Infinity

[Kermit1941]

Zeno Swijtink @ WaccoBB wrote:

From: Zeno Swijtink Supporting member
Category: WaccoTalk
Thread: Math Discussion

Quote:
Willie Lumplump wrote: View Post
At last something that I know the answer to. Yes.



Now is 2 to the power infinity equal to infinity?


:)

Yes, of course. Since infinity refers to all non-finite numbers.

There is an extensive theory of infinity.

This theory classifies different levels of infinity.

We have separate levels of infinity for cardinal numbers, the numbers we use to say how many we have;
one, two, three, ....


and ordinal numbers, the numbers we use to say what order things come in;
first, second, third, . . .



For the cardinal numbers, the generally accepted smallest infinity is the infinity of positive integers.
We call this infinity by the name Aleph-null.

2 raised to the Aleph-null is a larger infinity that we call Aleph-one.

Aleph-one counts the number of points on a line.


Kermit Rose < [email protected] ([email protected]) >

[Zeno Swijtink]

2 raised to the Aleph-null is a larger infinity that we call Aleph-one.


That 2 raised to the Aleph-null is a larger infinity than Aleph-null has been proven. It means that there cannot be a one-to-one correspondence between a set of size Aleph-null and a set of size 2 raised to the Aleph-null.

When you say "2 raised to the Aleph-null is a larger infinity that we call Aleph-one" are you saying you are a conventionalist?

That 2 raised to the Aleph-null is the next larger infinity beyond Aleph-one cannot be proven from basic set theoretical principles. So it functions as a additional hypothesis, called the Continuum Hypothesis.

By saying "2 raised to the Aleph-null is a larger infinity that we call Aleph-one" you seem to be saying that the Continuum Hypothesis is a matter of convention, of our naming things. That there is no independent mathematical reality that our thinking needs to answer to.

Which ties in with the strands about bullshit and lying, and whether the world is more than a text to be interpreted and to be given meaning.

[Kermit1941]

At 11:18 AM -0500 12/27/07, Kermit Rose wrote:


Zeno Swijtink @ WaccoBB wrote:

From: Zeno Swijtink Supporting member
Category: WaccoTalk
Thread: Infinity

Quote:
Kermit1941 wrote: View Post
2 raised to the Aleph-null is a larger infinity that we call Aleph-one.
That 2 raised to the Aleph-null is a larger infinity than Aleph-null has been proven.
It means that there cannot be a one-to-one correspondence between a set of size Aleph-null and a set of size 2 raised to the Aleph-null.

Yes.



When you say "2 raised to the Aleph-null is a larger infinity that we call Aleph-one" are you saying you are a conventionalist?

No.



That 2 raised to the Aleph-null is the next larger infinity beyond Aleph-one cannot be proven from basic set theoretical principles.
So it functions as a additional Hypothesis, called the Continuum Hypothesis.

Right.




By saying "2 raised to the Aleph-null is a larger infinity that we call Aleph-one" you see to be saying that the Continuum Hypothesis is a matter of what we call cardinalities.


No.

" is a larger infinity" does not mean the same thing as " is the next larger infinity".


The convention of naming Aleph-null, Aleph-one was created before we knew that there could be infinitely many levels of infinity less than Aleph-null, and between Aleph-null and
Aleph-one. I believe Cantor did not intend to convey the impression that there were no infinities between Aleph-null, and Aleph-one. I think it likely that he had in mind,
only that he did not know of any others.

When we count, 0, 1, 2, 3...
we are not denying that fractions exist.


In 1964 Joel Cohen proved that the Continuum hypothesis cannot be proven from the other axioms of set theory.




That there is no independent mathematical reality that our thinking needs to answer to.


I believe that mathematical reality is independent of our conventions. I believe that mathematical reality is the basis for physical reality.





Which ties in with the strands about bullshit and lying, and whether the world is more than a text to be interpreted and to be given meaning.


:) Did you want me to seem to be with those people?



Kermit Rose < [email protected] ([email protected]) >

[Zeno Swijtink]

The convention of naming Aleph-null, Aleph-one was created before we knew that there could be infinitely many levels of infinity less than Aleph-null, and between Aleph-null and Aleph-one.

I believe Cantor did not intend to convey the impression that there were no infinities between Aleph-null, and Aleph-one. I think it likely that he had in mind, only that he did not know of any others.

When we count, 0, 1, 2, 3... we are not denying that fractions exist.


