > On Mar 2, 12:48 am, RussellE <reaste...@gmail.com> wrote:
> > > > What is the purpose of your theory?
>
> > > I want to show it is possible to have a consistent, finite set theory.
>
> > What do you MEAN by a 'finite theory'? You seem to have your own
> > definition of 'finite'.
>
> The title of the thread is "An Ultrafinite Set Theory".
> I want a theory with a largest natural number.
> This is more restrictive than just a finite theory.
It's no problem to have a theory that has a theorem "there exists a
largest natural number". So what? What ELSE does the theory prove?
What interesting and/or useful mathematics can you prove with your
theory?
> > (1) Do you mean (given the ordinary definition of 'finite'), a theory
> > that has a theorem that there exist only finite sets? Then that is
> > easy: The theory, in the language of set theory, whose sole axiom is
> > "There exist only finite sets" is a consistent theory.
>
> > (2) Do you mean (given the ordinary definition of 'finite'), a theory
> > that has only models with finite domains? Again, that is easy. The
> > theory, in the langauge of set theory, whose sole axiom is "Axy(x=y)"
> > is consistent and has models only with finite domains.
>
> > So what?
>
> I have often been told there are no "consistent" ultrafinite set
> theories (UST).
Who told you that?
Here's a consistent "ultrafinite set theory":
Axy x=y.
So what?
> I suspect people don't mean we can always derive a contradiction from
> the axioms of a UST. I think they mean UST's aren't consistent with
> their idea of arithematic.
Yeah, okay. So it's not consistent with their idea of arithmetic. What
of it?
> > > What do you mean by "ordinary mathematics for the sciences"?
>
> > I have no PRECISE definition. It is left open to reasonable
> > interpretation. But a typical minimum would be some calculus for
> > predicting the motions of objects, for caclulating probabilites and
> > for statistics.
>
> The philosophy of science says truth can only be determined
> by measurement and repeatable experiments.
> In some ways, science is the antithesis of mathematics which
> says truth can be derived by pure reasoning.
For sake of argument, let's say you're right. So what?
> Could a UST predict the orbit of Mercury?
> I think one could. Assume a set of 2^500 natural numbers.
> Predicting the orbit of Mercury to within experimental error
> should not be much harder than writnig a video game for a
> really large computer monitor with a finite number of pixels.
For sake of argument, okay. It doesn't entail that infinite sets don't
provide a useful and easy to use caclulus.
> > > Can you derive E=MC^2 from ZFC?
>
> > As I understand, that is a statement of physics, not just of
> > mathematics. I'm referring to the mathematics uses for physics, not
> > the physics itself.
>
> The first experimental evidence for the theory of relativity
> was a deviation in the orbit of Mercury as calulated using
> Newton's laws. How do you draw a line between math and science?
I don't have a comprehensive answer. Personally, I would say that
(pure, or theoretical) mathematics concerns deductions about relations
among purely abstract objects. That is, objects whose properties are
entirely general and abstract.
> > > If you mean Peano arithematic,
>
> > No, I don't. First order PA by itself, is, as far as I know, not
> > adequate for a theory for the sciences.
>
> What does "adequate" mean? For the most part,
> science doesn't care about theory.
'adequate' in its ordinary English meaning. Science uses a certain
amount of mathematics. By adequate I mean such mathematics as needed
for ordinary calculus, finite combinatorics, probablity, statistics.
Then also for whatever other mathematics is needed for such constructs
as relativity and quantum mechanics.
> > > > > I don't need an axiom schema of specification.
> > > > > The singleton axiom and union axiom are enough to create any set.
>
> > > > Depends on what you think are "enough" sets.
>
> > > Do we really need a continuum number of sets to do "science"?
>
> > I don't know. But that doesn't obviate the sense of my question as to
> > what you consider enough sets. Why do you need sets at all?
>
> I don't need sets. I really want natural numbers.
You contradict below:
> > Why do you
> > need a set theory?
>
> To formalize intuitive notions of "natural numbers".
So do you want 'set' to be a concept in your theory or not?
If all you want are finitely many counting numbers, then maybe
something like this:
First order logic with identity.
Then use the language of identity theory to (theoretically) write out
the formula that says there exist exactly Y number of objects, where Y
is the number 2^500 or whatever you want, but we don't mention "2^500"
in the actual formula as instead we just write the HUGE formula of
identity theory that ensures all and every model of the theory has
exactly 2^500 elements. This is your sole non-logical axiom. Every and
only models that have exactly 2^500 elements are models of this
theory.
Done.
Now so what?
> > Why do you need any theory at all?
>
> So I can experiment on them. Before I started playing
> with this UST, I might not have considered the idea
> that arithmetic may not be defined for every pair of
> natural numbers. What if some numbers are so big
> they can't be added to another number. How would
> this change our intutions about natural numbers.
> Can I come up with a test for this in the real world?
Fine. But then first please find out how formal theories actually
work.
MoeBlee
MoeBlee
> On Mar 2, 10:58 am, MoeBlee <jazzm...@hotmail.com> wrote:
> > > 2) Axiom of finiteness - the set U = {u0,u1,u2,...,uk) exists.
> > > u is an element of U iff u is an urelement.
> > You skipped what I said about that.
>
> You said I needed to define finite ("r-nite").
> This may be difficult to do with one axiom considering
> the contortions we have to go through to define finite in ZFC.
Nope, you don't know what you're talking about. We can define 'is
finite' just as we do in ZFC but without using ANY set theory axioms
at all. I ALREADY told you: definition of predicates does not depend
on axioms.
> The simplest method does seem to be to define a set of urelements
> (you called it a ret). The axiom of infinity defines a set.
The axiom of infinity states that there exists a set (we don't even
have to say it is a 'set', as we could just say 'object' if we wanted
to) having certain properties. Then we prove that there is exactly one
set that has those properties and is a subset of any other set having
those properties.
However, as to definining 'is finite', we don't need the axiom of
infinity or any other set theory axioms.
> I could have an axiom schema for urelements. Each urelement
> is a new axiom. Then, I could require the theory to have
> a "finite" number of axioms.
Whatever you mean by a schema for urelements.
Also, if your theory is definite, you'd need to say not just that
there are finitely many such axioms but give the exact fintite number.