There are no infinitely many levels of infinity less than Aleph-null.

The indices with Aleph are ordinal numbers, so calling an infinity Aleph-null is in fact asserting that it is the first cardinal. Which is true and can be proven: There are no smaller infinities than the cardinal number of the set of natural numbers.

For the same reason Cantor did intend to convey the impression that there were no infinities between Aleph-null, and Aleph-one.

So, by definition, there are no infinities between Aleph-null and Aleph-one. Cantor's Continuum Hypothesis identifies Aleph-one as the cardinality of the continuum, the set of reals, which is the same as the cardinality of 2 to the power Aleph-null the cardinality of infinite strings of zeros and ones, which can be shown by a simple one-to-one mapping.

[Braggi]

I'm no expert at math, but one of the curiosities of math I learned is that .999999999... (to infinity) is exactly equal to 1.

Not almost one, but exactly equal. I've always had trouble with that, but it is, in consensus reality, true.

-Jeff

[Kermit1941]

There are no infinitely many levels of infinity less than Aleph-null.




:)

Now who is being a conventionalist?

Consider the potential infinity of calculus.

It is represented by a sideways 8.

limit(x) as x approaches infinity, is called potential infinity.

What about limit(2x) as x approaches infinity.

How may we justify distingishing between
limit (2x) as x approaches infinity,
and
lim(x) as x approaches infinity?

The answer is that we take their ratio.

limit ( (2x) / x) as x approaches infinity is 2.

Therefore we can consistently say that

limit(2x) as x approaches infinity is twice as large as
lim(x) as x approaches infinity.









The indices with Aleph are ordinal numbers, so calling an infinity Aleph-null is in fact asserting that it is the first cardinal. Which is true and can be proven: There are no smaller infinities than the cardinal number of the set of natural numbers.




No. Historically, the indices with Aleph are cardinal numbers.


https://en.wikipedia.org/wiki/Transfinite_number


From this web site I found that I had mis-remembered the meaning of
Aleph-one.

Cantor did indeed intend Aleph-one to be the next transfinite number.

I had mis-remembered that Aleph-one meant 2 raised to the aleph-null, which is why zeno began this argument with me.


And it is 2 to the aleph-null which counts the points on a line.

I now realize that aleph-1 is the name of the first uncountable cardinal number,
meaning it is the cardinal number of the set of all countable ordinal numbers.



I realize why I made that mistake. I found it reasonable to define a sequence of alephs by defining
aleph-one as 2 to the aleph-null, and aleph-2 as 2 to the aleph-one, etc.




Unless we choose to always use the axiom of choice ,
I considered it not reasonable to define aleph-one as the next transfinite number greater than aleph-null because we would never be able to agree on what that number is for nearly the same reason that we can never say what is the next rational number after 1.

That's probably why I mis-remembered the definition of aleph-1.


The axiom of choice permits us to well-order any set, thus matching any set with a set of ordinal numbers.

We might be interested in a different ordering of the set. For example, by the axiom of choice, the set of real numbers can be well ordered.

But we don't do calculus with the well-ordered version of the real numbers.
In a well ordered set, you always know what is the next element up, although you might not be able to name the next element down.

An important property of the real numbers with the order used in calculus is that we can define division. We can say how many times we must add one real number to itself to get another real number. ( add 1 to itself 6.5 times to get 6.5), ( add 2 to itself 6.5 times to get 13).







For the same reason Cantor did intend to convey the impression that there were no infinities between Aleph-null, and Aleph-one.




You are correct. My memory was over corrected by my logic.






So, by definition, there are no infinities between Aleph-null and Aleph-one.

Cantor's Continuum Hypothesis identifies Aleph-one as the cardinality of the continuum, the set of reals, which is the same as the cardinality of 2 to the power Aleph-null the cardinality of infinite strings of zeros and ones, which can be shown by a simple one-to-one mapping.




Exactly right.



And now we should reply to Jeff's comment.




I'm no expert at math, but one of the curiosities of math I learned is that .999999999... (to infinity) is exactly equal to 1.

Not almost one, but exactly equal. I've always had trouble with that, but it is, in consensus reality, true.

-Jeff





This is related to the fact that with real number arithmetic we cannot
distinguish between zero and infinitesimal.

Infinitesimals are not real numbers.

What would be the meaning of a decimal number consisting of an infinite number of zeros after the decimal point, and then after the infinite number of zeros we place a 1? Real number arithmetic would not permit us to distinguish such a number from zero.


The usual argument given that .999999..... is exactly equal to 1,
is:


.99999. . . means

9/10 + 9/100 + 9/1000 + 9/10^4 + 9/10^5 + . . .

= (9/10) * (1 + 1/10 + 1/10^2 + 1/10^3 + . . .)

and then the student is asked to apply the formula for infinite geometric sum

to get

that

(1 + 1/10 + 1/10^2 + 1/10^3 + . . .) = ( 1/( 1 - (1/10) )

= 1 / (9/10) = 10/9

Then the demostrator says, "Look. (9/10) * (10/9) = 1,

so, .999999..... must equal exactly 1.

Of course this is circular reasoning, ( which happens quite a lot in mathematics).

This argument presupposes that limit as m approaches infinity of
(1/10)^m is exactly zero.

If you don't accept this postulate, then you have defined a mathematical object different than the set of real numbers.


So, in summary,

the reason why the real number

.999999..... is exactly equal to 1, is that

we by our choice of what mathematical object the real numbers will be

had as a consequence that

limit as m goes to infinity of (1/10)^m shall be exactly equal to zero.

[Zeno Swijtink]

Zeno Swijtink wrote:

The indices with Aleph are ordinal numbers, so calling an infinity Aleph-null is in fact asserting that it is the first cardinal. Which is true and can be proven: There are no smaller infinities than the cardinal number of the set of natural numbers.



No. Historically, the indices with Aleph are cardinal numbers.

https://en.wikipedia.org/wiki/Transfinite_number



Aleph-null: the first transfinite cardinal number
Aleph-one: the second transfinite cardinal number
Aleph-two: the third transfinite cardinal number
etc.

"First," "second," "third" are ordinal numbers!

Your discussion of limits is from an other area of mathematics developed in the 19th century, and the epsilon/delta definition of limit by by Karl Weierstrass avoids talking directly about infinities.


Definition. The limit of f(x) as x approaches a is L

if and only if, given e > 0, there exists d > 0 such that 0 < |x - a| < d implies that |f(x) - L| < e.

These do not produce cardinal number smaller then Aleph-null.

In fact, limits such as you give


lim(x) as x approaches infinity

are not well defined.

There is an area of analysis developed by Abraham Robinson in the 1960s, which grew out of "model theory," that studies nonstandard models of analysis with "infinitely small" numbers or "infinitesimals."

[Kermit1941]

No. Historically, the indices with Aleph are cardinal numbers.





https://en.wikipedia.org/wiki/Transfinite_number (https://en.wikipedia.org/wiki/Transfinite_number)
Aleph-null: the first transfinite cardinal number
Aleph-one: the second transfinite cardinal number
Aleph-two: the third transfinite cardinal number
etc.





"First," "second," "third" are ordinal numbers!




Yes. Aleph-null is defined ( by Cantor ) to be the first transfinite Cardinal number.
Aleph-one is defined (by Cantor) to be the second transfinite Cardinal number.
Aleph-two is defined (by Cantor ) to be the third transfinite Cardinal number.

Perhaps confusing the issue is that
Aleph-null is the cardinal number of the set of finite positive integers. ( True of both the finite cardinals and the finite ordinals.)
Aleph-one is the cardinal number of the set of countable ordinal numbers.

Thus we can say, Aleph-null is the cardinal number of small omega, the set of natural numbers.
and Aleph-one is the cardinal number of large omega, the set of countable ordinals.

So you might see, in print, statements

Aleph-null = small omega
and
Aleph-one = large omega

When we equate a cardinal number to an ordinal number we mean the cardinal number is the number of elements in the set defined by the ordinal number.










Your discussion of limits is from an other area of mathematics developed in the 19th century, and the epsilon/delta definition of limit by by Karl Weierstrass avoids talking directly about infinities.

Definition. The limit of f(x) as x approaches a is L

if and only if, given e > 0, there exists d > 0 such that 0 < <CODE class=moz-txt-verticalline>|x - a|</CODE> < d implies that <CODE class=moz-txt-verticalline>|f(x) - L|</CODE> < e.



Yes, of course. Before non-standard analysis had been figured out, these definitions allowed us to consistently
do calculus without need of numbers that are less than every positive number.

Note that the concept of being less than every positive real number but still greater than zero,
has been replaced with the
concept of being less than any PREASSIGNED positive real number.