> > > 3) Axiom of well ordering - the urelements have the following order:
> > > u0 < u1 < u2 < ... < uk
>
> > This is not an axiom in the language of your system.
>
> Do I need something like a 1-place predicate?
Maybe another 2-place predicate (or you could use the 2-place
predicate you have, viz. 'e', and stipulate by axiom some kind of
ordering with it).
But you still need to straighten out other stuff for this to be
coherent.
> I want to have two types of urelements. One type is a natural number.
> The other type is "not a number".
Then 'r-tural number' might be a candidate for being a primitive
predicate. Then an axiom that says there exist r-tural numbers and an
axiom that says there exist objects that are not r-tural numbers.
> I think this makes sense for an UST.
> The largest "natural number" is not a natural number.
Once you get 'largest' straightened out, then the above could be
another axiom.
> > > For example, NaN+1 = NaN.
>
> > Freefloating mathematical verbiage.
>
> Arithmetic is defined for all urelements,
I don't know what you mean by 'arithemetic is defined for' in this
context. You haven't GIVEN any arithmetical operations.
> not just natural numbers.
> Assume addition is defined as a two place predicate.
You probably mean a 2-place operation.
> The sum of NaN and any other urelement is NaN.
So 'a' is the operation symbol? What is 'N' again?
> > > Arithmetic can be completely defined.
>
> > Whatever that means.
>
> It means I can define 2-place predicates that correspond
> to addition, multiplication, etc.
You probably mean 2-place operations.
If you can define them, then go ahead and do it, but you're not
allowed to use ANYTHING except your primitives and previously defined
predicates and operations. Also, to show that your definitions of
operations are "good" you need to prove the appropriate existence and
uniqueness theorems.
MoeBlee
Virgil
<be3d057d-1a58-4a17...@b5g2000prd.googlegroups.com>,
RussellE <reas...@gmail.com> wrote:
> Simpler is better.
>
> 1) Axiom of extensionality - two sets are equal if they have the same
> elements.
>
> 2) Axiom of finiteness - the set U = {u0,u1,u2,...,uk) exists.
> u is an element of U iff u is an urelement.
What are 0,1,2,...,n ?
You cannot use 0,1,2,...,n unless your world already contains them,
i.e., contains numerals for certain natural numbers.
>
> 3) Axiom of well ordering - the urelements have the following order:
> u0 < u1 < u2 < ... < uk
>
> 4) Axiom of singleton - if u is an urelement there exists a set with u
> as the only element.
>
> 5) Axiom of union - if A and B are sets there exists a set with all
> the elements of both A and B.
>
> 6) Axiom of intersection - if A and B are sets there exists a set with
> all elements common to A and B.
>
> 7) Axiom of complement - if A is a set there exists a set of
> urelements not in A.
>
>
> Define every urelement except the largest to be a natural number.
> The largest urelement is defined as NaN - not a number.
>
> Any arithmetic operation with NaN as an operand equals NaN.
> For example, NaN+1 = NaN.
If NaN - 1 = NaN (i..e., the predecessor of NaN is Nan), your arithmetic
is going to be bloody useless.
RussellE
> On Mar 2, 1:28 pm, RussellE <reaste...@gmail.com> wrote:
>
> > > > What do you mean by "ordinary mathematics for the sciences"?
>
> > > I have no PRECISE definition. It is left open to reasonable
> > > interpretation. But a typical minimum would be some calculus for
> > > predicting the motions of objects, for caclulating probabilites and
> > > for statistics.
>
> > The philosophy of science says truth can only be determined
> > by measurement and repeatable experiments.
> > In some ways, science is the antithesis of mathematics which
> > says truth can be derived by pure reasoning.
>
> For sake of argument, let's say you're right. So what?
>
> > Could a UST predict the orbit of Mercury?
> > I think one could. Assume a set of 2^500 natural numbers.
> > Predicting the orbit of Mercury to within experimental error
> > should not be much harder than writnig a video game for a
> > really large computer monitor with a finite number of pixels.
>
> For sake of argument, okay. It doesn't entail that infinite sets don't
> provide a useful and easy to use caclulus.
OK. But, Newton's calculus doesn't correctly predict the orbit of
Mercury.
> > > > Can you derive E=MC^2 from ZFC?
>
> > > As I understand, that is a statement of physics, not just of
> > > mathematics. I'm referring to the mathematics uses for physics, not
> > > the physics itself.
>
> > The first experimental evidence for the theory of relativity
> > was a deviation in the orbit of Mercury as calulated using
> > Newton's laws. How do you draw a line between math and science?
>
> I don't have a comprehensive answer. Personally, I would say that
> (pure, or theoretical) mathematics concerns deductions about relations
> among purely abstract objects. That is, objects whose properties are
> entirely general and abstract.
Which properties? I see math more as a science.
Natural numbers have properties we can observe.
Is geometry "entirely general and abstract"?
I get different predictions for the orbit of Mercury
depending on which geometry I choose.
> > > > If you mean Peano arithematic,
>
> > > No, I don't. First order PA by itself, is, as far as I know, not
> > > adequate for a theory for the sciences.
>
> > What does "adequate" mean? For the most part,
> > science doesn't care about theory.
>
> 'adequate' in its ordinary English meaning. Science uses a certain
> amount of mathematics. By adequate I mean such mathematics as needed
> for ordinary calculus, finite combinatorics, probablity, statistics.
> Then also for whatever other mathematics is needed for such constructs
> as relativity and quantum mechanics.
I think science needs mathematics as good as our ability
to measure things. If we can't measure it, science says
it doesn't exist.
Can I use this theory to better predict the orbit of Mercury?
I think such a system could have some interesting properties.
Can we prove such a system is consistent?
What are the differences between arithmetic on "small"
natural numbers and "large" natural numbers?
RussellE
NaN is an abbreviation used by many computer languages
to mean "not a number". Mathematical functions return
a number or "NaN" (which usually means an error).
I should have a single symbol for it.
I usually call the largest natural number "z".
MoeBlee
> On Mar 2, 12:54 pm, MoeBlee <jazzm...@hotmail.com> wrote:
> > For sake of argument, okay. It doesn't entail that infinite sets don't
> > provide a useful and easy to use caclulus.
>
> OK. But, Newton's calculus doesn't correctly predict the orbit of
> Mercury.
I never claimed otherwise. Maybe you're confusing 'sciences for the
mathematics' with the scientific theories that use mathematics?
> > I don't have a comprehensive answer. Personally, I would say that
> > (pure, or theoretical) mathematics concerns deductions about relations
> > among purely abstract objects. That is, objects whose properties are
> > entirely general and abstract.
>
> Which properties?
Those that are defined in a purely abstract way.
> I see math more as a science.
However you see mathematics, it doesn't vitiate anything I've said
here.
> Natural numbers have properties we can observe.
This subject has been beaten to death by other discussants.
In a nutshell: I am aware of experiences that impress me as sensory,
and then I am able to form constructs of objects that I categorize as
'physical objects', such as tables and chairs. However, I see no
physical object that is, e.g. the natural number 1.
> Is geometry "entirely general and abstract"?
Sure, pure abstract geometry is. It is not required to reference
physical objects or even pictures merely to study the abstract
relations of geometry.
> I get different predictions for the orbit of Mercury
> depending on which geometry I choose.
So?
> I think science needs mathematics as good as our ability
> to measure things. If we can't measure it, science says
> it doesn't exist.
You're welcome to have your view of things. But nothing you've said
vitiates anything I've said.
> > First order logic with identity.
>
> > Then use the language of identity theory to (theoretically) write out
> > the formula that says there exist exactly Y number of objects, where Y
> > is the number 2^500 or whatever you want, but we don't mention "2^500"
> > in the actual formula as instead we just write the HUGE formula of
> > identity theory that ensures all and every model of the theory has
> > exactly 2^500 elements. This is your sole non-logical axiom. Every and
> > only models that have exactly 2^500 elements are models of this
> > theory.
>
> > Done.
>
> > Now so what?
>
> Can I use this theory to better predict the orbit of Mercury?
I quite doubt it. That's my point.
> I think such a system could have some interesting properties.
> Can we prove such a system is consistent?
The one I mentioned. Of course it's consistent.
> What are the differences between arithmetic on "small"
> natural numbers and "large" natural numbers?
Whatever 'small' and 'large' mean to you in this context, I don't know
why you're asking ME about this.
MoeBlee
Jesse F. Hughes
> I have often been told there are no "consistent" ultrafinite set
> theories (UST).
Really? Have you been told so here on the newsgroup?
Can you point me to a single post in which someone said that?
> I suspect people don't mean we can always derive a contradiction from
> the axioms of a UST. I think they mean UST's aren't consistent with
> their idea of arithematic.
I suspect that you're making things up. Or, perhaps more charitably,
misremembering.
--
"Am I am [sic] misanthrope? I would say no, for honestly I never heard
of this word until about 1994 or thereabouts on the Internet reading a
post from someone who called someone a misanthrope."
-- Archimedes Plutonium
RussellE
> RussellE <reaste...@gmail.com> writes:
> > I have often been told there are no "consistent" ultrafinite set
> > theories (UST).
>
> Really? Have you been told so here on the newsgroup?
>
> Can you point me to a single post in which someone said that?
People have said that in this newsgroup (it might have been me).
Here is what Wikipedia says:
http://en.wikipedia.org/wiki/Ultrafinitism
but even constructivists generally view the philosophy as unworkably
extreme
and
the constructive logician A. S. Troelstra dismissed it by saying "no
satisfactory development exists at present."
Why are ultrafinite theories considered "unworkable"?
I would think an UST woiuld be similar to theories with universal
sets.
Russell
- Integers are an illusion
MoeBlee
> On Mar 2, 3:33 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>
> > RussellE <reaste...@gmail.com> writes:
> > > I have often been told there are no "consistent" ultrafinite set
> > > theories (UST).
>
> > Really? Have you been told so here on the newsgroup?
>
> > Can you point me to a single post in which someone said that?
>
> People have said that in this newsgroup (it might have been me).
You "told" yourself then?
> Here is what Wikipedia says:http://en.wikipedia.org/wiki/Ultrafinitism
I see no claim there that there any ultrafinite set theory must be
inconsistent.
> but even constructivists generally view the philosophy as unworkably
> extreme
So, that's not saying that any ultrafinite set theory must be
inconsistent.
> and
>
> the constructive logician A. S. Troelstra dismissed it by saying "no
> satisfactory development exists at present."
That's not saying that any ultrafinite set theory must be
inconsistent.
> Why are ultrafinite theories considered "unworkable"?
Whether they are unworkable or not (work for what purpose?), it seems
to me that what the article may be getting at is how difficult it is
to come up with axioms for such a theory that also provide us with
such results as we wish to have from a foundational theory.
> I would think an UST woiuld be similar to theories with universal
> sets.
I don't see the connection, though I'm not claiming there isn't one.
Why don't you first do some systematic, organized study on formal
theories, standard set theory, alternative theories, and the
philosophy of mathematics?
Right now you look like some guy thrashing about in a mental
cellophane bag.
MoeBlee
Jesse F. Hughes
> On Mar 2, 3:33 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>> RussellE <reaste...@gmail.com> writes:
>> > I have often been told there are no "consistent" ultrafinite set
>> > theories (UST).
>>
>> Really? Have you been told so here on the newsgroup?
>>
>> Can you point me to a single post in which someone said that?
>
> People have said that in this newsgroup (it might have been me).
You could have simply said "no". None of the below is any evidence in
your favor. You made a perfectly clear claim, you know.
>
> Here is what Wikipedia says:
> http://en.wikipedia.org/wiki/Ultrafinitism
>
> but even constructivists generally view the philosophy as unworkably
> extreme
>
> and
>
> the constructive logician A. S. Troelstra dismissed it by saying "no
> satisfactory development exists at present."
>
> Why are ultrafinite theories considered "unworkable"?
> I would think an UST woiuld be similar to theories with universal
> sets.
--
Jesse F. Hughes
"I'm not going to forget what I've seen. I understand the devastation
requires more than one day's attention."
-- G. W. Bush reassures Hurricane Katrina victims. Two days, minimum.