These do not produce cardinal number smaller then Aleph-null.



Right. They do not produce cardinal numbers at all. They are not infinities that can be used to count the elements of a set.
Therefore they are not cardinal numbers.

They are a third type of infinity, neither infinite cardinals nor infinite ordinals.

However, if we ask, "How do these infinities compare to the transfinite cardinals?"
it seems intuitively obvious to me that we should consider them to be less than aleph-null.





In fact, limits such as you give

lim(x) as x approaches infinity

are not well defined.



Let's talk more about this. Aim to prove inconsistency and I will aim to prove consistency.







There is an area of analysis developed by Abraham Robinson in the 1960s, which grew out of "model theory," that studies nonstandard models of analysis with "infinitely small" numbers or "infinitesimals."



Yes, and in 1977, a simpler way to look at non-standard analysis was developed.



Kermit Rose < [email protected] ([email protected]) >

[Kermit1941]

Aleph-null: the first transfinite cardinal number
Aleph-one: the second transfinite cardinal number
Aleph-two: the third transfinite cardinal number
etc.

"First," "second," "third" are ordinal numbers!






No. Historically, the indices with Aleph are cardinal numbers.




https://en.wikipedia.org/wiki/Transfinite_number (https://en.wikipedia.org/wiki/Transfinite_number)


Aleph-null: the first transfinite cardinal number
Aleph-one: the second transfinite cardinal number
Aleph-two: the third transfinite cardinal number
etc.


Aleph-null is defined ( by Cantor ) to be the first transfinite Cardinal number.
Aleph-one is defined (by Cantor) to be the second transfinite Cardinal number.
Aleph-two is defined (by Cantor ) to be the third transfinite Cardinal number.

Perhaps confusing the issue is that
Aleph-null is the cardinal number of the set of finite positive integers. ( True of both the finite cardinals and the finite ordinals.)
Aleph-one is the cardinal number of the set of countable ordinal numbers.

Thus we can say, Aleph-null is the cardinal number of small omega, the set of natural numbers.
and Aleph-one is the cardinal number of large omega, the set of countable ordinals.

So you might see, in print, statements

Aleph-null = small omega
and
Aleph-one = large omega

When we equate a cardinal number to an ordinal number we mean the cardinal number is the number of elements in the set defined by the ordinal number.








Your discussion of limits is from an other area of mathematics developed in the 19th century, and the epsilon/delta definition of limit by by Karl Weierstrass avoids talking directly about infinities.


Definition. The limit of f(x) as x approaches a is L



if and only if, given e > 0, there exists d > 0 such that 0 < |x - a| < d implies that |f(x) - L| < e.






Yes, of course. Before non-standard analysis had been figured out, these definitions allowed us to consistently
do calculus without need of numbers that are less than every positive number.

Note that the concept of being less than every positive real number but still greater than zero,
has been replaced with the
concept of being less than any PREASSIGNED positive real number.









These do not produce cardinal number smaller then Aleph-null.





Right. They do not produce cardinal numbers at all. They are not infinities that can be used to count the elements of a set.
Therefore they are not cardinal numbers.

They are a third type of infinity, neither infinite cardinals nor infinite ordinals.

However, if we ask, "How do these infinities compare to the transfinite cardinals?"
it seems intuitively obvious to me that we should consider them to be less than aleph-null.


It should be possible to construct infinities of this third class which are between aleph-null and 2^aleph-null.

I had not considered it before now.







In fact, limits such as you give


lim(x) as x approaches infinity
are not well defined.








Let's talk more about this. Aim to prove inconsistency and I will aim to prove consistency.









There is an area of analysis developed by Abraham Robinson in the 1960s, which grew out of "model theory," that studies nonstandard models of analysis with "infinitely small" numbers or "infinitesimals."






Yes, and in 1977, a simpler way to look at non-standard analysis was developed.






Kermit Rose < [email protected] ([email protected]) >

[Zeno Swijtink]

Quote:
Zeno Swijtink wrote: "In fact, limits such as you give

lim(x) as x approaches infinity

are not well defined."

Let's talk more about this. Aim to prove inconsistency and I will aim to prove consistency.



I did not claim inconsistency, only said that such limit expressions are not well defined in the Weierstrass sense.

If you give a new definition I can see whether it leads to consistency, but first we need to know what it means ...

[Kermit1941]

I did not claim inconsistency, only said that such limit expressions are not well defined in the Weierstrass sense.