Transfer Principle
> In article
> <be3d057d-1a58-4a17-89a7-e312ea28f...@b5g2000prd.googlegroups.com>,
> RussellE <reaste...@gmail.com> wrote:
> > Any arithmetic operation with NaN as an operand equals NaN.
> > For example, NaN+1 = NaN.
> If NaN - 1 = NaN (i..e., the predecessor of NaN is Nan), your arithmetic
> is going to be bloody useless.
In computer arithmetic (IEEE 754, which is of course where RE
got the idea of NaN from), NaN-1 is indeed NaN. Here's a link
which explicitly lists NaN-1 as being NaN:
http://users.tkk.fi/jhi/infnan.html
Therefore, by Virgil's standards, IEEE 754 arithmetic must be
"bloody useless," even though Virgil probably uses software
that adheres to IEEE 754 every time he turns on his computer.
Ironically, in another thread when I asked about ultrafinitist
theories, Fred Jeffries suggested that I consider the IEEE 754
standard as an example of ultrafinitism. RE appears to be
heading in that direction with his use of "NaN."
Jesse F. Hughes
> NaN is an abbreviation used by many computer languages
> to mean "not a number". Mathematical functions return
> a number or "NaN" (which usually means an error).
>
> I should have a single symbol for it.
> I usually call the largest natural number "z".
Using "NaN" is perfectly fine. Moe was presumably unfamiliar with the
notation, but I'm sure he has no complaint once it's explained.
--
"Come on people!!! The US just blew up a lot of people in Iraq, don't
you realize that a person with my exposure might just end up dead, by
mysterious circumstances?"
--James Harris, on the dangers of "proving" Fermat's last theorem
Jesse F. Hughes
> In computer arithmetic (IEEE 754, which is of course where RE
> got the idea of NaN from), NaN-1 is indeed NaN. Here's a link
> which explicitly lists NaN-1 as being NaN:
>
> http://users.tkk.fi/jhi/infnan.html
>
> Therefore, by Virgil's standards, IEEE 754 arithmetic must be
> "bloody useless," even though Virgil probably uses software
> that adheres to IEEE 754 every time he turns on his computer.
>
> Ironically, in another thread when I asked about ultrafinitist
> theories, Fred Jeffries suggested that I consider the IEEE 754
> standard as an example of ultrafinitism. RE appears to be
> heading in that direction with his use of "NaN."
Yes, it's very ironic when two wholly unrelated persons have a
difference of opinions.
--
"It is my opinion that since neither Spight nor Hughes can see or
understand their moral trespass [namely, quoting AP in .sigs], that
their degrees from whatever university they earned their degree should
be annulled." -- Archimedes Plutonium (12/1/09)
Transfer Principle
> > OK. The primitives are element and collection.
> > urelement - Only objects defined to be urelements can be elements of a
> > set
> > set - A collection of elements.
> > proper class - a collection of sets.
> 1-place predicate - 'x is a ret'
> 1-place predicate - 'x is an urment'
> 2-place predicate - 'xey' ('x is an element of y')
This is the second time that MoeBlee has played around with
rhyming words like "ret" and "urment" in trying to describe
RE's theory. (The first time was back in the second post of
this thread.) Also the standard theorists Patricia Shanahan
William Eliot have also criticized RE for trying to steal
terminology from the standard theories (such as ZFC and PA)
and use them in his own theory.
I don't agree with this notion that standard theories have
a monopoly on these terms. If we accept ZFC as the standard
theory, then what about the theory ZF (or to be explicit,
ZF+~AC)? Now ZF+~AC proves the existence of nonempty sets
without choice functions. But according to the standard
theory ZFC, every nonempty set has a choice function. So
what if I were to claim that therefore, these nonempty
objects in ZF+~AC that lack choice functions aren't really
sets, so we should call them "rets" or "tets" instead?
Similarly, ZFA proves the existence of illfounded sets. This
is in contrast with the standard theory ZFC, which proves
that every set is wellfounded. So what if I were to claim
that therefore, these illfounded objects in ZFA aren't really
sets, so we should call them "prets" or "vlets" instead?
Just as NFU proves the existence of non-Cantorian sets. This
is in contrast with the standard theory ZFC, which proves
that every set is Cantorian. So what if I were to claim
that therefore, these non-Cantorian sets in NFU aren't really
sets, so we should call them "nfets" or "wrets" instead?
Finally, bringing this back to ultrafinitism, we know that
there exist standard naturals n such that Y-V can't answer
yes to the question "Is n a natural number?" The sum or
product of two standard naturals is also a standard natural,
whereas the set of all naturals n such that Y-V can answer
"Is n a natural number?" isn't closed under either addition
or multiplication. So what if I were to claim that therefore,
Y-V isn't really talking about natural numbers, but something
called "yvatural numbers" instead?
Of course, this is silly. Adherents of ZF+~AC, ZFA, and NFU
aren't going to call their objects "rets" just because they
aren't sets in ZFC. The "yvatural numbers" example is even
worse, since the set of all "yvatural numbers" is a proper
subset of the set of all natural numbers, and so every
"yvatural number" literally _is_ a natural number, whether
the adherents of PA like it or not.
Thus, one shouldn't say that the objects that RE describes
in his theory aren't sets or natural numbers, unless one is
prepared to do the same with NFU's sets or Y-V's naturals.
Of course, at this point the standard anti-"cranks" are
likely thinking about a "slippery slope" argument -- if one
can call RE's objects "natural numbers," what's to stop
another so-called "crank" from calling Q the set of naturals,
or R the set of naturals, or {e, i, pi, 42} the set of
naturals, or some other crazy set? Where do we draw the line?
Here's where we draw the line: we can call an object defined
in a nonstandard theory by the same name as an object defined
in a standard theory, if the nonstandard object is an _analog_
of the standard object in the new theory, satisfying some
basic property of the standard object.
An example: RE wishes to define "urelement" in his theory. To
me, a basic property of "urelements" is that they contain no
elements (and aren't the empty set). Since RE's objects don't
contain elements, I believe that RE has the right to keep on
calling them "urelements." On the other hand, if RE were to
define "urelements" so that they have elements, then I'd agree
that RE would be disingenuous in calling them "urelements," so
that MoeBlee and the others would be justified in making him
change their name to "urments" or "burblements."
RE's "sets" can contain urelements. Sets in ZFCU and NFU may
contain urelements. So I see no reason for RE to change the
name "sets," unless we're going to make adherents of ZFCU and
NFU stop calling their objects "sets" too.
RE's "classes" can contain sets as elements. Classes in NBG
may contain sets. So I see no reason for RE to change the
name "classes," unless we're going to make adherents of NBG
stop calling their objects "classes" too.
Finally, ultrafinitists wish to work with a finite subset of
the set of standard naturals. So I see no reason for them to
stop calling their objects "natural numbers" simply because
there are only finitely many of them in their theories.
To repeat, ZFC/PA don't have a monopoly on the names "sets,"
"natural numbers," etc., no matter how much the standard
theorists may desire this.
Patricia Shanahan
> On Mar 2, 10:03 am, MoeBlee <jazzm...@hotmail.com> wrote:
>> On Mar 2, 12:48 am, RussellE <reaste...@gmail.com> wrote:
>>> OK. The primitives are element and collection.
>>> urelement - Only objects defined to be urelements can be elements of a
>>> set
>>> set - A collection of elements.
>>> proper class - a collection of sets.
>> Primitives:
>> 1-place predicate - 'x is a ret'
>> 1-place predicate - 'x is an urment'
>> 2-place predicate - 'xey' ('x is an element of y')
>
> This is the second time that MoeBlee has played around with
> rhyming words like "ret" and "urment" in trying to describe
> RE's theory. (The first time was back in the second post of
> this thread.) Also the standard theorists Patricia Shanahan
> William Eliot have also criticized RE for trying to steal
> terminology from the standard theories (such as ZFC and PA)
> and use them in his own theory.
I'm not a theorist at all, standard or otherwise. I'm a practical
programmer and computer architect. I do think it would reduce confusion
if the term "natural numbers" were used for a structure that does
conform to the Peano Postulates, and other terms were used for
structures that don't.
There are examples of errors in algorithms that may have been due to
thinking "integer", and applying a formula that works for integers, when
the reality is a bounded range number type.
Patricia
RussellE
http://en.wikipedia.org/wiki/Peano_axioms
I can define arithmetic the same way by
changing my definition of natural number.
PA defines natural numbers in "unary".
PA says 0, S(0), S(S(0)), ... are natural numbers.
We just count the calls to successor.
I can define natural numbers as sets just like PA.
With this definition, I don't assume the urlements
are natural numbers. I only assume they are ordered.
Define 0 as the singleton set containing the smallest urelement.
Define successor of set X to be the union of X and
the singleton set of the smallest urelement not in X.
Let U = {a,b,c,d}
Let a < b < c < d
0 = {a}
1 = {a,b}
2 = {a,b,c}
3 = {a,b,c,d}
The set U is closed under my successor function.
The successor of {a,b,c,d} is {a,b,c,d} U {}.
Marshall
>
> There are examples of errors in algorithms that may have been due to
> thinking "integer", and applying a formula that works for integers, when
> the reality is a bounded range number type.
One such example is Joshua Bloch's "Nearly All Binary Searches and
Mergesorts are Broken"
http://googleresearch.blogspot.com/2006/06/extra-extra-read-all-about-it-nearly.html
Marshall
Marshall
>
> Therefore, by Virgil's standards, IEEE 754 arithmetic must be
> "bloody useless," even though Virgil probably uses software
> that adheres to IEEE 754 every time he turns on his computer.
IEEE 754 is of course quite useful for doing calculations.
It's not something that qualifies as a model of anything
the least bit applicable to mathematical proof, which means
that *in context* Virgil's claim is entirely correct, even if
perhaps a bit dramatically phrased.
Algebraic properties so basic and fundamental as
associativity of addition and multiplication do not hold in
IEEE 754.
Marshall
Transfer Principle
> On Mar 2, 1:28 pm, RussellE <reaste...@gmail.com> wrote:
> > theories (UST).
> Here's a consistent "ultrafinite set theory":
> Axy x=y.
Ah yes, _that_ theory. The theory which spawned a long
debate between the standard theorists and Nam Nguyen over
whether "Axy (x+y=0)" is provable in the theory.
> > > No, I don't. First order PA by itself, is, as far as I know, not
> > > adequate for a theory for the sciences.
Of course, if MoeBlee doesn't even consider PA to be
adequate for the sciences, what chance does RE (or anyone
else) have in convincing him that a _weaker_ theory, such
as an ultrafinitist theory, is adequate for science?
> If all you want are finitely many counting numbers, then maybe
> something like this:
> First order logic with identity.
> Then use the language of identity theory to (theoretically) write out
> the formula that says there exist exactly Y number of objects, where Y
> is the number 2^500 or whatever you want, but we don't mention "2^500"
> in the actual formula as instead we just write the HUGE formula of
> identity theory that ensures all and every model of the theory has
> exactly 2^500 elements. This is your sole non-logical axiom. Every and
> only models that have exactly 2^500 elements are models of this
> theory.
> Done.
Perhaps the following is another way to grasp what RE is thinking,
in a way that also sheds light on what many so-called "cranks" are
thinking when they try to come up with new theories:
Let S be a set of natural numbers (and here we're returning to the
standard definition of "natural number"). Then the question is,
can we find a theory T such that (ZFC proves that) for every
natural number n, n is in S if and only if there exists a set M
such that the cardinality of M is n, and M is (a carrier set of) a
model of T?
Suppose S={1}. Then we need a theory T such that every model of T
has cardinality one. Obviously, the theory that MoeBlee mentions,
namely the one with lone axiom "Axy (x=y)", qualifies.
But suppose S is the set of even natural numbers. So we seek a
theory T such that there exists a model of T of each even
cardinality, but of no odd cardinality. We may try the following:
Assuming FOL with identity:
Language: Let Z be a one-place function symbol.
Axioms:
1. Ax ~(Zx = x)
2. Ax (ZZx = x)
Then every finite model of theory must have an even number of
objects since the objects appear in pairs, x and Zx.
Now suppose S is the set of odd natural numbers. Then we may
replace axiom 1 above with the following axiom:
1'. E!x (Zx = x)
But RE is probably thinking about letting S be a more
challenging set, such as the set of all powers of two:
S = {1, 2, 4, 8, ...}
So we need a theory T such that for every power of two there's
a model of T with that cardinality, and every finite model of
T has a power of two as its cardinality.