If you give a new definition I can see whether it leads to consistency, but first we need to know what it means ...



https://en.wikipedia.org/wiki/Surreal_number

describes a construction that John Conway of Princeton University created to generate natural numbers, integers, dyadic rationals, real numbers,
infinitesimals, transfinite numbers, etc.

John's construction enables us to create a new number < any already constructed number, create a new number > any already constructed number,

and create a new number between any two already constructed numbers.


We could apply his construction to create numbers between
0 and aleph-null.

But John's construction is quite different than what I had in mind.

I gave you John's construction only to show that it's possible to define
infinities less than aleph-null.

For my construction, I look at the theory of limits used in calculus.

I associate my new numbers with continuous monotonically non-decreasing functions of a real variable, defined on the open interval (0,infinity).


We can add and subtract, multiply, and divide those functions.

The constant functions in this class of functions correspond to the real numbers.

Some functions will correspond to infinitesimals.

and some functions will correspond to infinities.


I define function f(x) to be less than function g(x) if and only if

limit as x approach infinity of ( f(x) / g(x) ) is < 1.



Kermit Rose < [email protected] >

[Zeno Swijtink]

Thank you for the reference to the construction by John Conway. I do not see, however, how this can create a number larger then any n, for all n, but smaller then Aleph-zero. There is no embedding given of this extended field into Cantorian set theory in the reference.

Similarly, in your construction you have no proof that this creates cardinal numbers smaller then Aleph-zero but smaller then n, for all natural numbers. Your functional numbers are larger then x for all real x, but this may just be in an ordinal sense.

To show that your functional number is smaller then Aleph-zero you need to associate it with a set and show that there is a one-to-one mapping of this set into a set of cardinality Aleph-zero, but no one-to-one mapping of a set of Aleph-zero into this set. Similarly, you have to show that your set is larger then any finite set.

[Kermit1941]

Thank you for the reference to the construction by John Conway. I do not see, however, how this can create a number larger then any n, for all n, but smaller then Aleph-zero. There is no embedding given of this extended field into Cantorian set theory in the reference.



ω = [{ 1, 2, 3, 4, ...} | }]


Now consider the number defined by

A = [ { 1, 2, 3, 4, ... } | ω ]


A is between { 1, 2, 3, 4, ... } and ω .


A is larger than any integer, but less than ω .





Similarly, in your construction you have no proof that this creates cardinal numbers smaller then Aleph-zero but smaller then n, for all natural numbers. Your functional numbers are larger then x for all real x, but this may just be in an ordinal sense.



I don't consider the function numbers to be cardinal numbers. They are neither cardinal numbers nor ordinal numbers.

The function numbers are not being compared as cardinal numbers.
They are being compared as function numbers.
They cannot be compared as cardinal numbers because they are not cardinal numbers.


It's true that I have not said how to compare these function numbers to aleph-null.

In order to make this comparison, I would have to make a definition, or axiom that compared aleph-null to one particular function.

I make that assertion now.

I make the axiom that

aleph-null equals the function x --> x.


Does it bother you that I'm equating a cardinal number, aleph-null,
to a non-cardinal number, x --> x?


This should be no more of a problem than when we say

aleph-null = ω.

There we are equating a cardinal number to an ordinal number.

It means only that the ordinal number ω represents a set that necessarily has aleph-null elements in it.


The justification I give for defining aleph-null = x --> x,
is that we normally presume that

in the statement

let x approach infinity,

that we use a sequence of aleph-null terms to represent the approaching to infinity.

It also seems natural to set aleph-null to the identity function
x --> x.










To show that your functional number is smaller then Aleph-zero you need to associate it with a set and show that there is a one-to-one mapping of this set into a set of cardinality Aleph-zero, but no one-to-one mapping of a set of Aleph-zero into this set. Similarly, you have to show that your set is larger then any finite set.



No. I don't need to do 1- 1 mapping to compare the functional numbers.

I bypass the 1-1 mapping procedure by identifying aleph-null to be the function x --> x.


Borrowing from John Conway's construction, notice that:

There is no ordinal number that corresponds to

A = [ { 1, 2, 3, 4, ... } | ω ].


Therefore there will not be any describable set with the cardinal number of A.

A is neither a cardinal number nor an ordinal number.

We no longer need to think of numbers as necessarily being either ordinal or cardinal.



Kermit Rose < [email protected] >