I have yet to think of such a theory. Such a theory may be the
sort of theory that RE has in mind. The objects of this theory
may correspond to RE's notion of urelements and sets. (Also,
we need to find a way to make all the models _finite_.)
(Come to think of it, we may actually want S to be the set of
natural numbers of the form n+2^n, not merely 2^n, so that we
can have n urelements and 2^n sets.)
S = {1, 3, 6, 11, 20, 37, 70, ...}
This may be helpful for other "cranks," not just RE. Some
"cranks" don't believe in uncountable sets. Of course, with
theories with infinite models, we have to be worried about
Lowenheim-Skolem. So if we say, "Let T be a theory such
that every model of T is countable," we can't say (first-order)
PA since by L-S, there exists an uncountable model of PA. (I'm
not sure about second-order PA here.)
But it's been noted in previous threads that these models of
theories that exist via L-S don't necessarily map "e" to
anything resembling membership. So we might add a requirement
that "e" must be mapped to membership.
So one might ask the question:
Given a set U, find a theory T such that (ZFC proves that) U is
a model of T mapping "e" to membership.
The sets U=V_omega and U=V_(omega+omega) have well-known
solutions to this problem (viz., ZF-Infinity and ZF-Replacement
Schema, respectively.)
But what about V_(omega+1)? Notice that the all of the elements
of V_(omega+1) are countable, so it might work. (It's possible
that NBG-Infinity and Randall Holmes's PST work for V_(omega+1)
and V_(omega+2) respectively, but this may be undesirable to
the Cantor "cranks" because they want the countably infinite
objects to be _sets_, not classes as in these theories.)
To me, questions of this type (given a set S, find a theory T
such that S is a model of T, with certain requirements in order
to avoid trivial answers) are interesting, though they might not
be interesting to the standard theorists.
Marshall
> "It is my opinion that since neither Spight nor Hughes can see or
> understand their moral trespass [namely, quoting AP in .sigs], that
> their degrees from whatever university they earned their degree should
> be annulled." -- Archimedes Plutonium (12/1/09)
<zoidberg>What an honor!</zoidberg>
I've made it into Jesse Hughes' quotes file! I can now take my
rightful place, albeit as a junior member, alongside such giants
as Archimedes Plutonium and James Harris, as one of the
Greats of Usenet.
Let me now quote our lovable mascot John Jones, extraneous
comma and all:
"I am truly, a giant among mortals."
Marshall
Archimedes Plutonium
This is not the first time I complained about Hughes who constantly
plays this trick
of quoting me out of context, and uses that as his signature block.
This is FORGERY, since most new people would not be able to recognize
that I did
not write the body of the text. A signature block is meant for the
author of the text,
not for some jerk to fan hatred.
I have asked before that News Admin pull the plug on Hughes, since he
is never able
to stop his miscreant behaviour.
I do not like being quoted out of context, and I do not like being
forged with a text I never
wrote.
Please put the final hammer on Hughes and pull his plug.
Archimedes Plutonium
the text.
And NANA has strict rules.
This is about the umpteenth time Hughes has quoted out of context and
blurred
the signature block with a forgery.
So NANA, if you cannot abide by your own rules, I shall name you co-
responsible.
Virgil
<9ebc97a3-3dc7-4583...@u15g2000prd.googlegroups.com>,
RussellE <reas...@gmail.com> wrote:
> I looked at how Peano arithmetic is formalized:
> http://en.wikipedia.org/wiki/Peano_axioms
>
> I can define arithmetic the same way by
> changing my definition of natural number.
> PA defines natural numbers in "unary".
> PA says 0, S(0), S(S(0)), ... are natural numbers.
> We just count the calls to successor.
>
> I can define natural numbers as sets just like PA.
> With this definition, I don't assume the urlements
> are natural numbers. I only assume they are ordered.
You cannot even define order until you have a theory sufficient to allow
definition of relations .
>
> Define 0 as the singleton set containing the smallest urelement.
How do you know there is a "smallest" urelement?
>
> Define successor of set X to be the union of X and
> the singleton set of the smallest urelement not in X.
>
> Let U = {a,b,c,d}
> Let a < b < c < d
>
> 0 = {a}
> 1 = {a,b}
> 2 = {a,b,c}
> 3 = {a,b,c,d}
>
> The set U is closed under my successor function.
> The successor of {a,b,c,d} is {a,b,c,d} U {}.
By what definition of function? What is the domain of that "successor"
function, and what is its range, and is it a bijective function or not?
Jesse F. Hughes
> On Mar 2, 12:54 pm, MoeBlee <jazzm...@hotmail.com> wrote:
>> On Mar 2, 1:28 pm, RussellE <reaste...@gmail.com> wrote:
>> > I have often been told there are no "consistent" ultrafinite set
>> > theories (UST).
>> Who told you that?
>> Here's a consistent "ultrafinite set theory":
>> Axy x=y.
>
> Ah yes, _that_ theory. The theory which spawned a long
> debate between the standard theorists and Nam Nguyen over
> whether "Axy (x+y=0)" is provable in the theory.
You focus on the most inconsequential coincidences as if they were
central.
--
Jesse F. Hughes
Quincy (age 3 1/2, looking at a picture): Are these people Canadians?
Me: Uh, no, they're Australian Aborigines.
Quincy: Do they fight Canadians?
Jesse F. Hughes
> Let S be a set of natural numbers (and here we're returning to the
> standard definition of "natural number"). Then the question is,
> can we find a theory T such that (ZFC proves that) for every
> natural number n, n is in S if and only if there exists a set M
> such that the cardinality of M is n, and M is (a carrier set of) a
> model of T?
>
> Suppose S={1}. Then we need a theory T such that every model of T
> has cardinality one. Obviously, the theory that MoeBlee mentions,
> namely the one with lone axiom "Axy (x=y)", qualifies.
>
> But suppose S is the set of even natural numbers. [...]
As far as I can tell, Russell is only interested in taking finite
initial segments of N as his urelements and has not yet mentioned a
connection between cardinality and those urelements.
Maybe I've missed something, but what you're focusing on here doesn't
look at all like Russell's work.
--
"There's lots of things in this old world to take a poor boy down.
If you leave them be, you can save yourself some pain.
You don't have to live in fear, but you best have some respect,
For rattlesnakes, painted ladies and cocaine." -- Bob Childers
Jesse F. Hughes
> On Mar 2, 7:31 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>> "It is my opinion that since neither Spight nor Hughes can see or
>> understand their moral trespass [namely, quoting AP in .sigs], that
>> their degrees from whatever university they earned their degree should
>> be annulled." -- Archimedes Plutonium (12/1/09)
>
> <zoidberg>What an honor!</zoidberg>
>
> I've made it into Jesse Hughes' quotes file!
You were mentioned in a quote. You are not the author of a quote.
Sorry, but you haven't made the cut.
--
"After years of arguing I realize that your intellects are too limited
to fully grasp my work. [...] Still, no matter how child-like your
minds are, [...] since you have language, [...] there's a chance that
I'll be able to find something that your minds can handle." --JSH
Jesse F. Hughes, Love God
> Attention please: news admin
>
> This is not the first time I complained about Hughes who constantly
> plays this trick
> of quoting me out of context, and uses that as his signature block.
>
> This is FORGERY, since most new people would not be able to recognize
> that I did
> not write the body of the text. A signature block is meant for the
> author of the text,
> not for some jerk to fan hatred.
[...]
>> --
>> "Am I am [sic] misanthrope? I would say no, for honestly I never heard
>> of this word until about 1994 or thereabouts on the Internet reading a
>> post from someone who called someone a misanthrope."
>> -- Archimedes Plutonium
It's far worse than you can imagine! Sometimes, I use the following
.sig.
"'Every man who has ever lived holds tight to the belief that for him
alone the laws of probability are canceled out by love[...] Therefore,
you will marry Guinevere. You do not want advice --- only agreement.'
Merlin sighed..." -- John Steinbeck
Obviously, I am trying to fool people into believing that John
Steinbeck contributes to sci.math.
Other times I use the following .sig.
"Sexual love makes of the loved person an Object of appetite; as soon
as that appetite has been stilled, the person is cast aside as one
casts away a lemon which has been sucked dry." -- Immanuel Kant
"Squeeze my lemon til the juice runs down my leg." -- Robert Johnson
At those times, I'm trying to pass off my discussion of set theory as
a collaboration between Kant and Johnson.
The way I try to fool people with these forgeries is just a damn
shame. I suppose it's good that there's no way to change a post's
from address, or else my forgeries would be even more insidious.
--
Jesse F. Hughes
"To [mathematicians] amateur mathematicians are worse than scum, and
scarier than nuclear bombs."
-- James S. Harris on mathematicians' phobias
Marshall
> Marshall <marshall.spi...@gmail.com> writes:
> > On Mar 2, 7:31 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> >> "It is my opinion that since neither Spight nor Hughes can see or
> >> understand their moral trespass [namely, quoting AP in .sigs], that
> >> their degrees from whatever university they earned their degree should
> >> be annulled." -- Archimedes Plutonium (12/1/09)
>
> > <zoidberg>What an honor!</zoidberg>
>
> > I've made it into Jesse Hughes' quotes file!
>
> You were mentioned in a quote. You are not the author of a quote.
>
> Sorry, but you haven't made the cut.
Dammit!
Marshall
Jesse F. Hughes
> Attention please: news admin
>
> This is not the first time I complained about Hughes who constantly
> plays this trick
> of quoting me out of context, and uses that as his signature block.
>
> This is FORGERY, since most new people would not be able to recognize
> that I did
> not write the body of the text. A signature block is meant for the
> author of the text,
> not for some jerk to fan hatred.
[...]
>> --
>> "Am I am [sic] misanthrope? I would say no, for honestly I never heard
>> of this word until about 1994 or thereabouts on the Internet reading a
>> post from someone who called someone a misanthrope."
>> -- Archimedes Plutonium
Let me just be sure I understand. I omitted my name in the .sig block
because of an old and mostly ignored restriction that .sigs should be
no longer than four lines. I still prefer to keep my .sigs to four
lines, but I might be willing to make an exception for your sake.
So, suppose that I change the .sig so that it appears thus:
>> --
>> Jesse F. Hughes
>> "Am I am [sic] misanthrope? I would say no, for honestly I never heard
>> of this word until about 1994 or thereabouts on the Internet reading a
>> post from someone who called someone a misanthrope."
>> -- Archimedes Plutonium
In this case, even the most casual reader (the kind who somehow fails
to notice the From header which is prominently displayed by default in
every newsreader I've seen) would still see that the body of the text
was written by me.
I suppose, then, that you would be perfectly happy with the newly
amended .sig. You would have no complaint, since no one would mistake
the body as your writing and since the text attributed to you was
written by you (and, I suppose, you stand by that text).
Do I understand you correctly?
--
"I'm the guy. I have always been the guy. Your post will sit here for
a while, soon be ignored, except for people coming to read my reply,
and your satisfaction will fade as you move on, and I'll still be the
guy." -- James S. Harris will *always* be the guy. Duh.
MoeBlee
First, Transfer Prinicple, you blowhard, you lied again about me in
your previous posts. And, so far, you've not responded to my latest
requests that you stop doing that. Over a few years now, you've made
an actual pattern out of lying about my posts. And you continue
without blinking. You have my contempt for that.
> On Mar 2, 10:03 am, MoeBlee <jazzm...@hotmail.com> wrote:
> This is the second time that MoeBlee has played around with
> rhyming words like "ret" and "urment" in trying to describe
> RE's theory.
> I don't agree with this notion that standard theories have
> a monopoly on these terms.
Who said standard theories have such a monopoly?
I suggest different terms merely for the sake of avoiding confusion
with their more ordinary senses in mathematics. (Of course, though,
what I'm really trying to do is enforce a sinister program of mind
control.)
> If we accept ZFC as the standard
> theory, then what about the theory ZF (or to be explicit,
> ZF+~AC)?
Just to be clear, it is not the case that ZF = ZF+~AC. (Whether you
meant that or not.)
> Now ZF+~AC proves the existence of nonempty sets
> without choice functions. But according to the standard
> theory ZFC, every nonempty set has a choice function. So
> what if I were to claim that therefore, these nonempty
> objects in ZF+~AC that lack choice functions aren't really
> sets, so we should call them "rets" or "tets" instead?
Bad argument. Different theories do prove sets to have different
characteristics. But still there are some basic agreements so that
entirely new terminology is not needed.
Anyway, in ordinary mathematics we can find many differences in uses
of terms - incompatible from one author to the next. Sometimes fairly
serious confusions may result due to such differences. And so,
ideally, it would be nice if the community of mathematicians adopted a
more or less universal canonical system of terminology. But it's not
likely that will happen, especially as each author will adopt his own
terminology as best suits his own context, which makes sense in its
way too.
But that said, still, when someone uses ordinary terminology in a way
that is RADICALLY different, then it may make good sense to suggest
using different words so that we can discuss the new system without
confusing its terminology with that of more ordinary mathematics,
especially as we may wish to discuss comparisons between the new
system and more ordinary mathematics.
Also, for someone such as RussellE who does not understand the
axiomatic method, using words like 'ret' and 'rment' emphasizes that
his actual mathematical arguments may not make any use of the
connotations, suggested associations, and other non-formal baggage
associated with the terminology, but rather that the formal reasoning
must be purely from the axioms and definitions (i.e., Hilbert's famous
'tables and beer mugs' explanation).
(But of course, by suggesting distinct terminology, actually what I'm
trying to do is to impose a quite politically incorrect brainwashing
so that all alternatives to standard mathematics will, under my boot
of fascism, be crushed before they can barely be born.)
> Here's where we draw the line: we can call an object defined
> in a nonstandard theory by the same name as an object defined
> in a standard theory, if the nonstandard object is an _analog_
> of the standard object in the new theory, satisfying some
> basic property of the standard object.
>
> An example: RE wishes to define "urelement" in his theory. To
> me, a basic property of "urelements" is that they contain no
> elements (and aren't the empty set). Since RE's objects don't
> contain elements, I believe that RE has the right to keep on
> calling them "urelements." On the other hand, if RE were to
> define "urelements" so that they have elements, then I'd agree
> that RE would be disingenuous in calling them "urelements," so
> that MoeBlee and the others would be justified in making him
> change their name to "urments" or "burblements."
RussellE has a "right" to use terminology any way he wants to. In your
paranoid fantasy, you think this is some kind of battle between the
behemoth establishment of standard mathematics and poor plucky
underdogs like RussellE. Sorry, but, for my part, this is not a
territorial battle. Rather, my suggestion is just to most easily avoid
confusions (in fact, if even made such suggestions in conversations
regarding clashes of terminology WITHIN "standard mathematics").
As to your suggestion about analogs, there's no need even for that
amount of complication. Rather, since RussellE is proposing quite
different senses and since his system is in flux as he revises it
(thus we don't know where is terminology will finally end up), it's
just simpler to let him use a different set of terminology.
It's not as if the standard terminology is privileged in some unfair
way. If it would make you happy, I'd just as soon oblige by using
'zet' instead of 'set' and 'zember of' instead of 'member of', etc.,
in standard mathematics. In a strictly MATHEMATICAL sense it matters
not to me. The only problem though is that requires then a lot of
REwriting and EXPLAINING (that the terminology is now changed). So,
since RussellE is just STARTING to write his system, it makes more
sense to ask him to use distinct terminology. I'd think reasonable
people would understand such things. But you transcend reason, as in
your view from high above you can see that really what I'm trying to
do is impose a kind of thought control to exclude alternative
mathematics from the git-go.
MoeBlee
Marshall
>
> In this case, even the most casual reader (the kind who somehow fails
> to notice the From header which is prominently displayed by default in
> every newsreader I've seen)
Other than AP's assertions, is there any evidence whatsoever
that such people exist? Is there even one documented example
of someone confusing you with who you quote?
Marshall
MoeBlee
> I looked at how Peano arithmetic is formalized:http://en.wikipedia.org/wiki/Peano_axioms
>
> I can define arithmetic the same way by
> changing my definition of natural number.
> PA defines natural numbers in "unary".
No, FIRST ORDER PA doesn't define 'natural number' AT ALL.
(Well, trivially, we could define in PA:
x is a natural number <-> x=x
but that's not what's at stake here.)
> PA says 0, S(0), S(S(0)), ... are natural numbers.
No it doesn't. (We're talking about first order PA).
> We just count the calls to successor.
>
> I can define natural numbers as sets just like PA.
> With this definition, I don't assume the urlements
> are natural numbers. I only assume they are ordered.
>
> Define 0 as the singleton set containing the smallest urelement.
>
> Define successor of set X to be the union of X and
> the singleton set of the smallest urelement not in X.
Fine.
Now, we're waiting for your definition of 'natural number'.
> Let U = {a,b,c,d}
What are a, b, c, d?
(Below, I say what I think you're driving at.)
> Let a < b < c < d
>
> 0 = {a}
> 1 = {a,b}
> 2 = {a,b,c}
> 3 = {a,b,c,d}
I think what you mean to do is to say there is a unique least
urelement, then a unique least urelement greater than the unique least
urelement, etc. And then you form singletons and unions thereof.
Sure, no problem, and then you can one-by-one define 0, 1, 2, 3, etc.
(You could even simplify by taking a, b, c, etc., as defined above, to
be each 0, 1, 2, etc.)
But that is not a definition of the PREDICATE 'is a natural number'.
Damn, you don't understand any of this stuff, because you continue to
insist on remaining ignorant of what other human beings have done
previous to you to work on such problems.
MoeBlee
MoeBlee
> Of course, if MoeBlee doesn't even consider PA to be
> adequate for the sciences, what chance does RE (or anyone
> else) have in convincing him that a _weaker_ theory, such
> as an ultrafinitist theory, is adequate for science?
As far as I know, first order PA is not adequate. (I'm open to being
convinced otherwise, though.)
MoeBlee
Aatu Koskensilta
> As far as I know, first order PA is not adequate. (I'm open to being
> convinced otherwise, though.)
Sink your teeth into the mumblings of Feferman on predicativism, ACA_0,
proof-theoretic reductions, what not, then!
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus